Spin resonance and spin fluctuations in a quantum wire

This is a review of theoretical works on spin resonance in a quantum wire associated with the spin-orbit interaction. We demonstrate that the spin-orbit induced internal “magnetic field” leads to a narrow spin-flip resonance at low temperatures in the absence of an applied magnetic field. An applied...

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1. Verfasser:	Pokrovsky, V.L.
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spelling	irk-123456789-1293712018-01-20T03:05:11Z Spin resonance and spin fluctuations in a quantum wire Pokrovsky, V.L. К 100-летию со дня рождения И.М. Лифшица This is a review of theoretical works on spin resonance in a quantum wire associated with the spin-orbit interaction. We demonstrate that the spin-orbit induced internal “magnetic field” leads to a narrow spin-flip resonance at low temperatures in the absence of an applied magnetic field. An applied dc magnetic field perpendicular to and small compared with the spin-orbit field enhances the resonance absorption by several orders of magnitude. The component of applied field parallel to the spin-orbit field separates the resonance frequencies of right and left movers and enables a linearly polarized ac electric field to produce a dynamic magnetization as well as electric and spin currents. We start with a simple model of noninteracting electrons and then consider the interaction that is not weak in 1 d electron system. We show that electron spin resonance in the spin-orbit field persists in the Luttinger liquid. The interaction produces an additional singularity (cusp) in the spin-flip channel associated with the plasma oscillation. As it was shown earlier by Starykh and his coworkers, the interacting 1 d electron system in the external field with sufficiently large parallel component becomes unstable with respect to the appearance of a spin-density wave. This instability suppresses the spin resonance. The observation of the electron spin resonance in a thin wires requires low temperature and high intensity of electromagnetic field in the terahertz diapason. The experiment satisfying these two requirements is possible but rather difficult. An alternative approach that does not require strong ac field is to study two-time correlations of the total spin of the wire with an optical method developed by Crooker and coworkers. We developed theory of such correlations. We prove that the correlation of the total spin component parallel to the internal magnetic field is dominant in systems with the developed spin-density waves but it vanishes in Luttinger liquid. Thus, the measurement of spin correlations is a diagnostic tool to distinguish between the two states of electronic liquid in the quantum wire. 2017 Article Spin resonance and spin fluctuations in a quantum wire / V.L. Pokrovsky // Физика низких температур. — 2017. — Т. 43, № 2. — С. 259-282. — Бібліогр.: 57 назв. — англ. 0132-6414 PACS: 73.21.Hb, 76.20.+q, 71.70.Ej http://dspace.nbuv.gov.ua/handle/123456789/129371 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
topic	К 100-летию со дня рождения И.М. Лифшица К 100-летию со дня рождения И.М. Лифшица
spellingShingle	К 100-летию со дня рождения И.М. Лифшица К 100-летию со дня рождения И.М. Лифшица Pokrovsky, V.L. Spin resonance and spin fluctuations in a quantum wire Физика низких температур
description	This is a review of theoretical works on spin resonance in a quantum wire associated with the spin-orbit interaction. We demonstrate that the spin-orbit induced internal “magnetic field” leads to a narrow spin-flip resonance at low temperatures in the absence of an applied magnetic field. An applied dc magnetic field perpendicular to and small compared with the spin-orbit field enhances the resonance absorption by several orders of magnitude. The component of applied field parallel to the spin-orbit field separates the resonance frequencies of right and left movers and enables a linearly polarized ac electric field to produce a dynamic magnetization as well as electric and spin currents. We start with a simple model of noninteracting electrons and then consider the interaction that is not weak in 1 d electron system. We show that electron spin resonance in the spin-orbit field persists in the Luttinger liquid. The interaction produces an additional singularity (cusp) in the spin-flip channel associated with the plasma oscillation. As it was shown earlier by Starykh and his coworkers, the interacting 1 d electron system in the external field with sufficiently large parallel component becomes unstable with respect to the appearance of a spin-density wave. This instability suppresses the spin resonance. The observation of the electron spin resonance in a thin wires requires low temperature and high intensity of electromagnetic field in the terahertz diapason. The experiment satisfying these two requirements is possible but rather difficult. An alternative approach that does not require strong ac field is to study two-time correlations of the total spin of the wire with an optical method developed by Crooker and coworkers. We developed theory of such correlations. We prove that the correlation of the total spin component parallel to the internal magnetic field is dominant in systems with the developed spin-density waves but it vanishes in Luttinger liquid. Thus, the measurement of spin correlations is a diagnostic tool to distinguish between the two states of electronic liquid in the quantum wire.
format	Article
author	Pokrovsky, V.L.
author_facet	Pokrovsky, V.L.
author_sort	Pokrovsky, V.L.
title	Spin resonance and spin fluctuations in a quantum wire
title_short	Spin resonance and spin fluctuations in a quantum wire
title_full	Spin resonance and spin fluctuations in a quantum wire
title_fullStr	Spin resonance and spin fluctuations in a quantum wire
title_full_unstemmed	Spin resonance and spin fluctuations in a quantum wire
title_sort	spin resonance and spin fluctuations in a quantum wire
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2017
topic_facet	К 100-летию со дня рождения И.М. Лифшица
url	http://dspace.nbuv.gov.ua/handle/123456789/129371
citation_txt	Spin resonance and spin fluctuations in a quantum wire / V.L. Pokrovsky // Физика низких температур. — 2017. — Т. 43, № 2. — С. 259-282. — Бібліогр.: 57 назв. — англ.
series	Физика низких температур
work_keys_str_mv	AT pokrovskyvl spinresonanceandspinfluctuationsinaquantumwire
first_indexed	2025-07-09T11:14:44Z
last_indexed	2025-07-09T11:14:44Z
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fulltext	Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2, pp. 259–282 Spin resonance and spin fluctuations in a quantum wire V.L. Pokrovsky Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843-4242 Landau Institute for Theoretical Physics, Chernogolovka, Moscow Distr. 142432, Russia E-mail: valery@physics.tamu.edu Received August 5, 2016, revised November 28, 2016, published online December 26, 2016 This is a review of theoretical works on spin resonance in a quantum wire associated with the spin-orbit inter- action. We demonstrate that the spin-orbit induced internal “magnetic field” leads to a narrow spin-flip reso- nance at low temperatures in the absence of an applied magnetic field. An applied dc magnetic field perpendicu- lar to and small compared with the spin-orbit field enhances the resonance absorption by several orders of magnitude. The component of applied field parallel to the spin-orbit field separates the resonance frequencies of right and left movers and enables a linearly polarized ac electric field to produce a dynamic magnetization as well as electric and spin currents. We start with a simple model of noninteracting electrons and then consider the interaction that is not weak in 1d electron system. We show that electron spin resonance in the spin-orbit field persists in the Luttinger liquid. The interaction produces an additional singularity (cusp) in the spin-flip channel associated with the plasma oscillation. As it was shown earlier by Starykh and his coworkers, the interacting 1d electron system in the external field with sufficiently large parallel component becomes unstable with respect to the appearance of a spin-density wave. This instability suppresses the spin resonance. The observation of the electron spin resonance in a thin wires requires low temperature and high intensity of electromagnetic field in the teraherz diapason. The experiment satisfying these two requirements is possible but rather difficult. An alternative approach that does not require strong ac field is to study two-time correlations of the total spin of the wire with an optical method developed by S.A. Crooker and coworkers. We developed theory of such correlations. We prove that the correlation of the total spin component parallel to the internal magnetic field is dominant in systems with the developed spin-density waves but it vanishes in Luttinger liquid. Thus, the measurement of spin correlations is a di- agnostic tool to distinguish between the two states of electronic liquid in the quantum wire. PACS: 73.21.Hb Quantum wires; 76.20.+q General theory of resonances and relaxations; 71.70.Ej Spin-orbit coupling, Zeeman and Stark splitting, Jahn–Teller effect. Keywords: quantum wire, electron spin resonance, spin-orbit interaction, Luttinger liquid. 1. Introduction In my youth I had a happiness and privilege to be a stu- dent with Professor Ilya Lifshitz, the great scientists and a brilliant personality. His influence on my life was rather deep, sometimes decisive. I am deeply obliged to our teachers and first of all to Ilya Mikhailovich who displayed fearlessness and uncompromiseness in all what concerned the science. In this article dedicated to his memory I present a re- view on a topic associated with scientific interests of Ilya Mikhailovich: physical properties of extremely anisotropic materials, namely the so-called quantum wires (QW). Quantum wires are effectively one-dimensional systems that can be treated as waveguides for the de Broglie waves so narrow that only one mode propagates in it. It happens when the linear size of the wire cross section is about 1 nm. The QW can be created by special growth processes in which the wires appear suspended onto the relief of a sub- strate. Alternatively a narrow conducting channel can be created in a semiconductor film by applying a proper con- figuration of gate electrodes. In the beginning of the 21st century nanodevices have been engineered using the QW with predesigned properties [1–8]. These advances excited a new interest in comparatively weak electron interactions in nanowires, the most important weak spin-orbit interac- tion (SOI). The SOI is especially important in a media with violated reflection symmetry where it produces the mo- mentum-dependent effective field acting on the electron spin known as Dresselhaus interaction [9]. Another im- portant manifestation of the SOI proposed by Rashba [10,11] appears in thin films and wires due to the contact © V.L. Pokrovsky, 2017 V.L. Pokrovsky with the interface that violates the reflection symmetry and is referred as Rashba interaction. Together they give a combined SOI-induced effective “magnetic” field linearly dependent on the electron momentum. In a QW the direc- tion of this field does not vary and corresponding compo- nent of the total spin is conserved. In two and three dimen- sions all three components of the total spin are not conserved. If the electron gas (liquid) is degenerate, low-energy electron-hole excitations play the dominant role in the pro- cesses induced by low-frequency electromagnetic field. In this situation the value of the electron momentum p in the SOI can be replaced in 1d systems by the Fermi- momentum Fp . Thus, the absolute value of the SOI field is determined in addition to its direction and it becomes a well-defined vector. The absolute value of the SOI field determines the resonance frequency of a new type of reso- nance, electronic spin resonance on the SOI field. We will call it inner resonance. Sometimes it is called chiral reso- nance. For two dimensions such a resonance was predicted by Shekhter et al. [12], but only in the case when Dresselhaus SOI is absent. This condition is not realistic. In the combined Dresselhaus–Rashba SOI the resonance frequency in two dimensions depends on momentum direc- tion. It leads to the smearing of the inner resonance line. This smearing is absent in one-dimensional case. The resonance frequency can be regulated by an exter- nal magnetic field B . Its component \|\|B parallel to the SOI field creates difference between the resonance frequencies of the left and right movers and in this way the ac field induces direct electric and spin currents. The perpendicular component B⊥ induces the electric dipolar mechanism of the spin-flip transitions instead of magnetic dipole acting in its absence. Since electric dipole is much more efficient for resonance absorption than magnetic ones, the reso- nance absorption grows very