Non-Hermitian spin chains with inhomogeneous coupling
HTML articles powered by AMS MathViewer
- by
A. G. Bytsko
Translated by: the author - St. Petersburg Math. J. 22 (2011), 393-410
- DOI: https://doi.org/10.1090/S1061-0022-2011-01148-3
- Published electronically: March 17, 2011
- PDF | Request permission
Abstract:
An open $U_q(sl_2)$-invariant spin chain of spin $S$ and length $N$ with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter $\gamma$ are determined for which the spectrum of the model is real. For a certain range of $\gamma$, a universal metric operator is constructed, and thus, the quasi-Hermitian nature of the model is established. This universal metric operator is nondynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to all known homogeneous $U_q(sl_2)$-invariant integrable spin chains with nearest-neighbor interaction. In addition, the most general form of a metric operator for a quasi-Hermitian operator in finite-dimensional spaces is discussed.References
- Paulo E. G. Assis and Andreas Fring, Metrics and isospectral partners for the most generic cubic $\scr {PT}$-symmetric non-Hermitian Hamiltonian, J. Phys. A 41 (2008), no. 24, 244001, 18. MR 2455799, DOI 10.1088/1751-8113/41/24/244001
- Murray T. Batchelor and Michael N. Barber, Spin-$s$ quantum chains and Temperley-Lieb algebras, J. Phys. A 23 (1990), no. 1, L15–L21. MR 1034619
- Carl M. Bender and Stefan Boettcher, Real spectra in non-Hermitian Hamiltonians having $\scr {PT}$ symmetry, Phys. Rev. Lett. 80 (1998), no. 24, 5243–5246. MR 1627442, DOI 10.1103/PhysRevLett.80.5243
- Carl M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Progr. Phys. 70 (2007), no. 6, 947–1018. MR 2331294, DOI 10.1088/0034-4885/70/6/R03
- Carl M. Bender, Dorje C. Brody, and Hugh F. Jones, Complex extension of quantum mechanics, Phys. Rev. Lett. 89 (2002), no. 27, 270401, 4. MR 1950305, DOI 10.1103/PhysRevLett.89.270401
- D. Bessis and J. Zinn-Justin, 1993 (unpublished).
- Andrei G. Bytsko, On integrable Hamiltonians for higher spin $XXZ$ chain, J. Math. Phys. 44 (2003), no. 9, 3698–3717. MR 2003927, DOI 10.1063/1.1591054
- A. G. Bytsko, On $U_q(\textrm {sl}_2)$-invariant $R$-matrices for higher spins, Algebra i Analiz 17 (2005), no. 3, 24–46 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 17 (2006), no. 3, 393–408. MR 2167842, DOI 10.1090/S1061-0022-06-00910-1
- J. L. Cardy and R. L. Sugar, Reggeon field theory on a lattice. 1, Phys. Rev. D 12 (1975), 2514–2522.
- Olalla A. Castro-Alvaredo and Andreas Fring, A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field, J. Phys. A 42 (2009), no. 46, 465211, 29. MR 2552019, DOI 10.1088/1751-8113/42/46/465211
- J. Dieudonné, Quasi-hermitian operators, Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960) Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 115–122. MR 0187086
- V. G. Drinfel′d, Quantum groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 155 (1986), no. Differentsial′naya Geometriya, Gruppy Li i Mekh. VIII, 18–49, 193 (Russian, with English summary); English transl., J. Soviet Math. 41 (1988), no. 2, 898–915. MR 869575, DOI 10.1007/BF01247086
- V. G. Drinfel′d, Almost cocommutative Hopf algebras, Algebra i Analiz 1 (1989), no. 2, 30–46 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 2, 321–342. MR 1025154
- G. von Gehlen, Critical and off critical conformal analysis of the Ising quantum chain in an imaginary field, J. Phys. A 24 (1991), 5371–5400.
