To the theory of spin–charge separation in one-dimensional correlated electron systems
Spin—charge separation is considered to be one of the key properties that distinguish low-dimensional electron systems from others. Three-dimensional correlated electron systems are described by the Fermi liquid theory. There, low-energy excitations (quasiparticles) are reminiscent of noninteract...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1198522017-06-11T03:04:55Z To the theory of spin–charge separation in one-dimensional correlated electron systems Zvyagin, A.A. Низкоразмерные и неупорядоченные системы Spin—charge separation is considered to be one of the key properties that distinguish low-dimensional electron systems from others. Three-dimensional correlated electron systems are described by the Fermi liquid theory. There, low-energy excitations (quasiparticles) are reminiscent of noninteracting electrons: They carry charges e and spins 1/2. It is believed that for any one-dimensional correlated electron system, low-lying electron excitations carry either only spin and no charge, or only charge without spin. That is why recent experiments looked for such low-lying collective electron excitations, one of which carries only spin, and the other carries only charge. Here we show that despite the fact that for exactly solvable one-dimensional correlated electron models there exist excitations which carry only spin and only charge, in all these models with short-range interactions the low-energy physics is described by low-lying collective excitations, one of which carries both spin and charge. 2004 Article To the theory of spin–charge separation in one-dimensional correlated electron systems / A.A. Zvyagin // Физика низких температур. — 2004. — Т. 30, № 9. — С. 969–973. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 71.27.+a, 71.10.Fd, 71.10.Pm http://dspace.nbuv.gov.ua/handle/123456789/119852 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы |
| spellingShingle |
Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы Zvyagin, A.A. To the theory of spin–charge separation in one-dimensional correlated electron systems Физика низких температур |
| description |
Spin—charge separation is considered to be one of the key properties that distinguish low-dimensional
electron systems from others. Three-dimensional correlated electron systems are described
by the Fermi liquid theory. There, low-energy excitations (quasiparticles) are reminiscent
of noninteracting electrons: They carry charges e and spins 1/2. It is believed that for any one-dimensional
correlated electron system, low-lying electron excitations carry either only spin and no
charge, or only charge without spin. That is why recent experiments looked for such low-lying collective
electron excitations, one of which carries only spin, and the other carries only charge. Here
we show that despite the fact that for exactly solvable one-dimensional correlated electron models
there exist excitations which carry only spin and only charge, in all these models with short-range
interactions the low-energy physics is described by low-lying collective excitations, one of which
carries both spin and charge. |
| format |
Article |
| author |
Zvyagin, A.A. |
| author_facet |
Zvyagin, A.A. |
| author_sort |
Zvyagin, A.A. |
| title |
To the theory of spin–charge separation in one-dimensional correlated electron systems |
| title_short |
To the theory of spin–charge separation in one-dimensional correlated electron systems |
| title_full |
To the theory of spin–charge separation in one-dimensional correlated electron systems |
| title_fullStr |
To the theory of spin–charge separation in one-dimensional correlated electron systems |
| title_full_unstemmed |
To the theory of spin–charge separation in one-dimensional correlated electron systems |
| title_sort |
to the theory of spin–charge separation in one-dimensional correlated electron systems |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2004 |
| topic_facet |
Низкоразмерные и неупорядоченные системы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119852 |
| citation_txt |
To the theory of spin–charge separation in one-dimensional correlated electron systems / A.A. Zvyagin // Физика низких температур. — 2004. — Т. 30, № 9. — С. 969–973. — Бібліогр.: 20 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT zvyaginaa tothetheoryofspinchargeseparationinonedimensionalcorrelatedelectronsystems |
| first_indexed |
2025-07-08T16:46:42Z |
| last_indexed |
2025-07-08T16:46:42Z |
| _version_ |
1837098025951952896 |
| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, No. 9, p. 969–973
To the theory of spin–charge separation in
one-dimensional correlated electron systems
A.A. Zvyagin
Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str., 38 D-01187, Dresden, Germany
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy
of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: zvyagin@ilt.kharkov.ua
Received February 2, 2004, revised March 16, 2004
Spin—charge separation is considered to be one of the key properties that distinguish low-di-
mensional electron systems from others. Three-dimensional correlated electron systems are de-
scribed by the Fermi liquid theory. There, low-energy excitations (quasiparticles) are reminiscent
of noninteracting electrons: They carry charges �e and spins 1 2/ . It is believed that for any one-di-
mensional correlated electron system, low-lying electron excitations carry either only spin and no
charge, or only charge without spin. That is why recent experiments looked for such low-lying col-
lective electron excitations, one of which carries only spin, and the other carries only charge. Here
we show that despite the fact that for exactly solvable one-dimensional correlated electron models
there exist excitations which carry only spin and only charge, in all these models with short-range
interactions the low-energy physics is described by low-lying collective excitations, one of which
carries both spin and charge.