rapidly with B⊥. In Sec. 2 of this review we analyze the resonance phe- nomena employing a simple model of the ideal gas with SOI for electrons [13]. In Sec. 3 we consider a more realis- tic model of Luttinger liquid (LL) for electrons [14]. It is well known [15,16] that in 1d even a weak interaction de- stroys Fermi excitations in a vicinity of the Fermi point. Instead the collective Bose excitations — the charge and spin waves — play role of low-energy carriers of charge and spin, respectively. The SOI resonance can be treated as the excitation of a spin wave with the wave vector equal to the difference of the Fermi momenta for electrons with the same direction of the velocity but opposite directions of their spins. A new phenomenon caused by interaction together with the SOI is the appearance of a weak coupling between charge and spin channels and as a result the appearance of a new singularity (cusp) in the spin-flip response. Usally experimenters observe the resonance attenuation of the electromagnetic wave. In the QW this experiment is difficult because it requires sufficiently low temperature to distinguish the split Fermi points and rather high power of the incident electromagnetic wave. The high power is nec- essary to enhance a weak signal from a not very big num- ber of carriers in the wire. In a semiconductor wire of the length 10 µm with the bulk density of carriers 1018 cm–3 the number of electrons is about 1,000. The necessary power in the teraherz range is available only with free electron la- sers. The necessary temperatures are in the He range. These rather exotic requirements can be avoided employing quite different experimental tool: the real-time measurement of the two-time correlators of the total electron spin of the wire proposed and realized by S.A. Crooker [17]. This method employs the Faraday rotation of the light polarization by magnetized sample. High sensitivity of this method allows to avoid large power of incident light beam. With this method it is possible to distinguish two competing states of the elctron liquid in the QW. One of these states is the LL. The second one was proposed by O.A. Starykh and his coworkers [18] is the static spin-density wave with the wave vector 2 Fp . They proved that it wins at sufficiently large ratio \|\| /B B⊥ . We analyze the properties of the two- time spin correlators in the Sec. 4 of our review. 2. SOI spin resonance in ideal electron gas In this section we consider electron spin resonance (ESR) in nanowires with SOI. It is based on the article [13]. For ESR in metals or semiconductors an applied magnetic field B splits the unique Fermi surface for both spin directions into two different Fermi surfaces for up and down spins, with the same Zeeman splitting for all electrons. An almost uniform applied ac field of frequency equal to the Zeeman energy then induces sharp transitions between states with the same momentum and opposite spin. Even a weak SOI changes this picture. It creates an in- ternal “magnetic field” soB that depends linearly on the electron momentum for both Rashba and Dresselhaus SOI [9–11]. Therefore, for large enough SOI the ESR is smeared out. As indicated by Shekhter et al. [12], for 2D systems with only a Rashba interaction, the smearing is comparatively small at temperatures well below the Fermi energy, leading to a narrow ESR — a “chiral resonance”. However, noted that simultaneous presence of both Rashba and Dresselhaus interactions smears out the resonance since the SOI resonance frequency rω depends on the 2D momentum direction. This anisotropic broadening is completely absent for a QW (1D) as soB has the same direction for all right- moving particles and the opposite direction for the left- movers. Since the Rashba–Dresselhaus SOI is much less than the Fermi energy, the spin-flip energy is well defined. Thus the ESR line is narrow at low temperatures. The spin- flip resonance adsorption in the wire in the absence of an applied magnetic field B is very weak since it is magnetic dipole induced. The main predictions of the theory of ESR 260 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire in free-electron systems with the SOI are: i) A component of B perpendicular to soB activates electric dipole spin-flip transitions and therefore strongly enhances the resonance effects. Typically a B that is a tenth of soB increases the resonance absorption by 4 orders of magnitude, while changing rω by only 1%. ii) The component of B parallel to soB has little effect on the absorption, but it does sepa- rate the resonances of the right and left movers. Linearly polarized resonance radiation then produces a net magneti- zation and dc electric and spin currents. The SOI-induced dipole spin-flip excitation in 2D by an ac electric field E polarized in plane was considered in Ref. 12. Since, because of the SOI, spins in 2D are not collinear the excitation probability is almost independent of B . Rashba and Efros [19] considered a similar problem, but with an ac E polarized perpendicularly to the plane. To ensure a narrow resonance in this system, B must signifi- cantly exceed soB . The authors concluded that a tilted B is necessary to activate the electric-field-spin interaction. Due to the very high symmetry of their system, their spin-flip probability is proportional to the 6th power of B (instead of the square, as in the present case). The resulting proba- bility is very small for realistic field values. In the following subsections we analyze the electronic spectrum and eigenstates with SOI included; the effective interaction of the electron spin with an ac electric field; the dynamic generation of steady-state currents and magneti- zation and the relaxation processes. In conclusion some numerical estimates are made. 2.1. Electronic spectrum and eigenstates Weak SOI effects are better seen if the Fermi energy is not too large in comparison with the SOI energy. There- fore, it is reasonable to consider the SOI effects in semi- conductors rather than in metals. Here we consider type III–V semiconductors and only their electron bands, to avoid complications associated with degeneracy of the hole band. In p-doped semiconductors, analogous effects of the same order of magnitude should occur for the light holes with = 3/2J and = 1/2zJ ± similar to the present case of = 1/2S , = 1/2zS ± . But in the case of heavy holes with = 3/2J and = 3/2zJ ± photons cannot cause transitions between the two states. The 1D electron density n is as- sumed to be sufficiently large and the temperature suffi- ciently low to ensure a degenerate Fermi gas. In 1D the most general form of the SO interaction, in- cluding both Rashba and Dresselhaus terms, is = ( )so x yH pασ +βσ , where p is the 1D momentum along the wire direction x [20], and σ are the Pauli spin matrices. The total Hamiltonian, without impurity scattering, also includes the kinetic energy 2 /2p m and the Zeeman term −bσ , where = /2Bgµb B has dimensionality of energy. Let us introduce a unit vector n in the direction ˆ ˆx yα +β of soB and define the longitudinal and transverse components of magnetic field: \|\|= b ⊥+b n b . With 2 2γ ≡ α +β the SO velocity, the total Hamiltonian then reads ( )2 \|\|= /2 .H p m p b ⊥+ γ − −nσ b σ (1) Its eigenvalues are ( ) 2 , = / 2 ,E p p m qσ +σ (2) where 2 2 \|\|= ( )q p b ⊥γ − +b and = 1σ ± gives the projec- tion of the electron spin along the total effective magnetic field e so≡ +B B B and is the eigenvalue of the operator \|\| \|\| = . p b q p b ⊥ γ −   Σ −  γ −  bn σ (3) For a nonzero transverse magnetic field ⊥b , the direction of spin quantization depends on momentum. Figure 1 gives the energy vs magnetic field for small magnetic fields 2 /2Fp m<<b , with two slightly distorted Rashba parabo- las shifted vertically in opposite directions and an avoided crossing. The parallel component of the magnetic field \|\|b causes the reflection asymmetry, whereas its perpendicular component ⊥b causes the avoided crossing of energy lev- els. The four Fermi momenta correspond to the left and right movers and the two values of σ . For a typical experimental setup the SO velocity = /F Fp mγ << v . If Fp<< γb then the four Fermi mo- menta differ only slightly from the Fermi momentum in the absence of SOI and applied magnetic field = /2Fp n± ±π and are given by ( ) 2 \|\| \|\| = , 2F F F F b p p m p p p b ⊥ στ    τ −σ γ − τ +  γ − τ  b (4) Fig. 1. (Color online) Left part: Energy vs momentum according to Eq. (1). Shadowed regions of the spectrum are occupied. The spin-flip excitations of the occupied states by ac electric field are shown by long vertical arrows. Right part: Geometry and direc- tions of the applied magnetic field B and internal \|\|soB n. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 261 V.L. Pokrovsky where = 1τ ± indicates right (R) and left (L) movers. In the ground-state electrons with spin projection σ fill the mo- mentum interval from pσ− to pσ+ . All states in the interval ( , )p p−− ++ are doubly occu- pied. The states in the intervals ( , )p p++ −+ and ( , )p p+− −− are singly occupied (Fig. 1). A net spin-flip is possible only in the singly occupied intervals and requires energy ( )\|\|= 2 \| \| .sfE p bγ −τ (5) Thus, for \|\| 0≠b , there are two different resonance fre- quencies corresponding to the right and left movers = 1τ ± . For Fγ << v , the spin-flip energies are centered at 0 \|\|= 2( )sf FE p bγ − τ and lie in narrow bands of intrinsic width ∆, where ( ) 0 0* \|\|= 4 / = 2 / .F sf F sfm b p E E∆ γ γ − τ γ <<v (6) Spin-flip processes can be excited by a resonant applied field with frequency 0= /r sfEω . The temperature must satisfy < /r BT kω to avoid thermal smearing. 2.2. Transition rate due to linearly polarized ac electric field Let an ac field be linearly polarized along x: ( ) ( ) ( )0 00 0ˆ ˆ= e ei t i tE t xE t xE t− ω ω∗+ (7) and have spectral intensity ( )I ω centered at 0ω with extrin- sic width 0∆ω << ω . The symbols 0 ( )E t denote an enve- lope with frequencies in the interval ∆ω . Averaged over a time interval t′ satisfying 12 / r t −′π ω << << ∆ω , the two- time correlator of field can be represented by an integral: ( ) ( ) ( ) 0 0 1= e . 2 i t tE t E t I d ∞ ′ω −∗ ω −∞ ′ ω π ∫ In the presence of electric and magnetic fields the momen- tum p in the Hamiltonian (1) must be replaced by ep A c + , where A is the vector-potential. Thus, the SOI generates the spin dependence of the velocity: * = = ep Ac H ec e A cm +∂ γ + σ ∂ v . (8) Here we used the Weyl gauge where the electric potential = 0Φ , and thus ( ) ( )( )0 00 0 0 = e e .i t i ticA E t E t− ω ω∗− − ω (9) The interaction between the ac electric field and spin ap- pears from the SOI term of the Hamiltonian. It reads ( ) ( )( )0 0int 0 0 0 = e e .i t i tieH E t E t− ω ω∗γ − − ω nσ (10) For = 0⊥b , the interaction Hamiltonian is proportional to the same spin projection nσ that enters Eq. (1), and there- fore does not produce spin reversal. Then only magnetic dipole transitions can reverse the spin. Thus, there is no elec- tric dipole contribution to the spin-flip transition in the ab- sence of perpendicular component of the applied field ⊥b . This no-electric-dipole-spin-flip theorem was disputed by P. Upadhyaya et al. [21]. They noted that the SOI makes the magnetization and the internal “magnetic” field vary inside the wire in the direction perpendicular to its axis (x) and thus couple the electric field along y to the spin. However, in the article [13] we have found that to first and the second order in the small SOI parameter / Fγ v the contribution of this variation to the electric dipole cou- pling vanishes. The remaining, third order, coupling is comparable to or less than the magnetic dipole coupling and can be neglected. (See the details of our analysis in Appendix A.) This property is specific to 1D systems. In 2D the direction of eB changes along with the momentum direction. Thus almost any spin interacts with a linearly polarized electric field. In 1d the nonzero perpendicular component of the ap- plied field 0⊥ ≠b couples the electric field with the spin and induces spin reversals exceeding the magnetic dipole spin reversal at rather small values of b⊥. The matrix ele- ment 0= 2 / sfE⊥+ −nσ b of the operator nσ produces spin reversal between the two eigenstates of the operator .