×

zbMATH — the first resource for mathematics

On the thermodynamic limit of form factor expansions of dynamical correlation functions in the massless regime of the XXZ spin 1/2 chain. (English) Zbl 1410.82006
This paper derives the two-point dynamical correlation functions for the spin-1/2 XXZ chain in the massless regime. These correlation functions are expressed in terms of properly regularized series of multiple integrals (see Eq. 2.5 and explanations thereof), which are obtained by properly taking the thermodynamic limit of the finite volume form factor expansions. This allows to identify the contributions from conformal-type excitations and other massive excitations (such as bound states) and opens the possibility to study the (large-distance and long-time) asymptotic behaviors of dynamical correlation functions within the Bethe ansatz framework. With certain assumptions on the structure of the Hilbert space, the results obtained in this paper also shed light on the form factor expansions of other non-integrable models belonging to the Luttinger liquid universality class.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B30 Statistical thermodynamics
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T27 Continuum limits in quantum field theory
82B23 Exactly solvable models; Bethe ansatz
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aizenberg, I. A.; Yuzhakov, A. P., Integral Representations and Residues in Multidimensional Complex Analysis, (1978), American Mathematical Society
[2] Arikawa, M.; Kabrach, M.; Müller, G.; Wiele, K., Spinon excitations in the XX chain: Spectra, transition rates, observability, J. Phys. A: Math. Gen., 39, 10623-10640, (2006) · Zbl 1116.82303
[3] Babujian, H. M.; Foerster, A.; Karowski, M., The form factor program: A review and new results—the nested SU(N) off-shell Bethe Ansatz, SIGMA, 2, 082, (2006) · Zbl 1134.81391
[4] Barouch, E.; McCoy, B. M., Statistical mechanics of XY-model. 2. Spin-correlation functions, Phys. Rev. A, 3, 786-804, (1971)
[5] Beck, H.; Bonner, J. C.; Müller, G.; Thomas, H., Quantum spin dynamics of the antiferromagnetic linear chain in zero and nonzero magnetic field, Phys. Rev. B, 24, 1429-1467, (1981)
[6] Bethe, H., Zür theorie der metalle: Eigenwerte und eigenfunktionen der linearen atomkette, Z. Phys., 71, 205-226, (1931) · JFM 57.1587.01
[7] Bogoliubov, N. M.; Izergin, A. G.; Korepin, V. E., Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz, (1993), Cambridge University Press · Zbl 0787.47006
[8] Cardy, J. L., Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B, 270, 186-204, (1986) · Zbl 0689.17016
[9] Colomo, F.; Izergin, A. G.; Korepin, V. E.; Tognetti, V., Temperature correlation functions in the XX0 Heisenberg chain, Theor. Math. Phys., 94, 11-38, (1993)
[10] de Vega, H. J.; Woynarowich, F., Method for calculating finite size corrections in Bethe Ansatz systems—Heisenberg chains and 6-vertex model, Nucl. Phys. B, 251, 439-456, (1985)
[11] Delfino, G.; Mussardo, G.; Simonetti, P., Correlation functions along a massless flow, Phys. Rev. D, 51, R6620-R6624, (1995)
[12] des Cloizeaux, J.; Gaudin, M., Anisotropic linear magnetic chain, J. Math. Phys., 7, 1384-1400, (1966)
[13] des Cloizeaux, J.; Pearson, J. J., Spin-wave spectrum of the antiferromagnetic linear chain, Phys. Rev., 128, 2131-2135, (1962)
[14] Destri, C.; Lowenstein, J. H., Analysis of the Bethe Ansatz equations of the chiral invariant Gross-Neveu model, Nucl. Phys. B, 205, 369-385, (1982)
[15] Dugave, M.; Göhmann, F.; Kozlowski, K. K., Functions characterizing the ground state of the XXZ spin-1/2 chain in the thermodynamic limit, SIGMA, 10, 043, (2014) · Zbl 1291.82021
[16] Faddeev, L. D.; Takhtadzhan, L. A., What is the spin of a spin wave?, Phys. Lett. A, 85, 375-377, (1981)
[17] Fowler, M.; Zotos, X., Quantum sine-Gordon thermodynamics: The Bethe Ansatz method, Phys. Rev. B, 24, 2634-2639, (1981)
[18] Göhmann, F.; Klümper, A.; Seel, A., Integral representations for correlation functions of the XXZ chain at finite temperature, J. Phys. A: Math. Gen., 37, 7625-7652, (2004) · Zbl 1075.82004
[19] Hida, K., Rigorous derivation of the distribution of the eigenstates of the quantum Heisenberg-Ising chain with XY-like anisotropy, Phys. Lett. A, 84, 338-340, (1981)
[20] Ishimura, N.; Shiba, H., Effect of the magnetic field on the des Cloizeaux-Pearson spin wave spectrum, Prog. Theor. Phys., 57, 1862-1873, (1977) · Zbl 1098.82522
[21] Its, A. R.; Izergin, A. G.; Korepin, V. E.; Slavnov, N. A., Differential equations for quantum correlation functions, Int. J. Mod. Phys. B, 4, 1003-1037, (1990) · Zbl 0719.35091
[22] Its, A. R.; Slavnov, N. A., On the Riemann-Hilbert approach to the asymptotic analysis of the correlation functions of the quantum nonlinear schrödinger equation. Non-free fermion case, Theor. Math. Phys., 119, 2, 541-593, (1990) · Zbl 0991.81016
[23] Jimbo, M.; Miwa, T., QKZ equation with ∣q∣ = 1 and correlation functions of the XXZ model in the gapless regime, J. Phys. A, 29, 2923-2958, (1996) · Zbl 0896.35114
[24] Jimbo, M.; Miwa, T.; Mori, Y.; Sato, M., Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica D, 1, 80-158, (1980) · Zbl 1194.82007
[25] Karowski, M.; Weisz, P., Exact form factors in (1 + 1)-dimensional field theoretic models with soliton behaviour, Nucl. Phys. B, 139, 455-476, (1978)
[26] Kaufman, B.; Onsager, L., Crystal statistics. III. Short-range order in a binary Ising lattice, Phys. Rev., 76, 1244-1252, (1949) · Zbl 0035.42802
[27] Kerov, S.; Olshanski, G.; Vershik, A., Harmonic analysis on the infinite symmetric group. A deformation of the regular representation, C. R. Acad. Sci. Paris, Sér I, 316, 773-778, (1993) · Zbl 0796.43005
[28] Kirillov, A. N.; Smirnov, F. A., A representation of the current algebra connected with the SU (2)-invariant thirring model, Phys. Lett. B, 198, 506-510, (1987)
[29] Kitanine, N.; Kozlowski, K. K.; Maillet, J.-M.; Slavnov, N. A.; Terras, V., Algebraic Bethe Ansatz approach to the asymptotics behavior of correlation functions, J. Stat. Mech.: Theory Exp., 2009, 4, P04003 · Zbl 1189.45018
[30] Kitanine, N.; Kozlowski, K. K.; Maillet, J.-M.; Slavnov, N. A.; Terras, V., On the thermodynamic limit of form factors in the massless XXZ Heisenberg chain, J. Math. Phys., 50, 095209, (2009) · Zbl 1241.82029
[31] Kitanine, N.; Kozlowski, K. K.; Maillet, J.-M.; Slavnov, N. A.; Terras, V., A form factor approach to the asymptotic behavior of correlation functions in critical models, J. Stat. Mech.: Theory Exp., 2011, P12010 · Zbl 1241.82029
[32] Kitanine, N.; Kozlowski, K. K.; Maillet, J.-M.; Slavnov, N. A.; Terras, V., Thermodynamic limit of particle-hole form factors in the massless XXZ Heisenberg chain, J. Stat. Mech.: Theory Exp., 2011, P05028 · Zbl 1241.82029
[33] Kitanine, N.; Kozlowski, K. K.; Maillet, J.-M.; Slavnov, N. A.; Terras, V., Form factor approach to dynamical correlation functions in critical models, J. Stat. Mech.: Theory Exp., 2012, P09001 · Zbl 1189.45018
[34] Kitanine, N.; Kozlowski, K. K.; Maillet, J.-M.; Terras, V., Long-distance asymptotic behaviour of multi-point correlation functions in massless quantum integrable models, J. Stat. Mech.: Theory Exp., 2014, P05011
[35] Kitanine, N.; Maillet, J.-M.; Slavnov, N. A.; Terras, V., Dynamical correlation functions of the XXZ spin-1/2 chain, Nucl. Phys. B, 729, 558-580, (2005) · Zbl 1138.82309
[36] Kitanine, N.; Maillet, J.-M.; Terras, V., Form factors of the XXZ Heisenberg spin-1/2 finite chain, Nucl. Phys. B, 554, 647-678, (1999) · Zbl 0972.82014
[37] Kitanine, N.; Maillet, J.-M.; Terras, V., Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field, Nucl. Phys. B, 567, 554-582, (2000) · Zbl 0955.82010
[38] Klümper, A.; Batchelor, M. T., An analytic treatment of finite-size corrections of the spin-1 antiferromagnetic XXZ chain, J. Phys. A: Math. Gen., 23, L189, (1990) · Zbl 0716.60125
[39] Kojima, T.; Korepin, V. E.; Slavnov, N. A., Determinant representation for dynamical correlation functions of the quantum nonlinear Schrödinger equation, Commun. Math. Phys., 188, 657-689, (1997) · Zbl 0886.35138
[40] Korepin, V. E., Direct calculation of the S-matrix in the massive thirring model, Theor. Math. Phys., 41, 953-967, (1979)
[41] Korepin, V. E.; Slavnov, N. A., The time dependent correlation function of an impenetrable Bose gas as a Fredholm minor I., Commun. Math. Phys., 129, 103-113, (1990) · Zbl 0698.60095
[42] Korepin, V. E.; Slavnov, N. A., The new identity for the scattering matrix of exactly solvable models, Eur. Phys. J. B, 5, 555-557, (1998)
[43] Kozlowski, K. K., Riemann-Hilbert approach to the time-dependent generalized sine kernel, Adv. Theor. Math. Phys., 15, 1655-1743, (2011) · Zbl 1271.82015
[44] Kozlowski, K. K., Large-distance and long-time asymptotic behavior of the reduced denisty matrix in the non-linear Schrödinger model, Ann. Henri Poincare, 16, 437-534, (2015) · Zbl 1315.81037
[45] Kozlowski, K. K., Form factors of bound states in the XXZ chain, J. Phys. A: Math. Theor., 50, 184002, (2017) · Zbl 1373.82021
[46] Kozlowski, K. K., On condensation properties of Bethe roots associated with the XXZ chain, Commun. Math. Phys., 357, 3, 1009-1069, (2018) · Zbl 1391.82012
[47] Kozlowski, K. K., “On string solutions to the Bethe equations for the XXZ chain: A rigorous approach” (to be published).
[48] Kozlowski, K. K.; Maillet, J.-M.; Slavnov, N. A., Low-temperature limit of the long-distance asymptotics in the non-linear Schrödinger model, J. Stat. Mech.: Theory Exp., 2011, P03019
[49] Kozlowski, K. K.; Maillet, J. M., Microscopic approach to a class of 1D quantum critical models, J. Phys. A: Math and Theor., 48, 484004, (2015) · Zbl 1331.81259
[50] Kozlowski, K. K.; Ragoucy, E., Asymptotic behaviour of two-point functions in multi-species models, Nucl. Phys. B, 906, 241, (2016) · Zbl 1334.82014
[51] Kozlowski, K. K.; Terras, V., Long-time and large-distance asymptotic behavior of the current-current correlators in the non-linear Schrödinger model, J. Stat. Mech.: Theory Exp., 2011, P09013
[52] Lesage, F.; Saleur, H., Form-factors computation of Friedel oscillations in Luttinger liquids, J. Phys. A: Math. Gen., 30, L457-L463, (1997) · Zbl 0934.82003
[53] Lesage, F.; Saleur, H.; Skorik, S., Form factors approach to current correlations in one-dimensional systems with impurities, Nucl. Phys. B, 474, 602-640, (1996) · Zbl 0925.81444
[54] Lieb, E. H.; Mattis, D. C.; Schultz, T. D., Two dimensionnal Ising model as a soluble problem of many fermions, Rev. Mod. Phys., 36, 856-871, (1964)
[55] McCoy, B. M., Spin correlation functions in the XY model, Phys. Rev., 173, 531-541, (1968)
[56] McCoy, B. M.; Perk, J. H. H.; Shrock, R. E., Time-dependent correlation functions of the transverse Ising chain at the critical magnetic field, Nucl. Phys. B, 220, 35-47, (1983)
[57] Mejean, P.; Smirnov, F. A., Form factors for principal chiral field model with wess-zumino-novikov-witten term, Int. J. Mod. Phys. A, 12, 3383-3395, (1997) · Zbl 1161.81417
[58] Müller, G.; Shrock, R. E., Dynamic correlation functions for one-dimensional quantum-spin systems: New results based on a rigorous approach, Phys. Rev. B, 29, 288-301, (1984)
[59] Olshanski, G.; Duval, C.; Guieu, L.; Ovsienko, V., Point processes and the infinite symmetric group. Part I: The general formalism and the density function, The Orbit Method in Geometry and Physics: In Honor of A. A. Kirillov, (2003), Birkhäuser, Verlag: Birkhäuser, Verlag, Basel
[60] Orbach, R., Linear antiferromagnetic chain with anisotropic coupling, Phys. Rev., 112, 309-316, (1958)
[61] Perk, J. H. H.; Au-Yang, H., New results for time-dependent correlation functions in the transverse ising chain, J. Stat. Phys., 135, 599-619, (2009) · Zbl 1178.82048
[62] Ponsot, B., Massless N = 1 super-sinh-Gordon: Form factors approach, Phys. Lett., 575, 131-136, (2003) · Zbl 1094.81520
[63] Sakai, K., Dynamical correlation functions of the XXZ model at finite temperature, J. Phys. A: Math. Theor., 40, 7523-7542, (2007) · Zbl 1125.82008
[64] Slavnov, N. A., Non-equal time current correlation function in a one-dimensional Bose gas, Theor. Math. Phys., 82, 273-282, (1990)
[65] Slavnov, N. A., Differential equations for multipoint correlation functions in a one-dimensional impenetrable Bose gas, Theor. Math. Phys., 106, 131-142, (1996) · Zbl 0890.58026
[66] Slavnov, N. A., Integral equations for the correlation functions of the quantum one-dimensional Bose gas, Theor. Math. Phys., 121, 1358-1376, (1999) · Zbl 1005.82503
[67] Smirnov, F. A., Reductions of the sine-Gordon model as a perturbation of minimal models of conformal field theory, Nucl. Phys. B, 337, 156-180, (1990)
[68] Smirnov, F. A., Form factors in completely integrable models of quantum field theory, Advanced Series in Mathematical Physics, (1992), World Scientific · Zbl 0788.46077
[69] Takahashi, M., Thermodynamics of One Dimensional Solvable Models, (1999), Cambridge university press · Zbl 0998.82503
[70] Takahashi, M.; Suzuki, M., One-dimensional anisotropic Heisenberg model at finite temperatures, Prog. Theor. Phys., 48, 2187-2209, (1972)
[71] Tracy, C. A.; Vaidya, H. G., Transverse time-dependent spin correlation functions of the one-dimensional XY model at zero temperature, Physica A, 92, 1-41, (1978)
[72] Wu, T. T.; McCoy, B. M.; Tracy, C. A.; Barouch, E., The spin-spin correlation function of the two dimensional ising model: Exact results in the scaling region, Phys. Rev. B, 13, 316, (1976)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.