PACS: 71.27.+a, 71.10.Fd, 71.10.Pm
The spin—charge separation phenomenon arising
from interactions between electrons in low-dimen-
sional condensed matter models has attracted consid-
erable interest since the discovery of high-Tc supercon-
ductivity in low-dimensional cuprates [1,2]. Other
possible experimental realizations of spin—charge
separation appear to exist in organic conductors [3,4]
and in mesoscopic and nanoscale metallic systems [5],
e.g., quantum wires and carbon nanotubes. Physicists
believe that fundamental differences between low and
higher dimensions can be traced back to the reduced
phase space in the former, which results in the break-
down of the Fermi liquid description. The localization
of the electrons with even a small amount of disorder,
thermal fluctuations destroying long-range order at
any nonzero temperature (if only short-range interac-
tions are present), and quantum fluctuations tending
to suppress a broken continuous symmetry are among
other features of low-dimensional correlated electron
systems. For «standard» three-dimensional electron
systems an interaction between electrons is described
in the framework of Landau’s Fermi liquid theory [6].
A Fermi liquid consists of the Fermi sphere and a
rarefied gas of weakly interacting «quasiparticles» de-
fined via poles in one-particle Green’s functions.
Quasiparticles continuously evolve from free electrons
when the interaction is adiabatically switched on and,
hence, have the same quantum numbers and statistics
as noninteracting electrons. In one space dimension
the Fermi liquid quasiparticle pole disappears (its
residue vanishes) and is replaced by incoherent
structures, which follow from the global conformal
invariance [7,8]. These structures involve nonu-
niversal power-law singularities. Although the Fermi
surface is still properly defined, the discontinuity of
the momentum distribution at the Fermi surface disap-
pears as a consequence of the zero residue [8,9]. Sys-
tems displaying such breakdown of the Fermi liquid
picture and exotic low-energy spectral properties are
known as Luttinger liquids [7,8]. The theoretical un-
derstanding of two-dimensional electron systems is far
from being complete. They frequently manifest the ab-
sence of ordering, and, hence, mean-field-like approxi-
mations fail. Many of normal state properties of
two-dimensional high-Tc superconductors are very dif-
ferent from normal metals and cannot be reconciled
© A.A. Zvyagin, 2004
with a standard Fermi liquid theory. A marginal Fermi
or Luttinger liquid picture, similar to the one of one-di-
mensional conductors, which uses the spin—charge
separation, has been proposed to explain some of these
features [10].
An approximate description of one-dimensional corre-
lated electron models in the framework of the
bosonization approach [8] supports the Luttinger liquid
picture. The latter is characterized, among other fea-
tures, by a spin—charge separation: The charge and spin
contents of wave functions move with different speeds
and are related to different low-lying excitations of cor-
related electron systems. It is believed that for any
one-dimensional correlated electron system the low-ly-
ing electron excitations carry either only spin and no
charge, or only charge without spin [8,9]. However, the
range of applicability of the Luttinger liquid boso-
nization theory is limited to the continuous models and
weak enough interactions: The latter have to be smaller
than the Fermi velocity (related to the bandwidth of
electrons) by the consideration of only low-energy elec-
trons close to Fermi points, for which one can linearize
their dispersion law [7,9]. In recent experiment, which
imply the observation of a spin—charge separation these
conditions were often not satisfied [1–4]. Nonetheless,
for many one-dimensional correlated electron models
with strong interaction between electrons the powerful
Bethe ansatz method can be applied, in the framework
of which collective electron excitations also have differ-
ent velocities [11]. It is often stated that a spin—charge
separation persists in that description also (see, e.g.,
Refs. 1, 8, and 9), justifying in that way the applicabil-
ity of the Luttinger liquid picture to any one-dimen-
sional correlated electron systems with metallic behav-
ior. In our work we exactly prove that it is incorrect:
We show that in none of the known Bethe ansatz solv-
able models do the correlated electrons Dirac seas, and,
hence, low-energy excitations, pertain to only spin-car-
rying and only charge-carrying quasiparticles: For those
models necessarily one Dirac sea is formed by
quasiparticles, which carry both spin and charge. Un-
fortunately, this relatively simple statement has never
been clearly stated in published papers devoted to the
Bethe ansatz solvable correlated electron models, and
this has led to many misunderstandings.