Σ Time-dependent perturbation theory gives that the spin-flip transition rate for an electron with momentum p is 2 2 0 2 (2 / )2 2 0 0 4= ( / )sf p ew E I⊥ γ −ω γ ω b   . On resonance, 2 04 /( )I Eω ≈ ∆ω , which implies that ( )22 2 0 2 04 / / .sf Fw e E E p⊥≈ ∆ωb (11) The ratio of the electric and magnetic transition rates is 0 2( / )F sfc E⊥b v . For InGaAs the ratio / Fc v is about 310 . Thus, for 1 010 sfb E− ⊥  the transition rate (11) exceeds the magnetic dipole induced rate by four orders, whereas the resonance frequency changes by only 1%. The perturbation theory used above is valid if the aver- age excited electron occupation number exn is small, i.e. eff 1wτ << , where effτ is a characteristic lifetime. In the ballistic regime the time of flight = /f FLτ v plays the role of effτ . However, in 1D if the back-scattering time bτ is much less than fτ , then diffusion occurs, with lifetime 2 eff = /f b fτ τ τ >> τ . Saturation occurs for all excitation processes subject to recombination at a rate 1 eff −τ , eff 1wτ  , so the probability of excitation is effmin( , 1)wτ . For a nar- row spectral width ∆ω , Rabi oscillations occur. The density of right-moving states subject to spin-resonant excitation is = /4 .sr rn n∆ω ω (12) 262 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire 2.3. Dynamic generation of permanent currents and magnetization In the absence of applied longitudinal field the left movers with spin up and right movers with spin down have the same probability of spin-flip transition by ac field with the frequency close to the resonance. The resonance curves for both these groups of electrons are identical: they have rectangular shape with the width 28mα . The longitudinal field b⊥ shifts energy of the spins by \|\|b− σ . Thus, if 2* \|\| > 4b m γ , then the resonance lines for right and left movers do not overlap and can be excited separately. Thus a resonant linearly polarized ac field can produce a mag- netization as well as steady-state electric and spin currents. Consider a linearly polarized ac field that causes spin flips of right movers, so = =ex R Rn n n↑ ↓δ −δ . For electrons ex eff= min( ,1) ,srn w nτ (13) with equal hole density. The spin per electron is ex= / .s n n For eff 1wτ ≥ , ex = srn n . Thus, in the ballistic regime = ( ) =e R R R Rj e n n↑ ↑ ↓ ↓− δ + δv v 2 ex= 2 = /f Fen enw− γ − τ γ v . (14) Next we consider how diffusion affects the currents. For simplicity we neglect spin-flip back-scattering and energy relaxation, but retain backscattering by impurities. A set of kinetic equations for this simplified model reads 1 1 1 eff/ = ( ) ,e e e sr b bR R Ldn dt wn n n− − − ↑ ↑ ↑ − τ + τ δ + τ δ (15) 1 1 1 eff/ = ( ) .e e e b bL L Rdn dt n n− − − ↑ ↑ ↑ − τ + τ δ + τ δ (16) The ac field creates equal numbers of electrons and holes with parallel spins, and this property is maintained by the back- scattering if spin-flip processes are negligible. The pumped spin is polarized approximately along / Fp⊥+ γn b . Its steady- state absolute value per unit length is eff eff= 2 srs w nτ . The spin current sj is eff eff= /(2 )s B F sr b bj g wnµ τ τ τ + τv . The electric current ej is 2 eff eff eff 2 eff eff (4 ) = 2 . 2 (2 ) b b e sr sr b F b b j e wn ewn p ⊥τ τ τ τ + τ − γ + τ + τ γ τ + τ (17) Equation (17) shows that the electric current changes sign in the diffusive regime at eff= /2F bb p⊥ γ τ τ . This hap- pens because back-scattering equalizes the number of left and right moving excitations, whose velocities differ. For resonance of left movers, at frequency \|\|= 2( )/L r Fp bω γ +  , the magnetization and currents are reversed. The generation of currents by an ac field is similar to the photogalvanic effect predicted by Ivchenko and Pikus [22] and by Belinicher [23]. More recently many clever modifications of this effect have been proposed and exper- imentally observed (see review [24], and articles [25,26]). They are mostly realized in 2D systems, but more im- portantly, unlike 1D, nonresonant optical or infrared radia- tion is used. In most cases, dynamic magnetization and electric current generation require a circularly polarized pumping field, whereas for a quantum wire in ≠B 0 the same effect can be produced by a linearly polarized source. The 1D geometry implies a strong anisotropic dependence of the resonance line and transition probability on B . 2.4. Relaxation processes At low temperature the main mechanism for electron energy relaxation is phonon emission. If the correspond- ing relaxation time epτ becomes comparable to or less than fτ , energy relaxation occurs before electrons and holes leave the wire. The total spin is not changed but the excitation velocities may decrease because lower energy means lower p and lower v. On the other hand, energy relaxation removes particles from the excited states and fills the depleted states. This makes an increase of power in the applied ac field more effective. The electron- phonon interaction is modeled by a standard Hamiltonian †= ( ) ( ) ( )epH U ∇ ψ ψ∫ u x x x , where ( )u x is the displace- ment vector, ( )ψ x is the electron field operator and U is the deformation potential. Electrons in the wire are always 1D, but phonons can be 1D, 2D, or 3D depending on the experimental setup. Let M and a be the lattice cell mass and lattice constant, and let u be the sound velocity. Then for an electron with momentum deviating by ξ from the Fer- mi point, and emitting 3D phonons, the relaxation rate is 32 1 = . 6 F ep F aU Muv u − ξ τ  π    v (18) The detailed calculation can be found in the Appendix to the article [13]. Thus, even at such high density of energy, the dc elec- tric current through the QW is rather weak and magnetiza- tion per electron is small. Though the absolute value of the dc current is small, its density is large enough: 2,000 A/cm2 for the current 1 nA. In 2D and 3D systems elastic scattering (diffusion) leads to spin relaxation by the Dyakonov–Perel mechanism [27,28] because the direction of soB depends on the direc- tion of the p and is randomized by diffusion. In 1D for = 0⊥b the direction of soB is the same for all electrons, so the Dyakonov–Perel mechanism does not apply. A sup- pression of Dyakonov–Perel relaxation in 1D was found in numerical calculations [29]. However, for 0⊥ ≠b , spin flip does occur in back-scattering, but its probability is of the order of 0 2( / )sfE⊥b and can be neglected. Other spin relaxation mechanisms, such as phonon emission combined with SOI, are much weaker. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 263 V.L. Pokrovsky 2.5. Numerical estimates All numerical estimates are for In0.53Ga0.47As. We take 29* = 4.3 10 g 0.05 em m−⋅ ≈ , 6=1.08 10α ⋅ cm/s, and = 0.5g − [30–32]. A typical 2D electron density is 122 10⋅ g⋅cm–2. Take wire thickness = 5a nm and width =d 10 nm. Then we find 1D density 6= 10n cm–1, 21= 1.65 10Fp −⋅ g⋅cm/s and 8= 0.38 10F ⋅v cm/s. Assuming =α β, we have 12= 4.8 10rω ⋅ s–1 (~ 0.8 THz) and intrinsic width 11= / 3.8 10δ ∆ ≈ ⋅ s–1. The value 2 0E in Eq. (11) is deter- mined by the source power in the terahertz range. Alt- hough standard cascade lasers have power in the range 1 mW–1 W [33,34], the power can be strongly enhanced by non-linear devices, and in very short pulses (1 ps) it can reach 1 MW [35–38]. The free-electron laser at UCSB provides a continuous power of 1–6 kW for the frequency range 0.9–4.75 THz. On focusing, the energy flux rises to 40 kW/cm2 [38]. For the moderate flux = 1S kW/cm2, we find 2 0 = 4 / = 4.19E S cπ erg/cm3. For = 10B⊥ T we have 0/ = 0.05sfb E⊥ , and Eq. (11) yields 10= 0.92 10w ⋅ s–1. As noted above, w can be increased by changing the power or the focus area. For length = 1 10L − µm the time of flight is 12 11= 2.6 (10 –10 )f − −τ ⋅ s. The back- scattering time bτ can be estimated from typical mobilities 4 5= / = 2 10 –4 10e mµ τ ⋅ ⋅ cm2/(V⋅s) in the bulk or film [39]. Since the scattering cross-section area is much less than the wire cross-section area, τ can be identified with .bτ Typical values are 13 11= 5 10 –10b − −τ ⋅ s. In this case the regime is either diffusive or marginally diffusive-ballistic. First consider a ballistic regime with 11= 1.1 10f −τ ⋅ s. By Eq. (14) the electric current equals 1 nA. The ratio of spin current to the electric current in units of elementary charges per second is /(2 ) 12f γ ≈v . The magnetization per electron, in Bohr magnetons, is eff( / ) 0.004srn n wτ  . Now consider a diffusive regime with 12= 1.1 10b −τ ⋅ s and 11= 1.1 10f −τ ⋅ s, so 10 eff = 1.1 10−τ ⋅ s. By Eq. (17) the electric current is = = 0.12eI j nA and the magnetization per electron is 0.02 Bµ . The temperature must be maintained below 2 / 35 KF Bp kγ ≈ . effwτ in this case is approximately 1, indi- cating that saturation has been attained. For energy relaxation we assume 3D phonons. For InGaAs we take U = 16 eV, u = 3.3⋅105 cm/s [40], a = 5 Å, 22=1.8 10M −⋅ g and = mξ γ. Then by Eq. (8) 12= 1.4 10ep −τ ⋅ s. With 2D and 1D phonons the formulae differ, but numerical estimates give the same order of magnitude. This result shows that, even in the ballis- tic regime, epτ is usually much shorter than fτ , so energy relaxation is substantial, which decreases the currents. 3. SOI-induced resonance in Luttinger liquids In previous section we considered electron system in a QW as an ideal Fermi gas. However, in 1D systems the electron-electron interaction is known to be strong / \| ln( ) \| /( )FV na naε  , where Fε is the Fermi energy, n is the 1D electron density, 2 2= /( )a meκ is the Bohr’s radius in the material, = 0.05 em m is the effective electron mass, and κ is the dielectric constant. For typical values n ~ 106 cm–1 and 20κ  the ratio / 1FV ε  . Therefore, it is important to study the effect of interaction on electron spin resonance in a quantum wire with the SOI. In the present section we demonstrate that the resonance persists despite of disappear- ance of fermionic excitations. The ESR in the Luttinger electron liquid is the excitation of a spin wave by external ac electromagnetic field. This resonance would have a simple Lorentzian shape in the absence of interaction. As we mentioned earlier, the fermionic excitation does not exist in the Luttinger liquid. They are replaced by bosonic excitations: charge and spin waves [15,16]. The standard Luttinger liquid (LL) theory neglects the SOI and deviation of the electronic spectrum near the Fermi points from the linear behavior. In this approximation the charge and spin degrees of freedom do not interact (this is the so- called spin-charge separation). The SOI splits the Fermi points for different spin projections and makes possible the resonant spin-flip processes. It was shown that the inter- play of magnetic field, SOI, and electron-electron interac- tion leads to the formation of spin-density wave state when magnetic field is perpendicular to the effective SOI mag- netic field [18]. In this section we assume that the magnetic field has nonzero component along the SOI field. Such a field terminates the spin-density wave instability and sim- ultaneously separates the spin resonances for left and right movers (see the previous section). The Coulomb interac- tion in the Luttinger liquid usually changes the shape of the spin resonance line from simple Lorentzian to a power-like one [15,16]. but it does not violate spin-charge separation. It is the SOI that violates spin-charge separation and thus enables the excitation of the charge waves at spin reversal. This process be experimentally observed as a weak reso- nance (cusp) at a plasmon frequency instead of the spin- wave frequency. In this section we show that both these effects really take place, though both are weak for not too strong electron-electron interaction. 3.1. Model Hamiltonian To take in account simultaneously the Coulomb interac- tion, SOI and external magnetic field, we use a model fermionic Hamiltonian: = TL R ZH H H H+ + . (19) Here TLH is the standard Tomonaga–Luttinger Hamiltoni- an: 264 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire † † , ,, ,= ( )TL F x R x LR LH i dx σ σσ σ σ − ψ ∂ ψ −ψ ∂ ψ +∑∫v 2 , , ( )G dx xσ τ σ τ  + ρ ∑∫ . (20) The first term in the r.h.s. of this equation is the kinetic en- ergy, the second term is the interaction between electrons. Approximation accepted in Eq. (20) neglects the quadratic correction to the kinetic energy of an electron near the Fermi point and replaces the long-range Coulomb interaction by a short-range interaction with G being the interaction con- stant. The summation index σ corresponds to the spin of electrons, the index = ,L Rτ corresponds to the left or right movers. The Hamiltonian TLH is invariant with respect to the group of rotations SU(2) in the spin space. The spin-orbit (Rashba) Hamiltonian RH reads as follows [10]: †=R x zH p dxα ψ σ ψ∫ . (21) The Rashba SOI splits Fermi momenta of up and down spins so that four Fermi points , = Fp p mρ σ ρ −σα appear, but it leaves Fermi velocities unchanged. The Rashba SOI constant α has dimensionality of velocity and we assume Fα << v ; zσ is the Pauli matrix; Fp is the Fermi momen- tum at = 0α ; and , = 1ρ σ ± correspond to right (left) movers and up (down) spin projections, respectively. Though for brevity we speak about Rashba interaction, the Hamiltonian RH can include the Dresselhaus interac- tion as well. The only modification that the reader should have in mind is that the direction ẑ in the pure Rashba interaction is perpendicular to the wire, whereas in com- bined Rashba–Dresselhaus interaction this direction is tilted to the wire at some fixed angle different from 90°. In the absence of external magnetic field the Fermi-mo- menta splitting can be removed by a single-particle unitary transformation = exp ( / )zU i mx− σ α  which shifts the mo- menta by m±α . After this transformation the electronic spectrum becomes the