Boundary bound states in the Bose–Hubbard-like chain

Boundary bound states in the Bose–Hubbard-like chain

The degenerate Hubbard-like chain with open boundary conditions is studied with the help of the Bethe ansatz. The special case of the Bose–Hubbard-like chain is studied in detail. Boundary bound states, which appear as the consequence of the local potential(s), applied to the edge(s) of the open c...

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Datum:2007
1. Verfasser: Zvyagin, A.A.
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Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
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Низкоразмерные и неупорядоченные системы
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spelling irk-123456789-1278182017-12-29T03:03:17Z Boundary bound states in the Bose–Hubbard-like chain Zvyagin, A.A. Низкоразмерные и неупорядоченные системы The degenerate Hubbard-like chain with open boundary conditions is studied with the help of the Bethe ansatz. The special case of the Bose–Hubbard-like chain is studied in detail. Boundary bound states, which appear as the consequence of the local potential(s), applied to the edge(s) of the open chain are studied in the ground state 2007 Article Boundary bound states in the Bose–Hubbard-like chain / A.A. Zvyagin // Физика низких температур. — 2007. — Т. 33, № 5. — С. 597-600. — Бібліогр.: 19 назв. — англ. 0132-6414 PACS: 05.30.Jp, 71.10.–w http://dspace.nbuv.gov.ua/handle/123456789/127818 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
collection DSpace DC
language English
topic Низкоразмерные и неупорядоченные системы
Низкоразмерные и неупорядоченные системы
spellingShingle Низкоразмерные и неупорядоченные системы
Низкоразмерные и неупорядоченные системы
Zvyagin, A.A.
Boundary bound states in the Bose–Hubbard-like chain
Физика низких температур
description The degenerate Hubbard-like chain with open boundary conditions is studied with the help of the Bethe ansatz. The special case of the Bose–Hubbard-like chain is studied in detail. Boundary bound states, which appear as the consequence of the local potential(s), applied to the edge(s) of the open chain are studied in the ground state
format Article
author Zvyagin, A.A.
author_facet Zvyagin, A.A.
author_sort Zvyagin, A.A.
title Boundary bound states in the Bose–Hubbard-like chain
title_short Boundary bound states in the Bose–Hubbard-like chain
title_full Boundary bound states in the Bose–Hubbard-like chain
title_fullStr Boundary bound states in the Bose–Hubbard-like chain
title_full_unstemmed Boundary bound states in the Bose–Hubbard-like chain
title_sort boundary bound states in the bose–hubbard-like chain
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2007
topic_facet Низкоразмерные и неупорядоченные системы
url http://dspace.nbuv.gov.ua/handle/123456789/127818
citation_txt Boundary bound states in the Bose–Hubbard-like chain / A.A. Zvyagin // Физика низких температур. — 2007. — Т. 33, № 5. — С. 597-600. — Бібліогр.: 19 назв. — англ.
series Физика низких температур
work_keys_str_mv AT zvyaginaa boundaryboundstatesinthebosehubbardlikechain
first_indexed 2025-07-09T07:47:50Z
last_indexed 2025-07-09T07:47:50Z
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fulltext Fizika Nizkikh Temperatur, 2007, v. 33, No. 5, p. 597–600 Boundary bound states in the Bose–Hubbard-like chain A.A. Zvyagin 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 August 19, 2006 The degenerate Hubbard-like chain with open boundary conditions is studied with the help of the Bethe ansatz. The special case of the Bose–Hubbard-like chain is studied in detail. Boundary bound states, which appear as the consequence of the local potential(s), applied to the edge(s) of the open chain are studied in the ground state. PACS: 05.30.Jp Boson systems (for static and dynamic properties of Bose–Einstein condensates); 71.10.