The generalized Gibbs ensemble for Heisenberg spin chains. (English) Zbl 1456.82311
Summary: We consider the generalized Gibbs ensemble (GGE) in the context of global quantum quenches in XXZ Heisenberg spin chains. Embedding the GGE into the quantum transfer matrix formalism, we develop an iterative procedure to fix the Lagrange multipliers and to calculate predictions for the long-time limit of short-range correlators. The main idea is to consider truncated GGEs with only a finite number of charges and to investigate the convergence of the numerical results as the truncation level is increased. As an example we consider a quantum quench situation where the system is initially prepared in the Néel state and then evolves with an XXZ Hamiltonian with anisotropy \(\Delta > 1\). We provide predictions for short-range correlators and gather numerical evidence that the iterative procedure indeed converges. The results show that the system retains memory of the initial condition, and there are clear differences between the numerical values of the correlators as calculated from the purely thermal and generalized Gibbs ensembles.
MSC:
| 82B23 | Exactly solvable models; Bethe ansatz |
| 81Vxx | Applications of quantum theory to specific physical systems |
Software:
FORMReferences:
| [1] | Rigol M, Dunjko V and Olshanii M 2008 Thermalization and its mechanism for generic isolated quantum systems Nature 452 854 arXiv:0708.1324 [cond-mat.stat-mech] · doi:10.1038/nature06838 |
| [2] | Polkovnikov A, Sengupta K, Silva A and Vengalattore M 2011 Colloquium: nonequilibrium dynamics of closed interacting quantum systems Rev. Mod. Phys. 83 863 · doi:10.1103/RevModPhys.83.863 |
| [3] | Rigol M, Dunjko V, Yurovsky V and Olshanii M 2007 Relaxation in a completely integrable many-body quantum system: an ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons Phys. Rev. Lett. 98 050405 [arXiv:cond-mat/0604476] · doi:10.1103/PhysRevLett.98.050405 |
| [4] | Caux J-S and Essler F H L 2013 Time evolution of local observables after quenching to an integrable model Phys. Rev. Lett. 110 257203 arXiv:1301.3806 [cond-mat.stat-mech] · doi:10.1103/PhysRevLett.110.257203 |
| [5] | Barmettler P, Punk M, Gritsev V, Demler E and Altman E 2010 Quantum quenches in the anisotropic spin-1/2 Heisenberg chain: different approaches to many-body dynamics far from equilibrium New J. Phys. 12 055017 arXiv:0911.1927 [cond-mat.quant-gas] |
| [6] | Rossini D, Silva A, Mussardo G and Santoro G E 2009 Effective thermal dynamics following a quantum quench in a spin chain Phys. Rev. Lett. 102 127204 arXiv:0810.5508 [cond-mat.stat-mech] · doi:10.1103/PhysRevLett.102.127204 |
| [7] | Rossini D, Suzuki S, Mussardo G, Santoro G E and Silva A 2010 Long time dynamics following a quench in an integrable quantum spin chain: local versus nonlocal operators and effective thermal behavior Phys. Rev. B 82 144302 arXiv:1002.2842 [cond-mat.stat-mech] · doi:10.1103/PhysRevB.82.144302 |
| [8] | Calabrese P, Essler F H L and Fagotti M 2012 Quantum quench in the transverse field Ising chain: I. Time evolution of order parameter correlators J. Stat. Mech. P07016 arXiv:1204.3911 [cond-mat.quant-gas] |