same as without the SOI and the (2)SU symmetry is restored. An external permanent magnetic field breaks this sym- metry. It leads to additional splitting of the Fermi points and to a difference in Fermi velocities for up and down spins, which cannot be compensated by this unitary trans- formation. We consider in this article only the magnetic field, B⊥, perpendicular to the Rashba field (along z axis) and apply it for definiteness along x axis. The longitudinal field \|\|B destroys spin-density wave when its magnitude (in energy units) is much larger than the width of the spin res- onance 24mα [18]. The latter value is much smaller than the resonance frequency 2 Fpα . Therefore, it is possible to destroy SDW and still have resonance frequency very close to 2 Fpα , i.e., to neglect \|\|B in all following equa- tions. Thus, the Zeeman Hamiltonian reads † , , = ( ) , 2 B Z x g BH dx⊥ ′ ′ρσ σσ ρσ ′ρ σ σ µ − ψ σ ψ∑ ∫ (22) Further we consider the effect of the SU(2) violation by external magnetic field employing perturbation theory. Therefore it is convenient to divide the total Hamiltonian (19) into the SU(2)-invariant part 0 = TL RH H H+ and the perturbation. Moroz et al. [41,42] have shown that a veloc- ity difference , , , ,= =R R L L↑ ↓ ↓ ↑δ − −v v v v v appears also due to the Rashba SOI in the wires of finite width. The curvature of the bands near Fermi level [43–48] can also be effectively taken into account by the nonzero velocity difference δv on the upper and lower branches of the ener- gy spectrum. The later effect has a relative value of at most / Fα v . Figure 2 schematically shows the electron energy as a function of momentum in the presence of the trans- verse magnetic field. We assume that the magnetic field is weak, B Fg B p⊥µ << α , and further consider it perturbatively. The residual symmetry in the perpendicular field is the combined reflection , ,p pσ→ − −σ. It ensures that the right movers with the spin projection σ along z axis have the same veloci- ty as the left movers with the same energy and the opposite spin projection , ,=R Lσ −σv v , but , ,R Rσ −σ≠v v . 3.2. Resonant absorption at spin-flip in the LL To calculate the resonant absorption at SOI reso- nance in the LL we start with the Hamiltonian emH of the interaction between the ac electromagnetic field and electrons. It can be found in many different textbooks, for example [50]: 1= ,em xH jA dx c − ∫ (23) where † ˆ= ( ) ( )j e x v xψ ψ is the current, ˆ ˆ= / zp m +ασv is the velocity operator, and xA denotes the x component of the vector potential of the ac field. As in the Sec. 2 we em- Fig. 2. (Color online) The up-spin and down-spin branches of the electron spectrum with nonzero Rashba spin-orbit interaction α and magnetic field B⊥ . Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 265 V.L. Pokrovsky ploy the Coulomb gauge, = 0A and put the scalar poten- tial zero. In this gauge the electric field is = (1/ ) /c t− ∂ ∂E A . The part of the electric current responsible for the spin-flip processes is †( ) = ( ) ( ).s zj x e x xαψ σ ψ (24) The absorption power of electromagnetic field is deter- mined by the real part of the conductivity ωσ at the fre- quency ω of the field multiplied by the square of the field's amplitude 2\| ( ) \|xE ω . We employ the Kubo formula for the conductivity: ( ) 0 0 1= ( )e l l i t tdx dx dt t t l ∞ ′ω − ω −∞ ′ ′ ′σ − θ − × ω ∫ ∫ ∫  [ ]( , ), ( , ) ,s sj x t j x t′ ′×〈 〉 (25) where l is the length of the wire. According to Eq. (24), ( )sj x is proportional to the density of z component of the spin. Therefore, the spin-flip conductivity (25) can be rep- resented as [ ] 2 ( )4( )= ( ), ( ) e , t i t t z z e S t S t dt l ′ω − ω −∞ α ′ ′σ − 〈 〉 ω ∫  (26) where ( )zS t is the operator of the total spin projection at the moment of time t . In the absence of magnetic field B⊥ the z component of the total spin is conserved. Therefore, [ ( ), ( )] = 0z zS t S t′ and the conductivity associated with the spin flip is zero. The violation of this conservation law at small B⊥ in the first-order approximation of the time- dependent perturbation theory leads to [ ]( ) = ( ), ( ) , t z I z iS t V t S t dt −∞ ′ ′ ′δ − ∫  (27) where 1 0 0( ) = ( )( ) ( )I B xV t U t g B S U t− ⊥− µ with 0 ( ) =U t 0= exp ( / )iH t−  being the evolution operator in the ab- sence of magnetic field, and xS is the projection of the total spin on the x axis. It is convenient to write the Rashba Hamiltonian as a sum over electrons: ,=R i z iiH pα σ∑ . The kinetic and interaction energies commute with xS , and therefore the perturbation operator ( )IV t becomes ( ), ,( ) = cos sin , 2 B I x i i y i i i g BV t t t⊥µ − σ ω +σ ω∑ where = 2 /i ipω α  . Substituting this expression into Eq. (27), we obtain ( ), , 1= e e , 2 i t i tB i iz i i ii g BS p − ω ω⊥ + − µ δ σ + σ α ∑ (28) where = x yi±σ σ ± σ . Condition Fα << v makes it possible to replace the factor 1/ ip in Eq. (28) by 1/ Fp± . Then the expression for zSδ becomes proportional to the sum of the operators , ,( ) = exp( )i it i t± ±σ σ ω . In terms of secondary quantized operators it reads (we keep here only right movers): † ,,( ) = ( , ) ( , ) h.c. 2 B z RR F g BS t x t x t dx p ⊥ ↓↑ µ δ ψ ψ + α ∫ (29) The unitary transformation = exp ( / )zU i mx− σ α  that puts the split Fermi points together, modifies this equation by multiplying the integrand by factor exp ( 2 / )i mx− α  . As a result, we find for the conductivity (25) associated with the spin flip [51], 2 ( ) 2 0 0 ( ) = ( )e l l i t tB F eg B dx dx dt t t lp ∞ ′ω −⊥ ω −∞ µ ′ ′ ′σ − θ − × ω ∫ ∫ ∫  † † , ,, ,[ ( , ) ( , ), ( , ) ( , )]R RR Rx t x t x t x t↓ ↑↑ ↓ ′ ′ ′ ′×〈 ψ ψ ψ ψ 〉× 2 ( )/e .i m x x′− α −×  (30) This expression for spin-flip conductivity is the basic for the following calculation. It is convenient since the aver- age of four fermions in it must be calculated for the Ham- iltonian 0H . A little its modification taking in account a weak violation of SU(2) symmetry by the SOI will be discussed later. 3.3. Bosonization The particles propagating in a narrow channel in one di- rection with the same or close velocities interact a long time. Therefore, the interaction is not weak for fermions, whereas their collective excitations do not interact. The transformation from fermion to boson operators in 1d called bosonization was proposed in 1975 independently by elementary particle physicists S. Coleman and S. Man- delstam and by condensed matter theorists A. Luther and D. Mattis. It was repeatedly presented in numerous books and reviews. The standard and most informative references are already cited books [15,16]. The standard expression of fermionic operators in terms of bosonic operators reads [ ( ) ( ) ( ) ( )]/ 2 , , 0 e= e , 2 i k xF i x x x xc c s sU a ρ − ρϕ −θ +ρσϕ −σθ ρ σ ρ σψ π (31) where ,Uρ σ are the Klein factors which ensure the proper anticommutation relations between the fermion, and 0a is the ultraviolet cutoff length. The secondary quantized fermionic wave functions σψ can be represented by the linear combinations of right-moving and left-moving fer- mions ,ρ σψ with the momenta being close to Fk± , i.e., , ,= R Lσ σ σψ ψ +ψ . The advantage of this model is that the interaction energy becomes quadratic in the charge and spin density bosonic operators. The density of fermions becomes linear in bosonic fields ,c sϕ , 266 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire , , 2( ) = ( ).c s x c sx xρ − ∂ ϕ π (32) As we have seen already, the SOI leaves the velocities at four Fermi points equal in linear approximation, but the quadratic corrections make velocities of inner Fermi points different from the velocities of outer ones. After the canon- ical transformation inner and outer Fermi points merge, however the velocities all are different. Let they are 1v and 2v . Then the kinetic energy should be modified in compar- ison to that in Eq. (20): † † kin 1 , ,, ,= ( )x xR LR LH i dx ↑ ↓↑ ↓ − ψ ∂ ψ −ψ ∂ ψ −∫v † † 2 , ,, ,( ).x xR LR Li dx ↓ ↑↓ ↑ − ψ ∂ ψ −ψ ∂ ψ∫v (33) This simple modification of the Luttinger Hamiltonian was proposed by Moroz et al. [41,42]. A difference of veloci- ties 1 2=δ −v v v appears not only due to quadratic part of dispersion, but also due to the SOI effect in a wire of finite thickness [14,41,42]. After bosonization the total Hamilto- nian with kinetic energy given by Eq. (33) takes the form ( ) ( ) ( )2 2 2= 2 c c c x c x c s s x s c dxH K K K  ∂ θ + ∂ ϕ + ∂ θ +π  ∫ v v v ( ) ( )2s x s x c x s x s x c sK  + ∂ ϕ + δ ∂ ϕ ∂ θ + ∂ ϕ ∂ θ   v v , (34) where cv ( sv ) is the velocity of plasmons (spinons). We have omitted the term cos[2 2 ( )] /(2 )s x dxϕ π∫ as being irrelevant in the renormalization group procedure for the repulsive interactions ( < 1cK ) [42]. The reader is advised to find the derivation of the main part of the Hamiltonian (34) in the cited books on bosonization. The correction proportional to δv directly follows from bosonization transformation Eq. (31). 3.4. Spin resonance as spin-wave excitation The bosonization allows us to consider the SOI reso- nance as the spin-wave excitation. To find the conductivity (30) we need to calculate the retarded correlation function , ( , ) =RI x t ↑↓ ↓↑ † † , ,, ,= ( ) [ ( , ) ( , ), (0,0) (0,0)]R RR Ri t x t x t↓ ↑↑ ↓ − θ 〈 ψ ψ ψ ψ 〉 in the ground state of the Hamiltonian (34) with fermionic operators ρσψ given by Eq. (31). Since the perturbation theory is developed for time-ordered averages in the imag- inary time = itτ − [52], it is necessary to express ( , )RI x t in the Kubo formula (30) to the time-ordered product † † , ( , ) = ( , ) ( , ) (0,0) (0,0) .T R RR RI x T x xτ ↓ ↑↑↓ ↓↑ ↑ ↓ τ −〈 ψ τ ψ τ ψ ψ 〉 Applying the Wick theorem, we obtain in terms of bosonic operators: ( , ) 2, 0 e( , ) , (2 ) g x TI x a τ ↑↓ ↓↑ τ ∝ − π ( ) , ( , ) = [1 e ] ( , ) ( , ) ,i qx q g x Y q Y qωτ− ω τ − ω − −ω∑ (35) where we introduced †( , ) 0e /(2 ) = ( , ) ( , )Y x RRa x xτ ↓↑ π ψ τ ψ τ so that ( , ) = 2[ ( , ) ( , )]s sY x i x xτ ϕ τ − θ τ and > 0τ . After obtaining , ( , )TI x ↑↓ ↓↑ τ , it can be converted into retarded correlator using * , , ,( ) = ( )[ ( ) ( ( )) ]R T TI t i t I t I t ↑↓ ↓↑ ↑↓ ↓↑ ↓↑ ↑↓ θ − − [15]. Using standard techniques one can calculate the time-ordered fermionic correlation function in real time, see the Appendix B for the details of calculation of time-ordered fermionic correlation function in real time. It can be converted into retarded correlation function as , ,( ) = 2 ( )Im ( )R TI t t I t ↑↓ ↓↑ ↑↓ ↓↑ − θ , as shown in the Appendix C, and using Eq. (30) we obtain ( ) 0 = e [ ( ) ( )] ,i t qxdx K t i K t i dt ∞ ∞ ω − ω −∞ σ + δ − − δ∫ ∫ (36) 1( ) = , ( ) ( ) ( )c c s K t x t x t x tλ µ ν− + −v v v (37) where the constant ( )2 2 0 2 3= . 2 B F eg B a p λ+µ+ν− ⊥µ π α  (38) We recall that the wave vector q in the above integral is equal to 2 /mα  , cf. Eq. (30). The exact numerical factor  is obtained here from the comparison with the non- interacting result of Ref. 13. The integrand in the integral over x has two singularities in the lower half-plane, at = sx tv and = cx tv . The expressions for the exponents ,λ µ , and ν are as follows [53]: ( ) ( ) ( ) 2 2 2 1 = , 8 c c c s K K + λ δ − v v v (39) ( ) ( ) ( ) 2 2 2 1 = , 8 c c c s K K − µ δ + v v v (40) ( ) ( ) ( ) 2 2 2 2 22 2 1 = 2 . 2 c c c c s c s c c s K K K K + + + ν − δ − v v v v v v v (41) To approximate ωσ close to the spin resonance at frequen- cy res = = 2 /s sq mω α v v , we take res res\| \|γ ≤ ω−ω << ω with γ being the width of the resonance which we assume Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 267 V.L. Pokrovsky to be small. In the limit res( ) /( ) 1c s s− ω γ >>v v v , it is cal- culated in the Appendix D and given by 1 Re ( ) ( ) ( )c s c s qν− ω λ µ σ × − + Γ ν  v v v v 12 2 2res . ( ) λ+µ − γ ×  ω−ω + γ  (42) Physically Eq. (42) indicates that the chosen momentum of the spin wave excited by the electromagnetic field, 2 / ,mα  is associated with the spin-dependent translation in the momentum space needed to put together two SOI-split parabolas, see Fig. 2. This momentum defines the frequen- cy of the excited spin wave. In the absence of SOI this momentum is zero and there is no ESR. To evaluate ωσ close to the other singularity, = 2 /cmω α v , we use a similar approximation and obtain ( ) 12Re 2 ( ) (2 )c c s qλ− ω µ ν πλ σ × − −µ −ν  v v v 12 2 2 . ( )cq µ+ν − γ ×  ω− + γ v (43) The plasmon singularity has a character of a weak cusp that can be detected only at nonzero interaction. Physically Eq. (43) indicates that the spin-charge coupling in the presence of SOI is responsible for this cusp. Experimental- ly, even at weak interactions, the plasmon cusp can be re- solved if the maximal derivative of Eq. (43) with respect to frequency is larger than the derivative of Eq. (42) at the same frequency. This condition is satisfied near the plas- mon cusp if 2 res( ) ( )/ < 1cqδ −ω γv v . Equations (42) and (43) are obtained under the assump- tion of well separated spinon and plasmon peaks, ( )c s q− >> γv v [54]. In the opposite case corresponding to the limit of noninteracting fermions, the peaks at res=ω ω and = cqω v merge. According to Eq. (36), the combined power of the merged peaks is 2 2 2= 2 ( ) (1 ) /[8 ( ) ]c c c sK Kλ + ν + δ − +v v v . In the limit of noninteracting fermions c s→v v and 1cK → , so that the power becomes 2 which corresponds to the Lorentzian shape of the spin resonance. For small interaction 0g between fermions, 01 /cK g− π , = ,s Fv v and 0(1 / )c F g+ πv v , so that the power deviates from 2 by 2 2 0( ) gδ v . Therefore, in the framework of perturba- tion theory the shape of the resonance line near resω deviates slightly from Lorentzian. However, generally 2 0 = ( / ) \| ln \|Fg e qaκv can be of the order of 1. For repul- sive interactions 0 < < 1cK and for strong fermionic inter- action 0cK → . In this case the results (42) and (43) show that the shape of the absorption line may deviate signifi- cantly from Lorentzian at sufficiently strong interaction. 