The exact solution by means of Bethe’s ansatz of
numerous models of one-dimensional correlated elec-
tron systems provides deep insight into the ground
state of systems, complete classification of states,
thermodynamic properties, etc. The best known exam-
ples of models of strongly correlated electrons, solved
by the Bethe ansatz are the one-dimensional Hubbard
model and supersymmetric t—J model [11]. With this
method the Hamiltonian is diagonalized in terms of a
set of parameters (quantum numbers) known as
rapidities. A system with internal degrees of freedom
(such as a spin) requires a sequence of nested general-
ized Bethe ansätze for wave functions. Each internal
degree of freedom gives rise to one set of rapidities,
i.e., in the case of electrons, which carry spin, one has
two sets of rapidities. Rapidities parametrize the
eigenfunctions and eigenvalues of a stationary
Schrödinger equation. Independently of the symmetry
of the wave function and spin, energy eigenstates are
occupied according to the Fermi—Dirac statistics
(hard-core particles). There are many other solutions
of the Bethe ansatz equations for rapidities, which de-
scribe bound states of electrons with a complex struc-
ture. In the Bethe ansatz description in the ground
state (and at low temperatures) each internal degree
of freedom contributes with one Fermi sea. The Fermi
velocities of these Fermi seas are in general different,
giving rise to what is often also called the
spin—charge separation. However, the question ap-
pears: Whether these low-lying excitations really
carry only charge or only spin, as in the Luttinger liq-
uid picture? To answer this question, let us consider
the Bethe ansatz equations for two sets of rapidities
u j1, (j M� 1 1,..., , M1 being the total number of elec-
trons) and u j2, (j M� 1 2,..., , M2 is the number of
electrons with down spins). The Bethe ansatz equa-
tions for correlated electron chains with periodic
boundary conditions can be written in the form [11]:
2 0
11
2
� �J Lp u u u ii j i i j i m
l
l j
M
m
i j l m
l
, , ,
,
, ,( ) ( , ),� �
�
�
�
�� � 1 2, ,
(1)
with p u kj j1
0
1( ), � (where kj are quasimomenta),
p x x y2
0
11 0( ) ( , ),� �� , � �i j j ix y y x, ,( , ) ( , )� � , where
L is the number of sites, Ji j, are (half)integer numbers,
different from each other for each set. This solution is
valid in the domain of parameters 0 2 1 2� �M M / and
0 1� �M L. For the Hubbard model one has u j1, �
� sinkj , � �12
1
222 4, ,( , ) [ ( ) ], ( , )x y x y /U x y� � ��tan
� ��2 21tan [ ( ) ]x y /U (U is the constant of the Hub-
bard local interaction between electrons situated at the
same site of a ring but with different spin directions; the
hopping integral between neighboring sites is equal to
1.) For the supersymmetric t—J chain (for J t� �2 2)
these functions are
p u uj j1
0
1
1
12 2( ), ,� �tan ,
�12
12 2, ( , ) [ ( )]x y x y� ��tan ,
�22
12, ( , ) ( )x y x y� ��tan ( )k pj j� �2 21tan .
970 Fizika Nizkikh Temperatur, 2004, v. 30, No. 9
A.A. Zvyagin
The energy of the state with M1 electrons, M2 of
which having their spins down, is equal to
E E ui
j
M
i
i j
i
� �
��
��0
0
11
2
( ),, (2)
where E0 is the energy for Mi � 0,
�
� � � � �H/2
� 2 coskj , and for the repulsive Hubbard chain we
have
�( )x H� , for the attractive Hubbard model
2
0 22 4 1 4( ) [ ( )]x x i U/� � � � �Re ,
and for the supersymmetric t—J model one uses
�
0
22 2 2( ) ( )x a x� � � � .
Here a xn ( ) � ( ) [ ( ) ]n/ / x n/2 22 2� � ,
is the chemi-
cal potential, and H is an external magnetic field. So-
lutions (each set of ui j, corresponds to only one
eigenstate) of Eqs. (1) parametrize all eigenvalues
and eigenfunctions in the domain [12,13]
0 22 1� �M M / , 0 1� �M L. Actually, Eqs. (1) are
quantization conditions for rapidities. This can be re-
cognized, e.g., for the U � 0 case of the Hubbard
chain, which pertains to the one-dimensional free lat-
tice electron gas. «Separation» means that the Bethe
ansatz eigenstate is determined by two sets of quan-
tum numbers. However, Eqs. (1) imply that it is im-
possible to really separate their contributions: They
are, obviously, coupled to each other.