| [9] | Calabrese P, Essler F H L and Fagotti M 2012 Quantum quenches in the transverse field Ising chain: II. Stationary state properties J. Stat. Mech. P07022 arXiv:1205.2211 [cond-mat.stat-mech] |
| [10] | Blass B, Rieger H and Iglói F 2012 Quantum relaxation and finite-size effects in the XY chain in a transverse field after global quenches Europhys. Lett. 99 30004 arXiv:1205.3303 [cond-mat.stat-mech] · doi:10.1209/0295-5075/99/30004 |
| [11] | Essler F H L, Evangelisti S and Fagotti M 2012 Dynamical correlations after a quantum quench Phys. Rev. Lett. 109 247206 arXiv:1208.1961 [cond-mat.stat-mech] · doi:10.1103/PhysRevLett.109.247206 |
| [12] | Caneva T, Canovi E, Rossini D, Santoro G E and Silva A 2011 Applicability of the generalized Gibbs ensemble after a quench in the quantum Ising chain J. Stat. Mech. P07015 arXiv:1105.3176 [cond-mat.stat-mech] |
| [13] | Fagotti M and Essler F H L 2013 Reduced density matrix after a quantum quench Phys. Rev. B 87 245107 arXiv:1302.6944 [cond-mat.stat-mech] · doi:10.1103/PhysRevB.87.245107 |
| [14] | Calabrese P and Cardy J 2007 Quantum quenches in extended systems J. Stat. Mech. P06008 arXiv:0704.1880 [cond-mat.stat-mech] · Zbl 1456.81358 |
| [15] | Cassidy A C, Clark C W and Rigol M 2011 Generalized thermalization in an integrable lattice system Phys. Rev. Lett. 106 140405 arXiv:1008.4794 [cond-mat.stat-mech] · doi:10.1103/PhysRevLett.106.140405 |
| [16] | Gurarie V 2013 Global large time dynamics and the generalized Gibbs ensemble J. Stat. Mech. P02014 arXiv:1209.3816 [cond-mat.stat-mech] · Zbl 1456.82588 |
| [17] | Zhang J M, Cui F C and Hu J 2012 Dynamical predictive power of the generalized Gibbs ensemble revealed in a second quench Phys. Rev. E 85 041138 · doi:10.1103/PhysRevE.85.041138 |
| [18] | Barthel T and Schollwöck U 2008 Dephasing and the steady state in quantum many-particle systems Phys. Rev. Lett. 100 100601 arXiv:0711.4896 [cond-mat.stat-mech] · doi:10.1103/PhysRevLett.100.100601 |
| [19] | Kollar M and Eckstein M 2008 Relaxation of a one-dimensional Mott insulator after an interaction quench Phys. Rev. A 78 013626 arXiv:0804.2254 [cond-mat.str-el] · doi:10.1103/PhysRevA.78.013626 |
| [20] | Cazalilla M A, Iucci A and Chung M-C 2012 Thermalization and quantum correlations in exactly solvable models Phys. Rev. E 85 011133 arXiv:1106.5206 [cond-mat.stat-mech] · doi:10.1103/PhysRevE.85.011133 |
| [21] | Iucci A and Cazalilla M A 2009 Quantum quench dynamics of the Luttinger model Phys. Rev. A 80 063619 arXiv:0903.1205 [cond-mat.str-el] · doi:10.1103/PhysRevA.80.063619 |
| [22] | Caux J-S and Konik R M 2012 Numerical renormalization based on integrable theories: quantum quenches and their corresponding generalized Gibbs ensembles Phys. Rev. Lett. 109 175301 arXiv:1203.0901 [cond-mat.quant-gas] · doi:10.1103/PhysRevLett.109.175301 |
| [23] | Mossel J and Caux J-S 2012 Generalized TBA and generalized Gibbs J. Phys. A: Math. Gen. 45 255001 arXiv:1203.1305 [cond-mat.quant-gas] · Zbl 1243.82027 · doi:10.1088/1751-8113/45/25/255001 |