4. Spin fluctuations in a QW In previous sections it was shown that the SOI reso- nance shoud exist at low enough temperature. However its observation by the measurement of the ac electromgnetic field adsorption is difficult since the signal is too weak to be resolved unless the sourse has the highest power achievable with modern technique. In addition the sample must be maintained at helium temperature. Therefore we propose instead to measure the total spin fluctuation in real time employing the experimental technique developed by S.A. Crooker and his team at LANL [17]. In their method the fluctuating spin produces the rotation of the polariza- tion of light. The measurement of the polarization angle are rather sensitive. Therefore, the instant value of spin can be measured with high precision. In this method there is no need in the powerful incident electromagnetic wave. Instead a weak polarization rotation gives necessary information. In the original experiments they studied fluctuations of the spin in a quantum dot. The number of electrons in such a dot is comparable with that in the quantum wire. So far the time resolution was in the range about 0.1 ns that corresponds to the frequency 10 GHz. An experimental group in Germany has developed ultrafast spin spectroscopy and achieved fre- quency resolution up to hundreds GHz [56]. The SOI reso- nance will give a singularity in the spin fluctuation correlator in this range of frequency. But the measurement of the spin correlator of the QW in this range of frequency at helium temperature can give also another very interesting information on the state of electron system. As we mentioned earlier, Starykh and coworkers [18] have found theoretically that at large enough external magnetic field perpendicular to internal one the QW develops a static spin-density wave (SDW) structure with the wave vector 2 Fk , an inhomogeneous state with physical properties rather different from those of the LL. So far this prediction was not checked experimentally. In the bulk static magnetic structures with zero total spin can be found by applying slow neutron diffraction. In the QW the neutron scattering is too weak and the neutron diffraction cannot be applied. But spin correlation method is ideal to dis- tinguish two alternative states of the QW: LL and SDW. In this section we calculate spin correlators in the 1D interacting electron system with the SOI for both the ordi- nary LL and the SDW states. 4.1. Model We use the same model of the LL with the SOI of the Rashba–Dresselhaus type as in previous section. However, some limitation will be imposed on the parameters. Be- sides of that in this section we study the action of external 268 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire magnetic field that has both parallel and perpendicular components. Therefore we start this subsection again with the definition of the model and its Hamiltonian. We assume zero temperature and the length l of the QW to be much larger than any other scale of length. We remind that the most general SOI Hamiltonian has the form ˆ= ( )SOIH n pα ⋅σ . (44) The SOI coupling constant α has dimensionality of veloci- ty. We assume it to be small: Fα << v . Further in this sec- tion we put = 1 and apply everywhere wave vector k instead of momentum p. We denote the direction of inter- nal field as z axis in the spin space and assume that the external field lies in the x z− plane. Then the total Hamil- tonian of electrons reads 2 \|\| , , int= [ ( ) ] , 2 i i i z i x i k H k b b H m ⊥+ α − σ − σ +∑ (45) where i labels electrons; intH denotes the Hamiltonian of the electron-electron interaction and = Bb g Bµ . We as- sume external field to be weak: Fb k<< α . The time-ordered spin correlators are defined as \| \| \| \|( ) = ( ) (0) = e e ,T iH t iH t aa t a a a aS t T S t S S S−〈 〉 〈 〉 (46) where ( )aS t is the total spin projection operator along di- rection = , ,a x y z , and 〈〉 denotes the average over the ground state. The retarded correlators can be obtained from the time-ordered ones as ( ) = 2 ( )Im ( )R T aa aaS t t S t− θ [15], where ( )tθ is the Heaviside step function. In the absence of the SOI interaction the Hamiltonian of interacting electron system is invariant with respect to the group SU(2) of rotations in the spin space. In the presence of the SOI interaction (finite value of α) and/or finite parallel field \|\|b the symmetry is reduced to the group (1)U of rota- tions around direction of the internal field. In the presence of perpendicular field b⊥ the U(1) symmetry is also broken. 4.2. Ideal 1d Fermi gas Before calculating the spin correlators of interacting electrons, it is instructive to solve the same problem for the ideal 1D Fermi gas. In the presence of the SOI and the ex- ternal magnetic field, the spectrum consists of a pair of asymmetric parabola with avoided crossing as shown in Fig. 3. If Fb k<< α and Fα << v , the four Fermi momenta are approximately 2 \|\| \|\| = 2 ( )F F F F b bk k m k k k b ⊥ στ   τ −σ α − τ +  α − τ   , where =σ ± denotes the spin-up/down bands and =τ ± denotes right/left movers. At = 0T spin correlators can be obtained directly by calculating the ground state average. The results at = 0b read 22( ) = ( ) = ( ) sin(2 )sin(2 ), ( ) = 0, R R xx yy F R zz lS t S t t k t m t t S t θ α α πα (47) where l is the wire’s length, and spins are in units /2 1/2.≡ The Fourier transforms are 2 2 2 2 ( ) [2 ( )] ( ) = ( ) = log , 2 ( ) [2 ( )] R R F xx yy F i k mlS S i k m ω+ δ − α + α ω ω πα ω+ δ − α − α ( ) = 0,R zzS ω (48) where = 0+δ . The x and y components are equal and the z component vanishes, respecting the (1)U symmetry. The imaginary part of ( )R xxS ω has a narrow peak around = 2 Fkω α of the width 24mα . In the resonance interval of frequency 2 ( ) < \| \| < 2 ( )F Fk m k mα − α ω α + α , the absorp- tion intensity Im R xxS is constant, as is seen in Fig. 3. Disor- der can change this exotic shape of line. 4.3. Interacting electrons and bosonization For interacting electrons we apply the LL theory de- scribed in the previous section. As it was discussed earlier, in 1D the interaction between fermions near Fermi points is always strong enough to make the collective Bose exci- tations almost independent instead of fermionic excitations in the Landau–Fermi liquid. Before translation to the bosonic language (bosonization), the original quadratic spectrum of fermions is linearized around the Fermi points, and the infinite sea of negative energy levels is filled. The extension of the fermion spectrum to −∞ contrasts with the Fig. 3. (Color online). Total spin correlators for ideal 1D Fermi gas (Eq. (48)). ( )R xxS ω , both its real (blue) and imaginary (red) parts, are shown in unit /(2 )l πα as functions of /(2 )Fkω α at / = 0.2Fm kα . Note that ( ) = ( )R R yy xxS Sω ω , and ( )R zzS ω vanishes. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 269 V.L. Pokrovsky initial spectrum of fermions limited from below. It does not lead to mistakes in physical results if the substantial range of momenta is close to the Fermi points. However in some prob- lems a broader range of momentum is important. Namely this happens in the case of the total spin correlators as it will be shown in this subsection. In this situation the LL theory can be used only together with a proper cut-off of negative mo- menta. In this section we introduce an extension of the LL model to take in account more explicitly the interaction of electrons at opposite Fermi points. Thus, the interaction Ham- iltonian in the model has the following form: †† int , 1= ( ) ( ) ( ) ( ) ( ). 2 H dxdx U x x x x x x′σ σ σ′σ ′σ σ ′ ′ ′ ′− Ψ Ψ Ψ Ψ∑∫ (49) The interaction potential ( )U x x′− will not be specified apart of its repulsive short-ranged character. The Fermi- field operator is the sum of the fields related to right and left movers: , , = 1 ( , ) = ( , )e , ik x x t x t σ τ σ σ τ τ ± Ψ Φ∑ (50) where ,kσ τ are the four Fermi points for electrons with the spin projection σ , whereas = 1τ ± label right and left mov- ers. After the bosonization described in some details in the Appendix E, the resulting Hamiltonian is expressed in terms of the charge fields cϕ and cθ and spin fields sϕ and sθ . It differs from the Hamiltonian (34) of the Sec. 3 by the important term CH responsible for the formation of the SDW (it was omitted in Sec. 3 since it is irrelevant in the LL state): \|\| 2 0 4 = cos 8 2( ) C C s F bg H x dx a   πϕ −  π   ∫ v . (51) The connection between the Luttinger constants cK , sK and Fourier components ( )U q of the interaction potential are [15,16] { } 1/2 = 1 2 (0) (2 ) / ( ) ,c F FK U U k −  + − π    v (52) { } 1/2 = 1 (2 /( ) .s F FK U k − − π v (53) The velocities of the charge and spin waves are = /c F cKv v and = /s F sKv v , respectively. We neglected a correction \|\| / F fb kα <<v in the argument of U and ig- nored a term mixing charge and spin fields that has the relative order of magnitude / 1Fα << v . The charge and spin degrees of freedom in this Hamiltonian are separated. The charge Hamiltonian is quadratic, but the spin Hamilto- nian contains a cosine backscattering term CH . If the latter can be neglected, the remaining quadratic Hamiltonian sH describes the ordinary LL state. When CH dominates, the field sϕ becomes pinned to one of the minima of cosine, resulting in ordering in the spin sector of the SDW state. 4.4. SDW in weak magnetic field Starykh et al. [18] proved that static SDW appears when the external field is directed perpendicular to the internal one and strongly exceeds it. We consider a more realistic limit Fb k<< α , and first fix = 0b⊥ . The charge Hamiltonian cH is quadratic and does not change in the magnetic field. To renormalize the spin part, we define following [18] spin currents: †= ,i iJτ τ τΦ σ Φ (54) where iσ ( = , ,i x y z) are three Pauli matrices. The two- component spinors τΦ , with = ,R Lτ corresponding to the left and right movers, are defined by Eq. (50). The spin part of the Hamiltonian sH can be expressed in terms of these currents as follows: 2= 2 ( )z z z s s s R LH dx J y J Jτ τ ′π + +  ∑∫v ( )\|\|4 cos y yx x C R L R L F b x y J J J J  + + +  v ( )\|\|4 sin ,y yx x R L R L F b x J J J J + −  v (55) where 2 2= 2 (0) = [1 (2 )/(2 )]s s f F F FU k′ − − πv v v v v . The initial values of coupling constants are (0) = (0) =s cy y = (2 )/( )F sU k ′− π v . The contants ,s Cy y are connected to the constants sK and Cg by relations = (2 )/(2 )s s sK y y− + , =C s Cg y′−πv . At \|\| = 0b , the Hamiltonian sH becomes simplifies to the following form: ( )20 = 2 z z z s s s R LH dx J y J Jτ τ ′π + +  ∑∫v ( )}y yx x C R L R Ly J J J J+ + . (56) The renormalization group equations for this Hamiltonian in the one-loop approximation are 2= ; =s C C s C dy dy y y y d dλ λ . (57) Here λ is the standard renormalization group running pa- rameter (logarithm of the length scale l ). The renormalza- tion group equations have a simple integral of motion 2 2 = constC sy y− . Since it is zero at initial scale 0=l a , it re- mains zero at any length scale. The point = = 0s Cy y is the fixed point of the system (57), whereas the pair of the straight lines in the plane ,s Cy y defined by equation 2 2=s Cy y is the separatrix of the trajectories that are generally hyperboles. The motion along separatrix at increasing λ leads to the fixed 270 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire point. Since both charges sy and Cy are zero in the fixed point, according to the relations written above we find that at large length scale 0Cg → and 1sK → . Thus, at zero magnetic field the cosine interaction is irrelevant. The lim- iting value of sK shows that the SU(2) symmetry violated at small distance is restored on large distances. The system remains in the LL state. The same answer is correct in the presence of the paral- lel magnetic field since the oscillations associated with rotation of spin around this field stops renormalization at the scale \|\| \|\|= /Fl bv . However, if \|\| = 0b but 0b⊥ ≠ , the situation changes. More complicated analysis shows that in this case renormalization leads to = =s Cy Y −∞ at λ →∞. Though, this result can be treated as the requirement of very precise