Consider the case of the repulsive Hubbard chain.
The ground state and low-energy excitations pertain to
real rapidities ui j, [11]. What are the charges and spins
related to those quantum numbers? Suppose one
changes the number of rapidities u j2, , keeping the num-
ber of u j1, fixed. Such a process yields the change of the
total magnetic moment, while the total number of elec-
trons is not changed. Hence, such excitations carry
only spin, but not charge. [The redistribution of
rapidities with their number fixed can produce only
particle—hole excitations, which, by definition, carry
no spin and charge (but can change the energy).] The
simple way to see it without the connection to other
kinds of spin and/or charge-carrying excitations, re-
lated to the change of the number of u j1, , is to consider
the state with M L1 � fixed (i.e., at half-filling, which
belongs to the domain 0 22 1� �M M / , 0 1� �M L),
where the low-energy dynamics is known to be con-
nected with only spin excitations [11].
Let us now change the number of rapidities u j1,
with the number of u j2, being fixed. Such a process
produces a change of both the total charge and the to-
tal magnetic moment of the system. Thus, this kind of
excitation carries both spin and charge. Again, to
avoid connection to the other set of rapidities, one can
consider the state with M2 0� , which also belongs to
the domain 0 22 1� �M M / , 0 1� �M L. It is very dif-
ficult to believe that the fact that an excitation carries
spin depends on the number of such excitations.
We emphasize that namely the above-mentioned two
kinds of low-lying excitations are considered in the
conformal limit of the Bethe ansatz solvable theories to
compute correlation function exponents [14], and
namely those results of the Bethe ansatz considerations
are used to compare with the Luttinger liquid
(bosonization) approach [8]. This is why, namely these
two kinds of low-lying excitations are the most impor-
tant ones for the repulsive Hubbard chain.
Hence, low-lying excitations of the repulsive Hub-
bard chain are related to eigenstates, ones of which
carry only spin and others carry both charge and spin.
These states are often called spinons and unbound
electron excitations, respectively (sometimes, un-
bound electron excitations are called holons, which is,
unfortunately, misleading). The SO(4) symmetry of
the Hubbard Hamiltonian [15] implies the presence of
excitations which carry only charge and no spin. They
are spin-singlet bound states of electrons, e.g., local
pairs. However, for the repulsive Hubbard chain these
states have large energies and do not affect the low-en-
ergy thermodynamics [11]. These states (local
spin-singlet pairs) are low-lying states for the attrac-
tive Hubbard chain, and together with unbound elec-
tron excitations determine low-energy properties of
that model [11]. These low-energy excitations of the
attractive Hubbard chain carry only charge and both
spin and charge, respectively. Again, namely these
two kinds of low-lying excitations for the attractive
Hubbard chain are important, because they determine
the correlation function exponents in the conformal
limit [16].
The fact that excitations related to the change of
the number of rapidities u j1, for the Hubbard model
carry both spin and charge does not depend on the
value of U � 0. It is often misunderstood [8,9] that
the eigenfunction of the repulsive Hubbard chain with
U � � is reminiscent of the one of spinless fermions
multiplied by the wave function of the Heisenberg
spin1 2/ chain [17]. In fact, this limitU � � of the re-
pulsive Hubbard chain has been the only argument
used to prove the spin—charge separation for Bethe
ansatz solvable models, cf. Ref. 8. However, the care-
ful inspection of the expressions for energies of
charge-carrying excitations [11] in this limit shows
that they carry also spin-1 2/ (one can see it by taking
the derivative of the energy with respect to H, and
then putting H � 0). It turns out that charge-carry-
ing excitations (the wave function of which is reminis-
cent of spinless fermions) for the U � � repulsive
To the theory of spin–charge separation in one-dimensional correlated electron systems
Fizika Nizkikh Temperatur, 2004, v. 30, No. 9 971
Hubbard model pertain to the spin-polarized phase
(the critical field of the quantum phase transition to
the spin-polarized phase for the 0 � � �U case is zero
[11]). But namely in the spin-polarized case the fact
that charge-carrying excitations carry also spin is es-
pecially clear, see above. Hence, the claim that in the
U � � repulsive Hubbard chain one has factorization
of the wave function into only charge-carrying and
only spin-carrying parts is incorrect.
