The condition of quasi-periodicity in imaginary time as a constraint at the functional integration and the time-dependent ZZ-correlator of the XX Heisenberg magnet

K. L. Malyshev

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 18, Tome 317 (2004), pp. 142-173

Résumé

A functional integration approach for calculation of the longitudinal correlation functions of the XY Heisenberg magnet in the constant homogeneous magnetic field is considered in the present paper. The generating functionals of the correlators are defined in the form of the functional integrals over anti-commuting variables. The peculiarity of the functional integrals consists in the fact that the integration variables depend on the imaginary time quasi-periodically. The corresponding boundary conditions are accounted for as constraints which reduce the integration domain. For the XX Heisenberg magnet an application of the approach is given in the case of two-point correlator of third components of spins with an explicit dependence on time.

Bibliographie

[1] K. Malyshev, “Funktsionalnoe integrirovanie s “avtomorfnym” granichnym usloviem i korrelyatory tretikh komponent spinov v $XX$ modeli Geizenberga”, TMF, 136:2 (2003), 285–298 ; arXiv: /hep-th/0204007 | MR | Zbl

[2] C. Malyshev, “Functional integration with “automorphic” boundary conditions and correlators of z-components of spins in the XY and XX Heisenberg chains”, New Developments in Mathematical Physics Research, ed. Charles V. Benton, Nova Science Publishers, New York, 2004, 85–116 ; arXiv: /math-ph/0405009 | MR

[3] V. E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge, 1993 | MR | Zbl

[4] F. Colomo, A. G. Izergin, V. E. Korepin, V. Tognetti, “Temperature correlation functions in the $XXO$ Heisenberg chain, I”, Teor. Mat. Fiz., 94:1 (1993), 19–51 | MR

[5] F. H. L. Eßler, H. Frahm, A. G. Izergin, V. E. Korepin, “Determinant representation for correlation functions of spin-1/2 $XXX$ and $XXZ$ Heisenberg magnets”, Comm. Math. Phys., 174:1 (1995), 191–214 | DOI | MR

[6] A. G. Izergin, N. A. Kitanin, N. A. Slavnov, “O korrelyatsionnykh funktsiyakh $XY$-modeli”, Zap. nauchn. semin. POMI, 224 (1995), 178–191 | MR

[7] A. G. Izergin, V. S. Kapitonov, N. A. Kitanin, “Odnovremennye temperaturnye korrelyatory odnomernoi $XY$-tsepochki Geizenberga”, Zap. nauchn. semin. POMI, 245, 1997, 173–206 ; arXiv: /solv-int/9710028 | MR | Zbl

[8] N. Kitanine, J. M. Maillet, N. Slavnov, V. Terras, “Spin-spin correlation functions of the $XXZ$-$\frac12$ Heisenberg chain in a magnetic field”, Nucl. Phys. B, 641:3 (2002), 487–518 | DOI | MR | Zbl

[9] N. Kitanine, J. M. Maillet, N. Slavnov, V. Terras, “Correlation functions of the $XXZ$ spin-$\frac12$ Heisenberg chain at the free fermion point from their multiple integral representations”, Nucl. Phys. B, 642:3 (2002), 433–455 | DOI | MR | Zbl

[10] E. Lieb, T. Schultz, D. Mattis, “Two soluble models of an antiferromagnetic chain”, Ann. Phys., 16:3 (1961), 407–466 | DOI | MR | Zbl

[11] Th. Niemeijer, “Some exact calculations on a chain of spins 1/2, I, II”, Physica, 36:3 (1967), 377–419 ; 39:3 (1968), 313–326 | DOI | DOI

[12] V. N. Popov, V. S. Yarunin, Collective Effects in Quantum Statistics of Radiation and Matter, Kluwer, Dordrecht, 1988

[13] L. Alvarez-Gaumé, “Supersymmetry and index theory”, Supersymmetry, NATO ASI Ser., 125, eds. K. Dietz et. al., Plenum Press, New York, 1985, 1–44 | MR

[14] V. N. Popov, S. A. Fedotov, “Metod funktsionalnogo integrirovaniya i diagrammnaya tekhnika dlya spinovykh sistem”, Zhurn. Eksp. Teor. Fiz., 94:3 (1988), 183–194

[15] S. Katsura, T. Horiguchi, M. Suzuki, “Dynamical properties of the isotropic XY model”, Physica, 46:1 (1970), 67–86 | DOI | MR

[16] P. Mazur, Th. J. Siskens, “Time correlation functions in the a-cyclic XY model, I”, Physica, 69:1 (1973), 259–272 | DOI

[17] J. H. H. Perk, H. W. Capel, “Time-dependent $xx$-correlation functions in the one dimensional $XY$-model”, Physica A, 89:2 (1977), 265–303 | DOI | MR

[18] G. Müller, R. E. Shrock, “Dynamic correlation functions for one-dimensional quantum spin systems: New results based on a rigorous approach”, Phys. Rev. B, 29:1 (1984), 288–301 | DOI | MR

[19] F. A. Berezin, Metod vtorichnogo kvantovaniya, Nauka, Moskva, 1986 | MR | Zbl

[20] A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, London, 1987 | MR

[21] W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Springer, Berlin, 1966 | MR

[22] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, 1999 | MR

[23] F. Colomo, A. G. Izergin, V. Tognetti, “Correlation functions in the XXO Heisenberg chain and their relations with spectral shapes”, J. Phys. A: Math. Gen., 30:2 (1977), 361–370 | DOI | MR

[24] B.-Q. Jin, V. E. Korepin, “Entanglement, Toeplitz determinants and Fisher–Hartwig conjecture”, J. Stat. Phys., 116:1–4 (2004), 79–95 ; arXiv: /quant-ph/0304108 | DOI | MR | Zbl

[25] B.-Q. Jin, V. E. Korepin, “Correlation functions in spin chains and information theory”, Phys. Rev. A, 69:6 (2004), 062314 ; ; A. R. Its, B.-Q. Jin, V. E. Korepin, “Entanglement in XY spin chain”, J. Phys. A: Math. Gen., 38 (2005), 2975–2990 ; arXiv: /quant-ph/0309188arXiv: /quant-ph/0409027 | DOI | DOI | MR | Zbl