orientation of the external field for observa- tion of the SDW, this limitation is much more liberal for real QW because of their finite length. Indeed the field must be perpendicular if the length of wire is much larger than \|\|l , or \|\| /Fb l<< v . For the length 10 µm it requires parallel magnetic field less than 0.6 T. 4.5. Spin-density correlations In this subsection we calculate the spin-density correlators for the ordinary LL state and the SDW state. At Fb k<< α , the Fermi momenta are approximately , = .Fk k mσ τ τ −σ α In the ordinary LL state, the backscattering term HC can be dropped and the Hamiltonian becomes completely quad- ratic. The Luttinger parameters are given in the previous subsection, except of = 1sK at zero external field. Spin density operators read: † ,( ) = ( ) ( )a as x x x′ ′σ σσ σΨ σ Ψ , where = , ,a x y z . The time-ordered spin-density correlators are ( , ) = ( , ) (0,0)aa t a as x t T s x t s〈 〉 . Applying the bosonization one can express spin correlators as path integrals over bosonic fields. Details of calculation are placed in Appen- dix F. The results are 1 2 2 2 0 2 11 2 22 2 ( ) cos(2 ) ( , ) = ( , ) = ( ) Ks Ks s xx yy Ks Kss a y x m x s x s x x y + − + + − α τ τ + π + 1 2 0 2 /2 1/22 2 2 2 cos(2 )cos(2 ) , ( ) ( ) Kc Ks F K Kc sc s a k x m x x y x y + − α + π + + 2 2 2 2 2 2( , ) = ( ) s s zz s K y x s x x y − τ + π + 2 0 2 /2/22 2 2 2 cos(2 ) , ( ) ( )c K Kc s F KK sc s a k x x y x y + − + π + + (58) where / /( ) =s c s cy τ τv , τ is imaginary time, and 0a is an ultraviolet cut-off. Each correlator contains contributions from small q and from 2 Fq k . For weakly interacting case , 1c sK K ≈ , and both decay as 2x− and oscillate. The SDW state exists at completely perpendicular field, when Cy flows to the strong coupling limit Cy → −∞. CH is relevant and dominates the spin Hamiltonian. The field sϕ is pinned to 1= ( ) 2 2s N π ϕ + (N is an integer), whereas its conjugated field sθ is completely uncertain. Correlators of the charge fields remain the same as in ordinary LL. The correlators ( , )xxs x τ and ( , )yys x τ decay exponentially to zero being averaged with the oscillating factor ei sθ . But ( , )zzs x τ survives since sθ doesn’t appear in its expression: 0 2 2 20 2( , ) = cos(2 ) . ( ) Kc zz F c a s x k x a x y    τ  π +  (59) It is determined exclusively by the charge degrees of free- dom. It oscillates with the wave vector 2 Fk and decays power-like with 2 2 cx y+ . For 1cK ≈ it decays as 1x− which is slower than 2x− decay of the ordinary LL case. This is the result of ordering in the SDW state. 4.6. Total spin correlations The total spin correlator can be obtained by integration of spin-density correlators found in previous subsection over coordinates. Eqs. (58) and (59) present the time- ordered spin-density correlators for imaginary time τ. Let define the total spin-correlation function with imaginary time interval 0 0 ( ) = ( , ) ( , ) l l T aa aa aaS dx dx s x x l s x ∞ −∞ ′ ′τ − τ ≈ τ∫ ∫ ∫ and their Fourier transforms ( ) = e ( )T i T aa aaS S ∞ ωτ −∞ ω τ∫ . The Fourier transform of the retarded correlator ( )R aaS t is the analytic continuation of the time-ordered Fourier trans- form ( ) = ( )R T aa aaS S i iω ω→ω+ δ , where = 0+δ [15]. For details of calculation see Appendix G. In this point we are faced with a paradox: the integrated over x correlator ( , )zzs x τ in the absence of the transverse external magnetic field is not a constant in contradiction with the exact conservation of the z component of the total spin. For the SDW state ( )R zzS ω is also not a constant, but the SDW appears only in nonzero transverse field that vio- lates the zS conservation. Such a contradiction was first noted by Tennant et al. [57] (see their appendix) and they Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 271 V.L. Pokrovsky treated it phenomenologically assuming that the oscillating term is a complete derivative. This discrepancy originates in the filling of infinite Fermi sea, a crucial assumption in the LL model [15,16]. Electron and hole excitations in this model are completely symmetric. In real wires the relativ- istic particle-hole symmetry is violated. In particular, the momenta of holes cannot exceed Fk by modulus. This limitation is not important if only momenta close to Fk± are essential. This is the case for the spin-Peierls instability leading to the appearance of the SDW. However, the mo- menta far from Fk bring a significant contribution to the total spin. Therefore, the LL model does not respect the total spin conservation. Nevertheless, calculations for the non- interacting case within the Fermi gas model shows that the cut-off of the integration at some negative momentum Dk leads to conserving zS if <D Fk k . This cut-off produces additional terms in the spin-density correlator so that at = 0Dk , 2 2 2 2 2 2 2 2 2 cos(2 )1 1( , ) = ( ) F zz k xy xs x x y x y − τ + − π + π + 2 2 2 2 ( cos( ) sin( ))e2 , ( ) k yFF Fy y k x x k x x y −− − π + (60) where = Fy τv . The third term in Eq. (60) is the cut-off correction. After integration over x it completely cancels the contribution of the second term. The first term is con- tribution of small momentum transfer. Its integration gives zero. Though it is not clear how to introduce the momentum cut-off for interacting electrons, the main conclusion that we can extract from the calculations for free electrons is that the momentum transfer 2 Fk does not contribute anything to the zz correlation of the total spins at = 0B⊥ . Further we leave only small momentum con- tribution to this correlator. We then arrive at a simple result for the LL state: 2 2 0( ) = ( ) = [ ( ) ]R R xx yy sS S A iω ω ω + ω+ δ × 1 2 2 22 2[ ( ) ] , Ks Kss i + − × ω − ω+ δ ( ) = 0.R zzS ω (61) where = 2s smω αv and 1 2 0 0 1( ) (2 ) 2 2 2 = 1(1 ) 2 2 Ks K ss s s s s s a Kl K A K K + − Γ − − π Γ + + v v . The SDW state that appears only in the transverse field violating the total spin conservation does not require such a fine tuning. Its total spin correlators are 12 2 20 ( ) = ( ) = 0, ( ) = [ ( ) ] , R R xx yy Kc R zz SDW c S S S A i − ω ω ω ω − ω+ δ (62) where 0 = 2c F ckω v , and 202 ( ) (1 ) 2 2 = ( ) 2 K cc c SDW c c a Kl A K − Γ − π Γ v v . We summarize the major differences between these two states in Table 1. The line shapes of the non-vanishing spin correlators for both states are shown in Figs. 4 and 5. 4.7. Relation to the experiment The results given by Eqs. (61) and (62) show that meas- urements of the total spin correlators can be used as a diagnos- tic tool for identification of the state of the electronic liquid in the quantum wire, is it the LL or the SDW. Besides of that we predict that in the ordinary LL state the transverse correlators display the spin resonance at = 2 2s s Fm kω α ≈ αv . Table 1. Major differences of the total spin correlations be- tween the LL and SDW states State Vanishing component Position of singularity for nonvanishing component LL zzS = 2 2s Fm kω α ≈ αv SDW xxS , yyS = 2 F ckω v Fig. 4. (Color online). Total spin correlators for the LL state (Eq. (61)). Im ( )R xxS ω is shown in unit 0A as a function of / sω ω for = 1sK (blue), 1.2 (red) and 1.4 (black). The = 1sK curve is a δ-function spike at = sω ω . ( )R zzS ω vanishes for the LL state. 272 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire The position of resonance agrees with the previous non- interacting result (48). In the SDW state only the z correlator survives and it has a peak at a relatively high 0 = 2 2c F c F Fk kω ≈v v . A typical value for this frequency in semiconductors is 1410 Hz. At much lower frequency it is almost constant. Experimentally, the Faraday rotation method [17,55] measures directly the spin correlations in real time. At zero field the system is in the ordinary LL state, and we expect peaks at = 2 smω αv for directions perpendicular to the SOI axis. The direction of the SOI axis is not a priori known. It must be found utilizing the (1)U symmetry of the trans- verse spin correlations. Applying the magnetic field per- pendicular to the SOI axis, one can check whether the wire transits to the SDW state. At this transition the longitudinal correlator suppressed in the LL state becomes dominant, whereas the transverse correlators are suppressed. The impurity scattering does not change the results significantly if the mean free path is larger than 1/( )mα , typically 10–30 nm. The corresponding mobility is ~(1–3)⋅103 cm2/(V⋅s). 5. Conclusions Our theory is based on conservation of the total spin in 1d quantum electron system subject to the SOI. The effec- tive SOI field acting on electron spins has a definite direc- tion tilted to the direction of the wire. Projection of the total spin on this direction is conserved. Theory predicts the existence of the SOI-induced spin resonance in semi- conductors at low temperature 2< 4 3 KT mα  . The reso- nance can be regulated by external magnetic field. Its component perpendicular to the internal field switches on the electric dipolar mechanism of the spin-flip transition and strongly enhances the probability of the resonance absorption. The component of the external field parallel to the internal one may separate resonances for the left and right movers. In this way the permanent electric current can be generated by the resonance ac electromagnetic field. These phenomena obtained first in the model of ideal elctron gas are confirmed in a more realistic model of LL for elec- trons. In the framework of this model the resonance excites a spinon, an excitation of the spin component of the LL with the wave vector equal to the difference between Fermi mo- menta of the left and right movers due to the SOI splitting. An additional effect is the spin-flip process induced by a plasmon excitation due to a weak coupling between spin and charge degrees of freedom generated by the SOI. Experimentally the SOI-induced spin resonance mani- fests itself most directly in the resonance absorption of electromagnetic field with frequency about 1 THz. Such an experiment requires large power of the ac field (1 kW/cm2) so far available only with free elecron laser. Additional requirements of He temperature and magnetic fields in the range of several Tesla makes this experiment difficult. An alternative idea is to diagnose the LL state in a sem- iconductor (InGaAs) by measurements of the total spin correlation in real time proposed by Crooker at ANL [17]. These measurements are less direct. Their preference apart of a weak probe light field is that they can distinguish be- tween the LL state and the SDW state of electron liquid in the wire predicted by Starykh and his coworkers at suffi- ciently strong perpendicular component of external field. So far this state was not observed experimentally. Our analysis of the total spin correlations shows that the SDW supresses fluctuations of the perpendicular to the inner field magnetization in the wire leaving only parallel com- ponents, whereas in the LL state perpendicular fluctuations are dominant, whereas the parallel component is sup- pressed. We hope that this effect can be used for diagnostic of the state of electrons in the quantum wire. Acknowledgments I am indebted to the coauthors of publications on which this review is based Artem Abanov, Wayne Saslow, Peng Zhu, Oleg Tretyakov, Konstantin Tikhonov and Chen Sun. My thanks are due to A. Finkelstein, L. Glazman, M. Khodas, J. Kono, O. Starykh and A. Tsvelik for enlightening discus- sions. My special thanks is to Chen Sun for his invaluable help in the preparation of the manuscript. Appendix A. Spin-flip in a QW of a finite thickness In this appendix we present the calculation referred to Sec. 2 of the matrix element responsible for the spin flip- ping due to coupling to the transverse electric field. We treat the spin-orbit interaction as a perturbation and show Fig. 5. (Color online). Total spin correlators for the SDW state (Eq. (62)). Im ( )R zzS ω is shown in unit SDWA as a function of 0/ cω ω for = 1cK (blue), 0.8 (red) and 0.6 (black). ( )R xxS ω and ( )R yyS ω vanish for the SDW state. Note that if Fα << v and s c F≈ ≈v v v , we have 0c sω >> ω . Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 273 V.L. Pokrovsky that the matrix element is zero in the first and second order of the SOI coupling constants α and β. The first nonzero contribution comes in the third order in spin-orbit interac- tion. We start from the same Hamiltonian (1). We intro- duce the frame of reference with the x axis along the wire, y axis along the wide side of the cross-section whose linear size is denoted as .W Because we are only interested in the linear coupling to the y component of the electric field, we take xA , zA and the magnetic field B to be zero. Then the Hamiltonian, up to linear terms in the ac field, reads kin= so acH H H H+ + (63) where 2 2 kin = 2 x yp p H m∗ + , (64) = ( ) ( )so x x y y y xH p pασ +βσ − ασ +βσ , (65) =ac y y eH A c υ . (66) Here = ( )y y x y p m∗ υ − βσ +ασ and we assume that the ef- fective mass m∗ is the same in both x and y directions. Let us represent the Hamiltonian (63) without the last term as 0= ,H H V+ where 0 kin= ( )x x yH H p+ ασ +βσ and the perturbation is = ( ).y y xV p− ασ +βσ The stationary states 0, ,xn p τ of the Hamiltonian 0H are direct products of the eigenstates of 2,x yp p and the spin operator = ( )/ .z x yτ ασ +βσ γ The corresponding wave functions are (0) , , = ( )eip xxnn px f y ττψ χ (67) Here ( )nf y is the transverse part of the wave function, and τχ is an eigenspinor of zτ with the eigenvalue = 1.