–w Theories and models of many-electron systems. Keywords: Bose–Hubbard chain, boundary bound states. Last decade the interest in the behavior of low-dimen- sional interacting quantum systems has been grown con- siderably. In particular, the interest in the studies of inter- acting lattice bosonic systems is connected with the experimental realization of such a systems, e.g., in sys- tems of ultracold gases of atoms in optical lattices, see, e.g., [1,2], in some of which the Bose–Einstein condensa- tion is believed to be observed. Here we consider a Bo- se–Hubbard-like model, which properties can be obtained using non-perturbative methods. Several authors studied the properties of the so-called degenerate Hubbard-like chain with the help of the Bethe ansatz, see, e.g., [3–11]. The Hamiltonian of the model was proposed first in the form H P Pp s N j s j s j s s s s s N jc c U n n� � � � � �� � � � � 1 1 1 ( , † , , , ( ) H.c.) , ,s j L � � � � � � � �� 1 (1) where c j s, † (c j s, ) creates (destroys) a particle at site j with the spin index s, n c cj s j s j s, , † ,� , U � 0 is the Hubbard re- pulsion constant, and the hopping integral is taken to be equal to unity. In particular, for N � 2 one deals with the standard Hubbard chain of interacting electrons. On the other hand, the case N � has to be related to the so-called Bose–Hubbard model: each particle has infinite number of states. The operator P projects onto the sub- space of states having at most two particles at each site. The presence of such a projector is crucial for the applica- bility of the Bethe ansatz method (first studies of the model, e.g., [3,4] missed that fact). Bethe ansatz gives the possibility to find eigenvalues and eigenfunctions of the stationary Schr�dinger equation of the considered model. Eigenvalues and eigenfunctions are parametrized by, ge- nerally speaking, several sets of quantum numbers (one set for each degree of freedom), called rapidities [12]. Later, however, it was recognized that it was not enough to use the projectors. For example, forU � 0 Bethe ansatz describes free particles, while according to Eq. (1) there must be correlations between particles, caused by the projectors. It was shown [8] that the continuum limit of the scattering phase shifts, which follow from the Bethe ansatz equations, are those of particles, interacting via a potential of the form1 2� s inh r (where r is the distance be- tween particles). It implies the necessity to include a long-range interaction, which dynamically exclude ma- ny-particle configurations at each site, to the Hamiltonian Eq. ( 1). However, unfortunately, the precise form of the lattice Hamiltonian of the degenerate Hubbard-like mo- del, which can be solved by the Bethe ansatz, has not been found yet. Nevertheless, studies of the Bethe ansatz solv- able model are very important (even in the absence of the precise form of the lattice Hamiltonian), because the fea- tures of the Bethe ansatz exact solution, e.g., for the Bose–Hubbard-like case N � are reminiscent of those for the real lattice Bose–Hubbard model, which was con- © A.A. Zvyagin, 2007 sidered, e.g., in the mean-field approximation [13], or by the quantum Monte-Carlo techniques [14]. The most in- teresting feature of the lattice Bose–Hubbard-like chain (and, in fact, of the degenerate Hubbard-like chain for N � 2) is the quantum phase transition at the critical value of the couling constantU c between a metallic (superfluid) gapless phase and an insulator gapped one. This feature is absent for the standard electron Hubbard chain N � 2 (whereU c � 0). However, the Bethe ansatz solution of the degenerate Hubbard-like chain [3–10] reveals such a feature. Most of the previous Bethe ansatz studies of the model were related to the chain with periodic boundary condi- tions (i.e. L � 1 1etc.). Here we consider the chain with open boundary conditions, i.e. one has to omit the term with ( ) , † ,c c L s s1 � H.c. from the Hamiltonian ( 1). Also, we add the boundary potential � � �p n s s N 1 1 , to the considered Hamiltonian (in fact one can add another boundary poten- tial, which acts on the particles at the last site, but for sim- plicity we limit ourselves with only one boundary poten- tial) without destroying of the exact integrability. Bethe ansatz equations, for several sets of rapidities, which parametrize eigenfunctions and eigenvalues of the con- sidered system, have the form exp ( ) ( ) (sin ) (sin ( ) ( 2 1 1 1 1 1 1 ik L B k e k e kj j M j j� � � � � � � �� � ) ), j N p�1, ..., , e e e M s s s s Ms s 1 1 1 1 1 1 1 1 � � � � � � � � � � � � � � � � � �( — ) ( )( ) ( ) ( ) ( ) 1 1 1 1� � �� �( — ) ( )( ) ( ) ( ) ( )� � � �� � � � s s s s e � � � � � e e M s M s s s s s s 2 1 2 1 � � � � �� � � � �( — ) ( ), , ..., ,( ) ( ) ( ) ( ) � �1 1, ..., N , (2) where B k ik p ik pj j j( ) [exp ( ) ] [exp ( ) ]� � � � � , e xn ( ) � � � �( ) / ( )2 2x iUn x iUn , N p is the number of particles, M N p0 � , and � j jk ( ) sin 0 � . The energy of the corre- sponding state is equal to E k pNj j N p � � � � �2 1 1 cos , (3) where N 1 is the total number of particles at the first site of the chain. Obviously, the Bethe ansatz equations (2), (3) coincide with the known ones for the case of free parti- cles with N internal degrees of freedom (U � 0, from which case one can see that Bethe ansatz equations are quantization conditions), for the case of free spinless fer- mions (orthofermions) U �, and with the ones for the standard Hubbard chain for N � 2 [12]. In the continuum limit of small k j (here we replace cos k j �1 22k j / and sin k kj j the Bethe ansatz equations agree with the ones for particles with �-function local in- teractions between particles [15]. Below we shall consider only the case N �, related to the Bose–Hubbard chain with open boundary condi- tions. In this case only one set of quantum numbers, k j is essential [4]. In the thermodynamic limit L �, N p �, but with the ratio N Lp � being fixed, using the standard for Bethe ansatz solvable models procedure [12], we can write down integral equations for so-called «dressed ener- gies» �( )k and densities�( )k (�h k( ) is the density of holes) for eigenstates � � �( ) cos cos (sin — sin ) ( )k k dk k G k k k Q Q � � � � � � � �2 , � � � � ( ) ( ) cos ( sin ) ( ) cos ( ) k k L kG k i B k kB k h� � � � � ��1 2 1 2 2 2 2� � � � � � � � �cos (sin — sin ) ( )k dk G k k k Q Q � , (4) where� is the chemical potential, the kernel has the form G x U x U ( ) [ ] � � 2 4 2 2� . (5) Here, using the fact that dressed energies, densities, ker- nels and driving terms (those, which do not depend on �( )k , �( )k , and �h k( ) are even functions, we replaced dis- tributions of k over total intervals, like for periodic chains, instead of half-intervals for the open chain. This is why, the term G k L( sin ) /2 appeared in the equation for densities. One can see that driving terms in the equations for densities has the term of order of 1 and the contribu- tion of order of 1/ L. The integral equation for densities is linear. Hence, we can look for the solution of that equa- tion in the form � � �( ) ( ) ( / ) ( )( ) ( )k k L k� �0 11 . Equations 598 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 A.A. Zvyagin for � ( )0 and � ( )1 are separated from each other, and they describe the behavior of the bulk and boundaries, respec- tively. Obviously, the equations for the bulk part, � ( )0 , coin- cide with the ones for periodic case [3–11]. Quantum numbers k are distributed in the interval � � �Q k Q. It is easy to see that for U U c� � 4 3 the considered model is in the metallic phase with gapless low-lying excitations, while for U U c� the model is in the insulating phase [4,8]. From now on we concentrate in the insulating case with Q � � (N Lp � ). In that case it is easy to calculate the ground state energy (of the bulk), which is equal to E d J J f f UG U� � � � � � ��4 80 1 0 2� � � � � �( ) ( ) ( )/e E , (6) where f U� �4 162/ [ ], and E is the complete elliptic integral of the second kind. The energy of the low-lying excitation is E k d J kh U� � � � ��2 40 1 0 0 2cos ( ) cos ( sin ) /� � � � � �e , (7) where k 0 is related to the momentum of the excitation ph as p dk k d k Jh k U� � � � � � �� � � �2 1 2 0 0 0 2 0 cos cos ( sin ) ( ) /� � � �e � . (8) It is easy to see that Eh is the monotonically decreasing function of | |k 0 , and it has minima at k 0 � � and a maxi- mum at k 0 0� . Hence, the low boundary of the band of low-lying excitation is equal to � � � �2 8 42/ [ ]U U , and the upper boundary is equal to 2 8 42� � �/ [ ]U U . As for the chemical potential, it is then equal to � �� � �E kh ( )0 . Other bulk excitations (bound states, or strings) have higher energies. Now let us consider the boundary contribution. In the absence of the boundary potential the ground