| [24] | Yang C N and Yang C P 1969 Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction J. Math. Phys. 10 1115 · Zbl 0987.82503 · doi:10.1063/1.1664947 |
| [25] | Takahashi M 1999 Thermodynamics of One-Dimensional Solvable Models (Cambridge: Cambridge University Press) · Zbl 0998.82503 · doi:10.1017/CBO9780511524332 |
| [26] | Fioretto D and Mussardo G 2009 Quantum quenches in integrable field theories New J. Phys. 12 055015 arxiv:0911.3345 [cond-mat.stat-mech] · Zbl 1375.81169 |
| [27] | Pozsgay B 2011 Mean values of local operators in highly excited Bethe states J. Stat. Mech. P01011 arXiv:1009.4662 [hep-th] · Zbl 1456.82213 |
| [28] | Kormos M, Chou Y-Z and Imambekov A 2011 Exact three-body local correlations for excited states of the 1D Bose gas Phys. Rev. Lett. 107 230405 arXiv:1108.0145 [cond-mat.quant-gas] · doi:10.1103/PhysRevLett.107.230405 |
| [29] | Davies B and Korepin V E 2011 Higher conservation laws for the quantum non-linear Schroedinger equation arXiv:1109.6604 [math-ph] |
| [30] | Grabowski M P and Mathieu P 1994 Quantum integrals of motion for the Heisenberg spin chain Mod. Phys. Lett. A 9 2197 arXiv:hep-th/9403149 [hep-th] · Zbl 1020.82550 · doi:10.1142/S0217732394002057 |
| [31] | Grabowski M and Mathieu P 1995 Structure of the conservation laws in integrable spin chains with short range interactions Ann. Phys. 243 299 arXiv:hep-th/9411045 [hep-th] · Zbl 0834.35105 · doi:10.1006/aphy.1995.1101 |
| [32] | Klauser A and Caux J-S 2011 Equilibrium thermodynamic properties of interacting two-component bosons in one dimension Phys. Rev. A 84 033604 arXiv:1106.4971 [cond-mat.quant-gas] · doi:10.1103/PhysRevA.84.033604 |
| [33] | Klümper A 1993 Thermodynamics of the anisotropic spin-1/2 Heisenberg chain and related quantum chains Z. Phys. B 91 507 · doi:10.1007/BF01316831 |
| [34] | Klümper A 2004 Integrability of quantum chains: theory and applications to the spin-1/2 XXZ chain Quantum Magnetism(Springer Lecture Notes in Physics vol 645) ed U Schollwöck, J Richter, D J J Farnell and R F Bishop (Berlin: Springer) p 349 [arXiv:cond-mat/0502431] · doi:10.1007/BFb0119598 |
| [35] | Takahashi M, Shiroishi M and Klümper A 2001 Equivalence of TBA and QTM J. Phys. A: Math. Gen. 34 L187 · Zbl 1063.82011 |
| [36] | Boos H E, Damerau J, Göhmann F, Klümper A, Suzuki J and Weiße A 2008 Short-distance thermal correlations in the XXZ chain J. Stat. Mech. P08010 arXiv:0806.3953 [cond-mat.str-el] |
| [37] | Trippe C, Göhmann F and Klümper A 2010 Short-distance thermal correlations in the massive XXZ chain Eur. Phys. J. B 73 253 arXiv:0908.2232 [cond-mat.str-el] · doi:10.1140/epjb/e2009-00417-7 |
| [38] | Sato J, Aufgebauer B, Boos H, Göhmann F, Klümper A, Takahashi M and Trippe C 2011 Computation of static Heisenberg-chain correlators: control over length and temperature dependence Phys. Rev. Lett. 106 257201 arXiv:1105.4447 · doi:10.1103/PhysRevLett.106.257201 |
| [39] | Klümper A and Sakai K 2002 The thermal conductivity of the spin-1/2 XXZ chain at arbitrary temperature J. Phys. A: Math. Gen. 35 2173 [arXiv:cond-mat/0112444] · Zbl 1002.82019 · doi:10.1088/0305-4470/35/9/307 |
| [40] | Sakai K and Klümper A 2003 Non-dissipative thermal transport in the massive regimes of the XXZ chain J. Phys. A: Math. Gen. 36 11617 [arXiv:cond-mat/0307227] · Zbl 1039.82028 · doi:10.1088/0305-4470/36/46/006 |
| [41] | Zvyagin A A and Klümper A 2003 Quantum phase transitions and thermodynamics of quantum antiferromagnets with next-nearest-neighbor couplings Phys. Rev. B 68 144426 · doi:10.1103/PhysRevB.68.144426 |
| [42] | Trippe C and Klümper A 2007 Quantum phase transitions and thermodynamics of quantum antiferromagnets with competing interactions Low Temp. Phys. 33 920 arXiv:0709.0229 [cond-mat.str-el] · doi:10.1063/1.2747066 |
| [43] | Barmettler P, Punk M, Gritsev V, Demler E and Altman E 2009 Relaxation of antiferromagnetic order in spin-1/2 chains following a quantum quench Phys. Rev. Lett. 102 130603 arXiv:0810.4845 [cond-mat.other] · doi:10.1103/PhysRevLett.102.130603 |
| [44] | Bethe H 1931 Zur theorie der metalle Z. Phys. A 71 205 · JFM 57.1587.01 · doi:10.1007/BF01341708 |
| [45] | Orbach R 1958 Linear antiferromagnetic chain with anisotropic coupling Phys. Rev. 112 309 · doi:10.1103/PhysRev.112.309 |
| [46] | Walker L R 1959 Antiferromagnetic linear chain Phys. Rev. 116 1089 · doi:10.1103/PhysRev.116.1089 |
| [47] | Yang C N and Yang C P 1966 One-dimensional chain of anisotropic spin – spin interactions. I. Proof of Bethe’s hypothesis for ground state in a finite system Phys. Rev. 150 321 · doi:10.1103/PhysRev.150.321 |
| [48] | Yang C N 1967 Some exact results for the many-body problem in one dimension with repulsive delta-function interaction Phys. Rev. Lett. 19 1312 · Zbl 0152.46301 · doi:10.1103/PhysRevLett.19.1312 |
| [49] | Baxter R J 1982 Exactly Solved Models in Statistical Mechanics (London: Academic) · Zbl 0538.60093 |
| [50] | Lüscher M 1976 Dynamical charges in the quantized renormalized massive Thirring model Nucl. Phys. B 117 475 · doi:10.1016/0550-3213(76)90410-7 |
| [51] | Korepin V, Bogoliubov N and Izergin A 1993 Quantum Inverse Scattering Method and Correlation Functions (Cambridge: Cambridge University Press) · Zbl 0787.47006 · doi:10.1017/CBO9780511628832 |
| [52] | Göhmann F, Klümper A and Seel A 2004 Integral representations for correlation functions of the XXZ chain at finite temperature J. Phys. A: Math. Gen. 37 7625 [arXiv:hep-th/0405089] · Zbl 1075.82004 · doi:10.1088/0305-4470/37/31/001 |
| [53] | Boos H E, Göhmann F, Klümper A and Suzuki J 2007 Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field J. Phys. A: Math. Theor. 40 10699 [arXiv:0705.2716] · Zbl 1147.82316 · doi:10.1088/1751-8113/40/35/001 |
| [54] | Boos H, Jimbo M, Miwa T, Smirnov F and Takeyama Y 2006 Algebraic representation of correlation functions in integrable spin chains Ann. Henri Poincaré 7 1395 [arXiv:hep-th/0601132] · Zbl 1115.82008 · doi:10.1007/s00023-006-0285-5 |
| [55] | Tetelman M G 1982 Lorentz group for two-dimensional integrable lattice systems Sov. Phys.—JETP 55 306 |
| [56] | Thacker H B 1986 Corner transfer matrices and Lorentz invariance on a lattice Physica D 18 348 · Zbl 0596.58050 · doi:10.1016/0167-2789(86)90196-X |
| [57] | Vermaseren J A M 2000 New features of FORM arXiv:math-ph/0010025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