The same conclusions follow from the study of
low-energy properties of the supersymmetric t—J
chain. Here also one kind of low-energy excitations
carries both spin and charge. Again, namely these ex-
citations determine the correlation function exponents
in the conformal limit [18].
Hence, the statement that for any one-dimensional
correlated electron system the low-energy excitations
carry either only spin or only charge is invalid.
Why, then, does the Luttinger liquid description
manifest a spin—charge separation? [Note that Lut-
tinger liquid bosons formally do not carry either spin
or charge, because they are particle—hole excitations
of related correlated electron systems and, by defini-
tion, cannot carry spin and charge but can only change
energies and momenta.] One can see that the Lut-
tinger liquid approach and the Bethe ansatz approach
use different sets of quantum numbers. As we have
shown above, the Bethe ansatz eigenstates and eigen-
values are determined by two sets of quantum num-
bers, one of which is the total number of electrons,
and the other is the total number of spins down. Cont-
rary, in the Luttinger liquid approach one classifies
states with two sets of quantum numbers, one of
which is again the total number of electrons, but the
other is the total magnetic moment of electrons (not
the number of spins down!) [7]. Then, naturally, the
Luttinger liquid approach manifests the spin-charge
separation, unlike the low-energy behavior of the
Bethe ansatz solvable models of correlated electrons;
see above.
Then one is faced with an obvious contradiction.
The Bethe ansatz is the exact solution, and it does not
manifest the spin—charge separation, as we proved
above. The Luttinger liquid approach (an approxi-
mate one) manifests the spin—charge separation for
the same models of correlated electrons, e.g., for the
Hubbard model. Where, then, can the Luttinger liq-
uid approach be used? This can be seen, e.g., from the
conformal limit of the Bethe ansatz solution of the
Hubbard chain, where one linearizes the energy of
low-lying excitations about Fermi points. In this limit
for small U it is possible to rewrite the Bethe ansatz
answer [14,19] in such a way that states will be classi-
fied not by the total number of electrons and number
of spin-down electrons, but by the former and the
total magnetic moment. This answer, naturally, coin-
cides with the Luttinger liquid one [8,9]. However,
this approach is only correct in the limit of weak elec-
tron—electron interactions and linearized dispersion
laws of low-energy excitations. Simply by using the
corrections of higher order inU, one can see that it is
impossible to reformulate the Bethe ansatz conformal
limit of the Hubbard chain in terms of separated con-
tributions, which pertain to changes of the total num-
ber of electrons and the total magnetic moment.
Hence, one can conclude that the approximate boso-
nization (Luttinger liquid) procedure can be correctly
applied to exactly (Bethe ansatz) solvable correlated
electron models for only weak electron—electron cor-
relations; otherwise the results of the Luttinger liquid
approach contradict known exact Bethe ansatz results.
Experiments [1–5] have actually observed low-ly-
ing excitations, characterized by two different energy
scales. However, nothing permits one to conclude
from those experiments that one of these excitations
carries only spin while the other carries only charge.
Also, for the main issues of the Luttinger liquid-like
theory for the two-dimensional correlated electron
systems [10] it is not necessary that real spin—charge
separation occurs. One needs only the presence of two
low-lying excitations (with one of them carrying only
spin) instead of Fermi liquid quasiparticles. It seems
interesting to study the behavior of charge and spin
persistent currents in correlated electron rings with a
strong electron—electron interaction for nonzero H.
In such a case different (but nonzero) spins (charges)
of two kinds of low-energy excitations can yield an in-
terference of oscillations of persistent currents with
two different periods [20].
Summarizing, in this work we have shown that for
all known Bethe ansatz solvable one-dimensional
models of correlated electrons with short-range inter-
actions between electrons one of low-lying excitations
carries both spin and charge. This is very different
from what is often believed when considering the
spin—charge separation in low-dimensional correlated
electron models. This is why the applicability of the
conclusions of the Luttinger liquid approach to the
spin—charge separation for one-dimensional lattice
models with short-range electron—electron interac-
tions that are much stronger than the bandwidth of
the electrons is under question.
I appreciate a discussion with P.W. Anderson on
spin—charge separation theories for two-dimensional
correlated electron models. I thank H. Johannesson
for useful comments. This work was supported in part
by the Swedish Foundation for International Coopera-
tion in Research and Higher Education (STINT).
972 Fizika Nizkikh Temperatur, 2004, v. 30, No. 9
A.A. Zvyagin
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To the theory of spin–charge separation in one-dimensional correlated electron systems
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