state energy of free edges, or the surface energy (it is the difference be- tween the ground state energy of the chain with open and with periodic boundary conditions), is equal to E U U U i dk ks � � � � � ! � �2 2 4 2 2 2 � � � cos ! � � � � �� �� � (sin ( / )) (sin ( / ))k i U k i U4 1 4 12 2 . (9) Next, we turn to boundary bound states. In principle, one is free to leave boundary bound states empty [12]. However, they are important for, e.g., calculation of the exponents for correlation functions. There are no bound- ary bound states for | |p � 1, and we can use the previous analysis in that case. On the other hand, for | |p � 1bound- ary bound states, caused by the boundary potential p ap- pear with the roots k i pj � ln and the energy E p pb � � �( ) /2 1 . The (most important) levels of the low-energy boundary bound states are split off the upper and lower boundaries of the band of the low-lying excita- tions Eq. (7). Higher energy boundary bound states are also split off upper and lower boundaries of the bands for higher energy bulk excitations. Notice that we are inter- ested in decaying roots. The surface energy in the case with p � 0 is equal to E E pN E p dk ksb s b� � � � ! � �— (| | ) cos1 21" � � ! � � � � � � � � � iB k kB k G k i p G k i p ( ) cos ( ) (sin ln ) (sin ln ) 2� . (10) The last two terms denote the removing of the root, re- lated to the boundary bound state, from the continuum of bulk states, forming the vacuum. On the other hand, Eqs. ( 2) can be solved for the cases of free bosons (N �, U � 0), or free spinless fermions (orthofermions) (U �) [12]. Those cases are interest- ing, because we can understand the behavior of the boundary bound states for the metallic case with gapless excitations. Bethe ansatz equations for these two cases have the form exp ( ) ( )2 1ik B kj L j � , (11) where the «plus» sign is taken for the first case, and the «minus» one for the second case, respectively. Here one can obviously see that Bethe ansatz equations are quantization conditions for the rapidities k j (they are quasi-momenta for free particles). One can also see that for p � 0 ( p � 1) solutions for boundary bound states ap- pear even for free particles (they are split off the bands of bulk linear excitations). The surface energy at N Lp � is equal to E pN E p d S p sp f b ( ) (| | ) [ ( )] ( � � � � � � � �1 0 1 1 2 2 2 2 " � � sinh sinh x x) ( )cosh , (12) where S p p p( ) ( / ) [( )/| | ]� � �2 1 1� arctan . The surface en- ergy for p � 0 is equal to Es � �1 2( / )� . It turns out that it is incorrect to take the limit U of Eqs. ( 9) and ( 10) to ob- tain this result, because those equations were obtained for the insulating phase U U c� . In summary, we have studied the Bethe ansatz solution of the degenerate Hubbard-like chain with open boundary Boundary bound states in the Bose–Hubbard-like chain Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 599 conditions. The most interesting case of the Bo- se–Hubbard-like chain is analysed for the ground state. The surface energy (the energy of the free edges of the chain) is calculated. We have shown that for large values of boundary potentials boundary bound states are split off the bands of the bulk states. The most important states ap- pear in the gap, where no bulk excitations exist for the in- sulating phase with U U c� . Our results can be applied, e.g., for ultracold atoms in optical lattices, or photonic lat- tices, where discrete surface low-lying excitations were recently observed [16–19]. 1. T. St�ferle, H. Moritz, C. Schori, M. K�hl, and T. Esslin- ger, Phys. Rev. Lett. 92, 130403 (2004). 2. B. Paredes, A. Widera, V. Murg, O. Mandel, S. F�lling, I. Cirac, G.V. 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Lett. 96 073901 (2006). 17. C.R. Rosberg, D.N. Neshev, W. Krolikowski, A. Mitchell, R.A. Vicencio, M.I. Molina, and Yu.S. Kivshar, unpub- lished. 18. M. Matuszewski, W. Kvolikowski, M. Trippenbach, and Yu.S. Kivshar, Phys. Rev. A73, 063621 (2006). 19. B. Eirmann, Th. Anker, M. Albiez, M. Taglieber, P. Treut- lein, K.-P. Marzlin, and M.K. Oberthaler, Phys. Rev. Lett. 92, 230401 (2004). 600 Fizika Nizkikh Temperatur, 2007, v. 33, No. 5 A.A. Zvyagin

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