- Harro Heuser, Über Eigenwerte und Eigenlösungen symmetrisierbarer finiter Operatoren, Arch. Math. 10 (1959), 12–20 (German). MR 102020, DOI 10.1007/BF01240752
- Timothy Hollowood, Solitons in affine Toda field theories, Nuclear Phys. B 384 (1992), no. 3, 523–540. MR 1188363, DOI 10.1016/0550-3213(92)90579-Z
- A. G. Izergin and V. E. Korepin, The inverse scattering method approach to the quantum Shabat-Mikhaĭlov model, Comm. Math. Phys. 79 (1981), no. 3, 303–316. MR 627054
- A. N. Kirillov and N. Yu. Reshetikhin, Representations of the algebra ${U}_q(\textrm {sl}(2)),\;q$-orthogonal polynomials and invariants of links, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988) Adv. Ser. Math. Phys., vol. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 285–339. MR 1026957
- Christian Korff and Robert Weston, PT symmetry on the lattice: the quantum group invariant $XXZ$ spin chain, J. Phys. A 40 (2007), no. 30, 8845–8872. MR 2344527, DOI 10.1088/1751-8113/40/30/016
- P. P. Kulish, On spin systems related to the Temperley-Lieb algebra, J. Phys. A 36 (2003), no. 38, L489–L493. MR 2006441, DOI 10.1088/0305-4470/36/38/101
- P. P. Kulish and A. A. Stolin, Deformed Yangians and integrable models, Czechoslovak J. Phys. 47 (1997), no. 12, 1207–1212. Quantum groups and integrable systems, II (Prague, 1997). MR 1608809, DOI 10.1023/A:1022869414679
- Ali Mostafazadeh, Pseudo-Hermiticity versus $PT$-symmetry. III. Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries, J. Math. Phys. 43 (2002), no. 8, 3944–3951. MR 1915636, DOI 10.1063/1.1489072
- —, Pseudo-Hermitian quantum mechanics, arXiv:0810.5643 [hep-th].
- William T. Reid, Symmetrizable completely continuous linear transformations in Hilbert space, Duke Math. J. 18 (1951), 41–56. MR 45314
- F. G. Scholtz, H. B. Geyer, and F. J. W. Hahne, Quasi-Hermitian operators in quantum mechanics and the variational principle, Ann. Physics 213 (1992), no. 1, 74–101. MR 1144600, DOI 10.1016/0003-4916(92)90284-S
- J. P. O. Silberstein, Symmetrisable operators, J. Austral. Math. Soc. 2 (1961/1962), 381–402. MR 0149299
- J. P. O. Silberstein, Symmetrisable operators. II. Operators in a Hilbert space ${\mathfrak {H}}$, J. Austral. Math. Soc. 4 (1964), 15–30. MR 0162138
- A. N. Varchenko and V. O. Tarasov, Jackson integral representations for solutions of the Knizhnik-Zamolodchikov quantum equation, Algebra i Analiz 6 (1994), no. 2, 90–137 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 6 (1995), no. 2, 275–313. MR 1290820
- Eugene P. Wigner, Normal form of antiunitary operators, J. Mathematical Phys. 1 (1960), 409–413. MR 117557, DOI 10.1063/1.1703672
- A. C. Zaanen, Ueber vollstetige symmetrische und symmetrisierbare Operatoren, Nieuw Arch. Wiskunde (2) 22 (1943), 57–80 (German). MR 0015661
- Miloslav Znojil and Hendrik B. Geyer, Construction of a unique metric in quasi-Hermitian quantum mechanics: nonexistence of the charge operator in a $2\times 2$ matrix model, Phys. Lett. B 640 (2006), no. 1-2, 52–56. MR 2245629, DOI 10.1016/j.physletb.2006.07.028
Bibliographic Information
- A. G. Bytsko
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, 27 Fontanka, St. Petersburg 191023, Russia
- Email: bytsko@pdmi.ras.ru
- Received by editor(s): December 18, 2009
- Published electronically: March 17, 2011
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 393-410
- MSC (2010): