This work constructs a well-defined and operational form factor expansion in a model having a massless spectrum of excitations. More precisely, the dynamic two-point functions in the massless regime of the XXZ spin-1/2 chain are expressed in terms of a properly regularised series of multiple integrals. These series are obtained by taking, in an appropriate way, the thermodynamic limit of the finite volume form factor expansions. The series are structured in a way allowing one to identify directly the contributions to the correlator stemming from the conformal-type excitations on the Fermi surface and those issuing from the massive excitations (deep holes, particles, and bound states). The obtained form factor series opens up the possibility of a systematic and exact study of asymptotic regimes of dynamical correlation functions in the massless regime of the XXZ spin 1/2 chain. Furthermore, the assumptions on the microscopic structure of the model’s Hilbert space that are necessary so as to write down the series appear to be compatible with any model—not necessarily integrable—belonging to the Luttinger liquid universality class. Thus, the present analysis also provides the phenomenological structure of form factor expansions in massless models belonging to this universality class.
REFERENCES
1.
Aizenberg
, I. A.
and Yuzhakov
, A. P.
, Integral Representations and Residues in Multidimensional Complex Analysis
, Volume 58 of Graduate Texts in Mathematics (American Mathematical Society
, 1978
).2.
Arikawa
, M.
, Kabrach
, M.
, Müller
, G.
, and Wiele
, K.
, “Spinon excitations in the XX chain: Spectra, transition rates, observability
,” J. Phys. A: Math. Gen.
39
, 10623
–10640
(2006
).3.
Babujian
, H. M.
, Foerster
, A.
, and Karowski
, M.
, “The form factor program: A review and new results—the nested SU(N) off-shell Bethe Ansatz
,” in Proceedings of the O’Raifeartaigh Symposium on Non-Perturbative and Symmetry Methods in Field Theory, Budapest, Hungary, 2006
[SIGMA
2
, 082
(2006
)].4.
Barouch
, E.
and 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.
, and 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
).7.
Bogoliubov
, N. M.
, Izergin
, A. G.
, and Korepin
, V. E.
, Quantum Inverse Scattering Method, Correlation Functions and Algebraic Bethe Ansatz
, Cambridge Monographs on Mathematical Physics (Cambridge University Press
, 1993
).8.
Cardy
, J. L.
, “Operator content of two-dimensional conformally invariant theories
,” Nucl. Phys. B
270
, 186
–204
(1986
).9.
Colomo
, F.
, Izergin
, A. G.
, Korepin
, V. E.
, and Tognetti
, V.
, “Temperature correlation functions in the XX0 Heisenberg chain
,” Theor. Math. Phys.
94
, 11
–38
(1993
).10.
de Vega
, H. J.
and 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.
, and Simonetti
, P.
, “Correlation functions along a massless flow
,” Phys. Rev. D
51
, R6620
–R6624
(1995
).12.
des Cloizeaux
, J.
and Gaudin
, M.
, “Anisotropic linear magnetic chain
,” J. Math. Phys.
7
, 1384
–1400
(1966
).13.
J.
des Cloizeaux
and Pearson
, J. J.
, “Spin-wave spectrum of the antiferromagnetic linear chain
,” Phys. Rev.
128
, 2131
–2135
(1962
).14.
Destri
, C.
and 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.
, and Kozlowski
, K. K.
, “Functions characterizing the ground state of the XXZ spin-1/2 chain in the thermodynamic limit
,” SIGMA
10
, 043
(2014
).16.
Faddeev
, L. D.
and Takhtadzhan
, L. A.
, “What is the spin of a spin wave?
,” Phys. Lett. A
85
, 375
–377
(1981
).17.
Fowler
, M.
and Zotos
, X.
, “Quantum sine-Gordon thermodynamics: The Bethe Ansatz method
,” Phys. Rev. B
24
, 2634
–2639
(1981
).18.
Göhmann
, F.
, Klümper
, A.
, and Seel
, A.
, “Integral representations for correlation functions of the XXZ chain at finite temperature
,” J. Phys. A: Math. Gen.
37
, 7625
–7652
(2004
).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.
and Shiba
, H.
, “Effect of the magnetic field on the des Cloizeaux-Pearson spin wave spectrum
,” Prog. Theor. Phys.
57
, 1862
–1873
(1977
).21.
Its
, A. R.
, Izergin
, A. G.
, Korepin
, V. E.
, and Slavnov
, N. A.
, “Differential equations for quantum correlation functions
,” Int. J. Mod. Phys. B
4
, 1003
–1037
(1990
).22.
Its
, A. R.
and 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
).23.
Jimbo
, M.
and 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
).24.
Jimbo
, M.
, Miwa
, T.
, Mori
, Y.
, and Sato
, M.
, “Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent
,” Physica D
1
, 80
–158
(1980
).25.
Karowski
, M.
and 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.
and Onsager
, L.
, “Crystal statistics. III. Short-range order in a binary Ising lattice
,” Phys. Rev.
76
, 1244
–1252
(1949
).27.
Kerov
, S.
, Olshanski
, G.
, and 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
).28.
Kirillov
, A. N.
and 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.
, and Terras
, V.
, “Algebraic Bethe Ansatz approach to the asymptotics behavior of correlation functions
,” J. Stat. Mech.: Theory Exp.
2009
(04
), P04003
.30.
Kitanine
, N.
, Kozlowski
, K. K.
, Maillet
, J.-M.
, Slavnov
, N. A.
, and Terras
, V.
, “On the thermodynamic limit of form factors in the massless XXZ Heisenberg chain
,” J. Math. Phys.
50
, 095209
(2009
).31.
Kitanine
, N.
, Kozlowski
, K. K.
, Maillet
, J.-M.
, Slavnov
, N. A.
, and Terras
, V.
, “A form factor approach to the asymptotic behavior of correlation functions in critical models
,” J. Stat. Mech.: Theory Exp.
2011
, P12010
.32.
Kitanine
, N.
, Kozlowski
, K. K.
, Maillet
, J.-M.
, Slavnov
, N. A.
, and Terras
, V.
, “Thermodynamic limit of particle-hole form factors in the massless XXZ Heisenberg chain
,” J. Stat. Mech.: Theory Exp.
2011
, P05028
.33.
Kitanine
, N.
, Kozlowski
, K. K.
, Maillet
, J.-M.
, Slavnov
, N. A.
, and Terras
, V.
, “Form factor approach to dynamical correlation functions in critical models
,” J. Stat. Mech.: Theory Exp.
2012
, P09001
.34.
Kitanine
, N.
, Kozlowski
, K. K.
, Maillet
, J.-M.
, and 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.
, and Terras
, V.
, “Dynamical correlation functions of the XXZ spin-1/2 chain
,” Nucl. Phys. B
729
, 558
–580
(2005
).36.
Kitanine
, N.
, Maillet
, J.-M.
, and Terras
, V.
, “Form factors of the XXZ Heisenberg spin-1/2 finite chain
,” Nucl. Phys. B
554
, 647
–678
(1999
).37.
Kitanine
, N.
, Maillet
, J.-M.
, and Terras
, V.
, “Correlation functions of the XXZ Heisenberg spin-1/2 chain in a magnetic field
,” Nucl. Phys. B
567
, 554
–582
(2000
).38.
Klümper
, A.
and 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
).39.
Kojima
, T.
, Korepin
, V. E.
, and Slavnov
, N. A.
, “Determinant representation for dynamical correlation functions of the quantum nonlinear Schrödinger equation
,” Commun. Math. Phys.
188
, 657
–689
(1997
).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.
and 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
).42.
Korepin
, V. E.
and 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
).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
).45.
Kozlowski
, K. K.
, “Form factors of bound states in the XXZ chain
,” J. Phys. A: Math. Theor.
50
, 184002
(2017
). Special Issue “Emerging talents.”46.
Kozlowski
, K. K.
, “On condensation properties of Bethe roots associated with the XXZ chain
,” Commun. Math. Phys.
357
(3
), 1009
–1069
(2018
).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.
, and 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.
and Maillet
, J. M.
, “Microscopic approach to a class of 1D quantum critical models
,” J. Phys. A: Math and Theor.
48
, 484004
(2015
). Baxter anniversary special issue.50.
Kozlowski
, K. K.
and Ragoucy
, E.
, “Asymptotic behaviour of two-point functions in multi-species models
,” Nucl. Phys. B
906
, 241
(2016
).51.
Kozlowski
, K. K.
and 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.
and Saleur
, H.
, “Form-factors computation of Friedel oscillations in Luttinger liquids
,” J. Phys. A: Math. Gen.
30
, L457
–L463
(1997
).53.
Lesage
, F.
, Saleur
, H.
, and Skorik
, S.
, “Form factors approach to current correlations in one-dimensional systems with impurities
,” Nucl. Phys. B
474
, 602
–640
(1996
).54.
Lieb
, E. H.
, Mattis
, D. C.
, and 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.
, and 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.
and 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
).58.
Müller
, G.
and 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.
, “Point processes and the infinite symmetric group. Part I: The general formalism and the density function
,” in The Orbit Method in Geometry and Physics: In Honor of A. A. Kirillov
, Volume 213 of Progress in Mathematics, edited by Duval
, C.
, Guieu
, L.
, and Ovsienko
, V.
(Birkhäuser, Verlag
, Basel
, 2003
).60.
Orbach
, R.
, “Linear antiferromagnetic chain with anisotropic coupling
,” Phys. Rev.
112
, 309
–316
(1958
).61.
Perk
, J. H. H.
and Au-Yang
, H.
, “New results for time-dependent correlation functions in the transverse ising chain
,” J. Stat. Phys.
135
, 599
–619
(2009
).62.
Ponsot
, B.
, “Massless N = 1 super-sinh-Gordon: Form factors approach
,” Phys. Lett.
575
, 131
–136
(2003
).63.
Sakai
, K.
, “Dynamical correlation functions of the XXZ model at finite temperature
,” J. Phys. A: Math. Theor.
40
, 7523
–7542
(2007
).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
).66.
Slavnov
, N. A.
, “Integral equations for the correlation functions of the quantum one-dimensional Bose gas
,” Theor. Math. Phys.
121
, 1358
–1376
(1999
).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
, (World Scientific
, 1992
), Vol. 14.69.
Takahashi
, M.
, Thermodynamics of One Dimensional Solvable Models
(Cambridge university press
, 1999
).70.
Takahashi
, M.
and Suzuki
, M.
, “One-dimensional anisotropic Heisenberg model at finite temperatures
,” Prog. Theor. Phys.
48
, 2187
–2209
(1972
).71.
Tracy
, C. A.
and 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.
, and 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
).© 2018 Author(s).
2018
Author(s)
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