τ ± The energy of the state 0, ,xn p τ is 0 0 , , ,= Fn p n px x E E pτ + γ , where 0 0 kin, 0= , , , ,x xn px E n p H n p〈 τ τ . The first-order perturbation theory correction to the wave function (67) is (1) , , 0 , = , ,xn px m n m pτ ′≠ τ ′ψ − τ ×∑ 0 0 0 0 , , , , ( ) , ,x y x y x n p m px x m p p n p E E ′〈 τ ασ +βσ τ〉 × − , (68) where we neglect the contribution of the first SOI term ( )x x yp ασ +βσ to the energies in denominator retaining the leading term 0 0 , ,n p m px x E E− . In order to calculate the sum in Eq. (68) we note that kin= [ , ].y imp H y ∗  Then this equation can be simplified as follows: (1) , , 0 , = , ,xn px m n im m p ∗ τ ′≠ τ ′ψ τ ×∑  0 0, , ( ) , ,x y x xm p y n p′× 〈 τ ασ +βσ τ〉 . (69) Finally by choosing the frame of coordinates so that = 0n y n and using the completeness relation 0 0 , , , , , = ,x x m m p m p I ′τ ′ ′τ τ∑ we find the first order cor- rection to the eigenstate 0, ,xn p τ 1 0, , = ( ) , ,x y x x imn p y n p ∗ τ ασ +βσ τ  . (70) Now we consider the effect of the transverse ac electric field. We are interested in the off-diagonal term of the operator y y e A c υ for the states of a fixed band, i.e., in the matrix element , , , ,y x y x e A n p n p c − υ + , (71) where + and − represent the up and down eigenstates of the spin operator zτ , respectively. Again employing the Heisenberg equation kin= [ , ],y so i H H yυ +  one can trans- form the matrix element (71) as follows: , , , ,x y xn p n p− υ + = , , , ,= ( ) , , , ,n p n p x xx x i E E n p y n p− +− − +  . (72) The energies belonging to a fixed band n and different spin projections differ only because of the SOI. Therefore, an ex- pansion of the difference , , , ,n p n px x E E− +− in terms of SOI coupling constants begins with a linear term of α and β: , , , , 2n p n p xx x E E p− +− ≈ − γ . (73) Thus, to obtain the contribution to the matrix element (72) linear in SOI coupling constants, it is necessary to calcu- late the matrix element of y with the zeroth-order wave function 0, , ( = 1)xn p τ τ ± for which the space and spin variables are factored. Therefore, the matrix element of y contains the scalar product − + , which equals zero. To find the matrix element in Eq. (72) to second order in α and β, we need to use 1, ,xn p τ for the matrix element of .y Then, using Eq. (70), one finds 1 00 1, , , , , , , ,x x x xn p y n p n p y n p− + + − + = 0 0= , , ( )[ , ] , ,x y x x im n p y y n p ∗ − ασ +βσ +  . (74) 274 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire Since this matrix element is zero, the y component of the electric field produces no spin-flip processes in the second order in α and β as well. The quickest way to calculate the Hamiltonian acH in the third order or, equivalently, the matrix element of y in the second order goes through a unitary transformation = eFU , where = ( ).y x mF i y ∗ − ασ +βσ  Applying it to the Hamiltonian (without the ac field term), and truncating the Baker–Hausdorff series at the second order of α and β (recall that 2 2=γ α +β ), we find 1 kin= ( ) [ , ]U soH U H H U H F H−+ ≈ + = 0= constUH V+ + , (75) where 2 2 0 = 2 x y x z p p H p m∗ + + γτ (76) is the starting approximation Hamiltonian introduced earli- er and 2 22= ( )U x z mV yp ∗ β −α σ  (77) is the transformed perturbation that is proportional to squares of the SOI constants. Note that the transformed eigenstate , ,x Un p τ obeys the same boundary conditions as the initial one. Thus, the transformed state , ,x Un p τ differs from zero approximation state 0, ,xn p τ by the second order correction 2 2 ,2 0 , 2 , , = ( ) , ,x x xU m n m p n p m p ∗ ′≠ τ ′τ β −α τ ×∑  0 0 0 0 , , , , , ,x z x n p m px x m p y n p E E ′τ σ τ × − . (78) The state vector we are looking for , , =xn p τ 1= , ,x UU n p− τ has an additional term of the second or- der equal to the operator 2 / 2F acting on the zeroth order state. Since 2F is proportional to unit operator, it does not contribute to the matrix element in Eq. (72). With Eqs. (67), (72), (73), (78), we obtain the matrix element to the third order in α and β: 2 2 28 , , , , = ( )x x y x m n m pin p n p ∗ ≠ − υ + − γ β −α ×∑   2 0 0 0 0 , , , ,x x n p m px x m p y n p E E × − , (79) where we use matrix elements for spin operator zσ be- tween the eigenstates of the operator zτ = 0z± σ ± and = 1.zσ ± The zero order off-diagonal matrix element reads 0 0 2 2 2 2 8 sin( ) 2, , = ( ) x x m nnmW m p y n p m n + π − π − , (80) and 2 2 2 2 0 0 , , 2 ( )= . 2n p m px x n mE E m W∗ π − −  (81) Thus, the matrix element of the velocity yυ from Eq. (72) can be expressed in terms of an infinite series: 2 4 2 2 2 2 4 6 1024 , , , , = ( ) x x y x m W p n n p n p i ∗ − υ + β −α γ × π 2 2 2 5 = ( )m n odd m m n+ × − ∑ . (82) For = 1n the sum is 2 2 2 2 5 =1 4 (15 )= 0.01648. 3072(4 1)k k k ∞ π − π ≈ − ∑ We can now obtain the coefficient at / y ye cA υ for the spin- flip amplitude induced by the transverse electric field and it is given by 2 4 2 2 2 4 ( ) 0.0176xm W p i ∗ β −α γ ⋅  . (83) If we let = 10 nm,W 21= 1.65 10 g cm/sFp −⋅ ⋅ , 6= 1.08 10 cm/sα ⋅ and 8= 0.38 10 cm/s,F ⋅v then we find that the upper bound for the coefficient at γ given by Eq. (83) is 510− γ . It is by 5 decimal orders less than the coefficient (γ ) of ( / ) x ze c A τ . However, the latter does not produce a spin-flip transition. The transverse magnetic field makes spin-flips possible but decreases the coefficient by a factor 2( / ) 0.01Fb p⊥ γ  . Nevertheless the anisotropy ratio of the amplitudes is about 0.001. The anisotropy of the spin-flip probability is about 610− . Thus, resonant exci- tation by an ac electric field polarized along the y axis is very ineffective and practically unobservable. However, the amplitude (83) depends very strongly on W . Therefore, the width can not be increased significantly. On the other hand, a significant change of W would violate the condi- tion of one channel. Appendix B. Calculation of time-ordered averages in real time To find the correlation functions of fields sϕ and sθ in Eq. (35) we use the generating functional [ ]J : 1= exp . 2i i d dx M J  ϕ θ τ − Φ Φ + Φ     ∫ ∫ ∫   (84) Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 275 V.L. Pokrovsky This expression is written in a matrix form with four vectors of the field = ( , , , )c s c sΦ ϕ ϕ θ θ and “current” 1 2 3 4= ( , , , )J J J J J . The 4 4× matrix M describes the sys- tem Lagrangian and is presented below. After the standard Gaussian integration we find ( ) 1/2 11[ ] = det exp . 2 M JM J− −      J (85) The bosonic correlation functions from Eq. (35) are repre- sented in terms of the elements of matrix M as 2 =0 ln( , ) (0,0) = ( , ) (0,0)i j i j x J x J δ Φ τ Φ = δ τ δ J  1= e ( , ). 2 2 iqx i ij d dq M q− ωτ −ω ω π π∫ ∫ (86) The matrix M is symmetric and has the following nonzero elements 2= /( ),c cc c M q Kϕ ϕ πv 2= /( ),s ss s M q Kϕ ϕ πv 2= / ,c cc c M K qθ θ πv 2= / ,s ss s M K qθ θ πv = = / , c c s s M M iqϕ θ ϕ θ ω π and 2= = /(2 ). c s s c M M qϕ θ ϕ θ δ πv With these expressions, ( , )g x t in Eq. (35) takes the form 1 2( , ) = 2 (1 e )[ ( , ) (2 ) iqx i s s dqdg x i M q− ωτ − ϕ ϕ ω τ − ω + π∫∫ 1 1 1( , ) ( , ) ( , )]. s s s s s s M q M q M q− − − θ θ ϕ θ θ ϕ+ ω − ω − ω (87) At zero SOI ( = 0α ) and magnetic field ( = 0B⊥ ), the system has (2)SU symmetry of spin rotation. This sym- metry prevents the renormalization of the interaction constant in the spin channel and therefore = 1sK [48]. A weak SOI ( )Fα << v and magnetic field ( /F BB p⊥ << α µ ) only slightly violate the (2)SU symmetry [49], so that 21 ( / )s FK − α v [49]. Therefore, in what follows we put = 1sK up to small corrections of order 2α . Thus, with this precision up to quadratic in δv terms we find 1 1 1 1( , ) ( , ) ( , ) ( , ) s s s s s s s s M q M q M q M q− − − − ϕ ϕ θ θ ϕ θ θ ϕω + ω − ω − ω  2 2 2 2 2 2 ( 1) 22 ( ) . ( ) 4 ( ) ( ) c c c s c s c K q iKi q q i q K i q q + + ωπ π − δ ω+ ω+ ω +  v v v v v Performing the integration over frequencies, one finds the correlator as a function of imaginary time. Because of fac- tor e i− ωτ only the poles in the lower half-plane of the com- plex plane ω contribute to the integral. After the integra- tion and analytical continuation, expression (87) turns into a sum of logarithms of the type ,ln ( )c sC x t± v , where C is a constant. Inserting this result in Eq. (35) we find the corresponding time-ordered fermionic correlator but in real time. Appendix C. Relation between retarded and time-ordered spin correlators The perturbation theory is valid for time-ordered ave- rages in imaginary time, whereas what we need to calcu- late is a retarded average , ( )RI t ↑↓ ↓↑ . Therefore, we need a relationship between ( , ) = ( ) [ ( , ), (0,0)]R BAI x t i t B x t A− θ 〈 〉 and ( , ) = ( , ) (0,0)T BAI x T B x Aττ −〈 τ 〉 for imaginary time τ where † ,,( , ) = ( , ) ( , )RRB x t x t x t↓↑ ψ ψ and † ,,(0,0) = (0,0) (0,0)RRA ↑↓ ψ ψ are boson-like operators. These two types of averages are related by equality [15] * † †( ) = ( ) ( ) ( ) ,R T T BA BA A B I t i t I t I t   θ − −       (88) which follows from ( ) = ( ) ( ) (0) (0) ( ) ,R BAI t i t B t A A B t− θ  −   (89) ( ) = ( ) ( ) (0) ( ) (0) ( ) .T BAI t t B t A t A B t− θ + θ −   (90) For positive time > 0t , = ( ) (0) ,T BAI B t A− (91) * † † † † ( ) = (0) ( ) ,T A B I t B A t − − −    (92) and † †(0) ( ) = ( ) (0) =B A t A t B− − † †= (0) ( ) = ( ) (0) .A B t B t A In our case, due to the above definitions of A and B , † †( ) (0)B t A differs from ( ) (0)B t A by changing the spin components σ→ −σ. It is equivalent to ( , ) ( , )Y x t Y x t→ − since we introduced †( , ) 0e /(2 ) = ( , ) ( , )Y x RRa x xτ ↓↑ π ψ τ ψ τ . However, Y enters in all correlation functions quadratically, see Eq. (35) in the main part, and we con- 276 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire clude that this transformation does not change the correlator. Then ( ) = ( )[ ( ) ( ( )) ]R T T BA BA BAI t i t I t I tθ − , and we find ( ) = 2 ( )Im ( ).R T BA BAI t t I t− θ (93) Appendix D. Evaluation of conductivity in Eqs. (42) and (43) To find the absorption power of electromagnetic field we need to calculate Re( )ωσ , where the conductivity is given by Eq. (36) in the main text. The integral in Eq. (36) is ( )= [ ( ) ( )] ,i t qxdx e K t i K t i dt ∞ ∞ ω − ω −∞ −∞ σ + δ − − δ∫ ∫ (94) 1( ) = , ( ) ( ) ( )c c s K t x t x t x tλ µ ν− + −v v v (95) where ( )2 2 0 2 3= .B F eg B a p λ+µ+ν− ⊥µ π α  Since q is positive, the exponent e iqx− vanishes at large x in the lower half-plane of the complex variable x . There- fore, the integral over real axis x is equal to the sum of two contour integrals in the lower half-plane of x along contours winding around two branch cuts shown in Fig. 6. The contour 1C winds around the branch cut from the point = sx t i− δv to =x i+∞ − δ, and the contour 2C winds around the branch cut from = cx t i ′− δv to =x i ′+∞ − δ . (We ignore here the potential singularity at = cx t−v which is associated with the inverse processes of spin flip from up to down on the right branch. These pro- cesses should be suppressed in the approximation of small excited-state occupation numbers which we em- ploy.) We can estimate the integrals over x around each branch cut separately. The conductivity ωσ has two sin- gularities: at = sqω v and at = cqω v . As we show below, close to the singularity near = sqω v the main contribution to the integral comes from contour 1C , and the other (plasmon) singularity is dominated by the integral over 2C . First, we estimate the integral 1I over the contour 1C . After the change of variable = su x t− v the contour that maps 1C in a complex plane u winds around the branch cut from = 0u to =u +∞ and will be denoted by the same symbol 1C . Thus, the integral 1I can be written as follows: ( ) ( ) 1 1 e= . iqu i qts C s c s c I du u u t u t − − λ µν    + − + +    ∫ v v v v v We aim to approximate ωσ very close to the resonance at res = sqω v . The closeness is determined by inequalities relating to the detuning res= \| \|δω ω−ω : res ,γ << δω << ω where γ is the attenuation rate mainly due to Cherenkov emission of phonons [13]. Then the time during which the resonance absorption is accumulated is large enough, 1/t δω . In this limit, ( ) 1c s qt− >>v v , we approximate the above integral as ( ) ( ) 1 1 e ( )= e , i qts z Cs c s c iqI z dz t t − ν− −ν λ µ −    − +    ∫ v v v v v (96) where we introduced new variable =z iqu− . As a result of this change of variables, the contour 1C turns into contour C winding around a branch cut going from = 0z to =z i− ∞. The contour C can be rotated clockwise together with the branch cut until the latter coincides with the left half of the real axis < 0z . Then the contour integral turns into the Hankel's representation of the inverse Gamma function and gives 2 / ( )iπ Γ ν . After that we can integrate over time with the following final result: ( ) ( ) ( ) 1 , 1 2 (1 ) . ( ) q s c s c s i q iI q i ν− −ν−λ−µ ω λ µ −λ−µ − π Γ −λ −µ − + Γ ν ω− + γ  v v v v v (97) Then the real part of conductivity near = sqω v becomes 1(1 )Re ( ) ( ) ( )c s c s qν− ω λ µ Γ −λ −µ σ × − + Γ ν  v v v v 12 2 2res , ( ) λ+µ − γ ×  ω−ω + γ  (98) which for small λ and µ is approximated by Eq. (42). Here we used that c s≥v v . The integral 2I over the contour 2C does not contribute to the singularity at = sqω v and there- fore can be neglected. Fig. 6. The integration contour in the lower half-plane of the complex variable x. Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 277 V.L. Pokrovsky In addition, there is also a small part of Re ωσ which is independent of γ : 2 1 1 (1 ) ( 2) . ( ) ( ) ( ) \| \|c s c s res q v v v v ν− λ µ −λ−µ π Γ −λ −µ ν + λ +µ − − + Γ ν ω−ω  (99) It is small in positive parameter 2 2 22 = ( ) ( 1) /[4 ( ) ]c c c sK Kν + λ +µ − δ − +v v v . Nevertheless, 2I contributes to a singularity at plasma frequency = cqω v . Next we analyze this singularity. We consider the integral 2I over contour 2C similarly to what we did for 1I . We change variable in 2I to = cu x t− v . The mapped contour 2C winds around the branch cut from = 0u to =u +∞ . As a result of winding around the branch cut we obtain factor 2 ( 1)(1 e ) 2i i− π λ−− ≈ π λ and the integral over u from 0 to infinity: ( ) ( ) [ ] 2 0 e= 2 . 2 ( ) iq u tc c c s I i du u u t u t − +∞ µ νλ π λ + + − ∫ v v v v (100) At small detuning =\| \|cqδω ω− v from the plasma reso- nance we expect that similarly to what we observed for 1I the accumulation time for the resonance absorption is large, 1/ 1/( )ct qδω >> v , and therefore in the factors 2 cu t+ v , ( )c su t+ −v v it is possible to neglect 1/u q . After this procedure the resulting integral over t diverges at = 0t . This divergence however is spurious. It has hap- pened because at small < 1/[( ) ]c st q−v v , the variable u cannot be neglected. It means that the integration over t is effectively cut off at 0 1/[( ) ]c st q< −v v . To estimate the singular part on the background of nonsingular contribu- tion originated from small t , we represent the exponent ( )ei q tcω−v as a sum, ( ) ( )e = [e 1] 1i q t i q tc cω− ω− − +v v , and divide the integral over time into two parts: ( ) 0 0 e 1 . i q tc t t dt dt t t ω− ∞ ∞ µ+ν µ+ν  −   +∫ ∫ v The second integral is approximately equal to 1 0 /( 1)t −µ−ν ν+µ− and has no singularity. The first integral converges and can be extended to = 0t if < 2µ + ν . This condition is satisfied in a broad range of not too strong inter- action as it can be readily checked from Eqs. (18)–(20). The first integral after the change of variable = ( )cq tτ ω− v turns into ( ) 1 0 ( 1) . i c e dq ∞ τ µ+ν− µ+ν − τ ω− τ∫v (101) The integral in Eq. (101) is a large number 1(2 )i −≈ −µ −ν proportional to 2[ /( )]c s −δ −v v v . The ratio of the first term to the second has the order of magnitude 2 1[ /( )] [\| \| /( )]c s c cq q− µ+ν−δ − ω−v v v v v . Thus, the nonresonant contribution is comparable with the resonant one only in a narrow region close to the resonance 2[ /( )]c c sqδω ≤ δ −v v v v . Combining all the results, we arrive at the expression for the singularity due to spin-flip processes at the plasmon frequency: ( ) 1 1 1 , 2 (1 ) . 2 (2 ) ( ) c q c c s q i I i q µ+ν− λ− λ− ω µ ν ω− + γπλΓ −λ − −µ −ν −  v v v v (102) The calculation of the real part gives the following result: ( ) 1 , 2 (1 )Re 2 ( ) (2 ) q c c s qI λ− ω µ ν πλΓ −λ × − −µ −ν  v v v 12 2 2 , ( )cq µ+ν − γ ×  ω− + γ v (103) c.f. (43) in the main text for Re ωσ . The plasmon singulari- ty has a character of a weak cusp that can be detected only at large enough interaction. Appendix E. Bosonization The left and right chiral fermionic fields are expressed in terms of the bosonic fields as follows: 4 , , 0 = e , 2 i a πϕσ σ τ σ τ η Φ π (104) where ση are Klein factors (see Sec. 2) and Bose fields ,σ τϕ obey the following commutation relations valid for two operators of the Bose field at the same moment of time, but different positions: , , , ,( ), ( ) = sign ( ) 4 ix y x y′ ′ ′ ′σ τ σ τ σ σ τ τ τ ϕ ϕ δ δ −  . (105) Finally the Klein factors obey the following anticom- mutation and algebraic relations: { } † ,, = 2 ; = ; = i′ ′σ σ σ σ σ σ +η η δ η η η η− . (106) Appendix F. Spin-density correlations Here we present calculations of the spin density correla- tions for the ordinary LL state. In terms of the bosonic fields, the spin-density operators reads 278 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire 0 0 1 1( ) = sin[ 2 ( ) 2 ] sin[ 2 ( ) 2 ]x s s s ss x m x m x a a π θ −ϕ − α + π θ + ϕ − α + π π 0 0 1 1sin[ 2 ( ) (2 2 ) ] sin[ 2 ( ) (2 2 ) ],s c F s c Fk m x k m x a a + π θ −ϕ + − α + π θ + ϕ − + α π π 0 0 1 1( ) = cos[ 2 ( ) 2 ] cos[ 2 ( ) 2 ]y s s s ss x m x m x a a π θ −ϕ − α + π θ + ϕ − α + π π 0 0 1 1cos[ 2 ( ) (2 2 ) ] cos[ 2 ( ) (2 2 ) ],s c F s c Fk m x k m x a a + π θ −ϕ + − α + π θ + ϕ − + α π π 0 0 2 1 1( ) = ( ) sin[ 2 ( ) 2 ] sin[ 2 ( ) 2 ].z x s c s F c s Fs x x k x k x a a − ∂ ϕ − π ϕ −ϕ + + π ϕ + ϕ + π π π (107) ________________________________________________ Let us define the partition function as a functional integral: 0 = ( , )exp ( ( , )) ,Z x d dx x β   Φ τ τ Φ τ    ∫ ∫ ∫  (108) where = sgn ( )it tτ + ε ( = 0+ε ) is the imaginary time, = 1/( )Bk Tβ , = ( , , , )c c s sΦ ϕ θ ϕ θ is the four vector of fields, and ( ( , ))xΦ τ is the Lagrangian associated with the Hamiltonian H . Note that for the ordinary LL state H is completely quadratic and thus invariant under a uniform translation of any bosonic fields: ( ) ( )i i ix x AΦ →Φ + , a symmetry which we use later. In the functional integral language, the time-ordered correlation for operators ( )A Φ and ( )B Φ is ( ) (0) =T A Bτ〈 τ 〉 0 1= ( , ) ( ( )) ( (0))exp ( ( , )) .x A B d dx x Z β   Φ τ Φ τ Φ τ Φ τ    ∫ ∫ ∫  (109) Later we will drop the time ordering symbol Tτ and use directly 〈〉 to denote the time-ordered average. The Lagrangian  can be written as 1 1( ) = = 2 2 i ij jM MΦ − Φ Φ Φ Φ , where the Fourier transform of the matrix ( , )M x τ is 2 2 2 2 0 0 0 0 ( , ) = . 0 0 0 0 c c c c s s s s q iq K iq K q M q q iq K iq K q   ω      ω ω    ω     ω  v v v v (110) Note that here ω is the imaginary frequency associated with τ. The inverse of ( , )M q ω reads 2 2 2 2 1 2 2 2 2 0 0 0 0 ( , ) = , 0 0 0 0 c c c c c c c c s s s s s s s s K i q i q K M q K i q i q K − ω −  Ω Ω   ω −  Ω Ω ω   ω − Ω Ω   ω −  Ω Ω  v v v v (111) where we denoted 2 2 2 2 / /=c s c sqΩ +ωv . Let ( , )i qΦ ω be the Fourier transform of ( , )i xΦ τ . Then 1( , ) ( , ) = ( , ).i j ijq q lM q−〈Φ ω Φ − −ω 〉 β ω (112) Correlations for / ( , )s c xϕ τ and / ( , )s c xθ τ can be obtained from Eq. (112) by inverse Fourier transform. The results at zero temperature are 2 2 2 / / / / 2 0 ( ) ( ( , ) (0,0)) = log , 2 s c s c s c s c K x y x a + τ 〈 ϕ τ −ϕ 〉 π (113a) 2 2 2 / / / 2 / 0 ( )1( ( , ) (0,0)) = log , 2 s c s c s c s c x y x K a + τ 〈 θ τ − θ 〉 π (113b) / / /( , ) (0,0) = Arg [ ( ) ], 2s c s c s c ix y ix〈ϕ τ θ 〉 − τ + π (113c) where / / 0( ) = sgn ( )s c s cy aτ τ + τv . The argument in Eq. (113c) is defined with a branch cut at ( ,0]−∞ . When calculating the spin-density correlators ( , )aas x t employing Eq. (107), there appear terms of three types: (a) ( , ) (0,0)x s x sx〈∂ ϕ τ ∂ ϕ 〉 , (b) (0,0) ( , )e i Ai i x s x Φ 〈∂ ϕ τ 〉∑ , and (c) ( , ) (0,0) e e i B x i Ci i i iΦ τ Φ 〈 〉∑ ∑ , where , ,i i iA B C are Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 279 V.L. Pokrovsky numerical coefficients. For their calculation we employ the invariance of H and  under the uniform translation of iΦ . For terms of type (b) with 0iA ≠ , the translation /i i iAΦ →Φ + π changes the sign of the averaged value leaving the Lagrangian invariant. Thus, the average (0,0) ( , )e i Ai i x s x Φ 〈∂ ϕ τ 〉∑ must be zero if at least one 0iA ≠ . For terms of type (c), a similar argument shows that ( , ) (0,0) e e = 0 i B x i Ci i i iΦ τ Φ 〈 〉∑ ∑ if at least one of the sums 0i iB C+ ≠ . As a result, ( , )zzs x τ reduces to _____________________________________________________ 2 ( ( , ) ( , )) 2 ( (0,0) (0,0))2 2 2 0 2 1( , ) = ( , ) (0,0) [e e e h.c.] 4 i x x ii k x c s c sFzz x s x ss x x a π ϕ τ −ϕ τ − π ϕ −ϕτ 〈∂ ϕ τ ∂ ϕ 〉 + 〈 〉 + + π π 2 ( ( , ) ( , )) 2 ( (0,0) (0,0))2 2 2 0 1 [e e e h.c.]. 4 i x x ii k x c s c sF a π ϕ τ +ϕ τ − π ϕ +ϕ+ 〈 〉 + π (114) ________________________________________________ From Eq. (113a) it follows that 2 ( , ) (0,0) =x s x sx〈∂ ϕ τ ∂ ϕ 〉 π 2 =0 2 1= ( ( , ) ( ,0)) \| 2x x s s xx x′ ′′∂ ∂ 〈− ϕ τ −ϕ 〉 = π 2 2 =02 0 ( ) ( )2= [ log[ ]] \| = 4 s s x x x K x x y a ′ ′ ′− + τ ∂ ∂ − π π 2 2 2 2 2 2= . ( ) s s s K y x x y − π + (115) In Eq. (115) we applied the formula 1 2 2e = e AiA − 〈 〉 〈 〉 valid for any Gaussian distributed variable A . Let calculate for example an average: 2 ( ( , ) ( , )) 2 ( (0,0) (0,0))e ei x x ic s c sπ ϕ τ −ϕ τ − π ϕ −ϕ〈 〉 = 2 ( ( , ) ( , )) 2 ( (0,0) (0,0))= e =i x x ic s c sπ ϕ τ −ϕ τ − π ϕ −ϕ〈 〉 2[ ( , ) ( , ) (0,0) (0,0)]= e x xc s c s−π〈 ϕ τ −ϕ τ −ϕ +ϕ 〉 = 2 2 2 2( ) ( ) [ log log ]2 22 2 0 0= e = K x y K x yc c s s a a + τ + τ −π + π π 0 0 2 2 2 2 = . K Kc s c s a a x y x y            + +    (116) Similar calculations can be done for other terms in ( , )zzs x τ and for the other two spin density correlators, which lead to the results (7) in the main text. The z direction spin den- sity correlators of the SDW state can also be calculated in the same way, with sϕ replaced by a constant that mini- mizes CH . Appendix G. Total spin correlations We calculate the total spin correlators by integration the spin density correlators over coordinate. The integrals that must be evaluated are of the forms 2 2 1 2 2= e cos ( ) , ( ) i s a s y x I d dx kx x y ∞ ∞ ωτ −∞ −∞ − τ +∫ ∫ 2 2 2 2 2 1= e cos ( ) , ( ) ( ) i b c c s I d dx kx x y x y ∞ ∞ ωτ −∞ −∞ τ + +∫ ∫ 3 2 2 1= e cos ( ) , ( ) i d c I d dx kx x y ∞ ∞ ωτ −∞ −∞ τ +∫ ∫ (117) where 0k ≥ and , , ,a b c d are constants, and ω is imagi- nary frequency associated with τ. The integrals 1I or 2I are parts of the correlations with small q or 2 Fq k of the ordinary LL state, and 3I corresponds to the z -component total spin correlations of the SDW state. The calculations are simpler in polar coordinates ( , )r ϕ related in the standard way to cartesian coordinates: = cosx r ϕ and = sinsy r ϕ. For example, 1I reads 2 ( cos sin ) 2 2 2 1 2 1 0 0 1 ( )sin cos= e ir k s a s rI d dr r ωπ ∞ ϕ+ ϕ − ϕ− ϕ ϕ =∫ ∫ v v 22 cos( arctan )2 2 2 3 0 0 1 cos(2 )= e ir k k ss a s d dr r ω ω + ϕ−π ∞ − ϕ − ϕ =∫ ∫ vv v 22 2 2 2 2 2 2 2 2 2 (3 )= . ( ) 4 a s s s s s k ka a k −  −ω +ωπΓ −   Γ +ω   v v v v v (118) The calculation of the integrals 2I , and also 3I is simi- lar, but for them the approximation = =c s Fv v v is suffi- cient. The results are 280 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 Spin resonance and spin fluctuations in a quantum wire 12 2 2 2 2 (1 )= , ( ) 4 b c F F F kb cI b c + −  −ωπΓ − −   Γ +   v v v 12 2 2 3 2 (1 )= . ( ) 4 d c c c kdI d −  −ωπΓ −   Γ   v v v (119) Applying these results to the correlations and analyti- cally continuing to real frequency by i iω→ω+ δ, we ob- tain the correlations for the ordinary LL state: ( ) = ( )R R xx yyS Sω ω = 1 2 2 22 2 2 2 0= ( )( ) Ks Kss sA + − ω +ω ω −ω + 1 11 1 2 2 2 22 2 2 2 2 1 [( ) ( ) ], 2 K Kc c K Kx s s kF A + − + − + −+ ω −ω + ω −ω 1 1 2 22 2 02( ) = ( ) , Kc KR z szz kF S A + − ω ω −ω (120) and those for the SDW state: ( ) = ( ) = 0,R R xx yyS Sω ω 12 2 20( ) = ( ) . Kc R zz SDW cS A − ω ω −ω (121) In the latter two equations we have defined the frequen- cies = 2s smω αv , 0 = 2 F Fkω v , = 2( )F Fk m±ω ± α v , 0 = 2c F ckω v , and the amplitudes 1 2 0 0 1( ) (2 ) 2 2 2 = , 1(1 ) 2 2 Ks K ss s s s s s a Kl K A K K + − Γ − − π Γ + + v v 1 2 0 2 1( ) (1 ) 2 2 2 = , 1( ) 2 2 Kc K cs x F s kF c F s a Kl K A K K + − Γ − − π Γ + v v 20 2 ( ) (1 ) 2 2 2 = , ( ) 2 2 K K c sc s z F kF c s F a K Kl A K K + − Γ − − π Γ + v v 202 ( ) (1 ) 2 2 = . ( ) 2 K cc c SDW c c a Kl A K − Γ − π Γ v v (122) In these equations ω is assumed to have a small imaginary part. Note that the expressions for the 2 Fq k parts of the ordinary LL correlations are only approximations when cv and sv are both close to Fv . We remind that the conservation of zS requires that ( ) = 0R zzS ω at any ω in the absence of the transverse exter- nal magnetic field. 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Spin-flip in a QW of a finite thickness Appendix B. Calculation of time-ordered averages in real time Appendix C. Relation between retarded and time-ordered spin correlators Appendix D. Evaluation of conductivity in Eqs. (42) and (43) Appendix E. Bosonization Appendix F. Spin-density correlations Appendix G. Total spin correlations

Spin resonance and spin fluctuations in a quantum wire

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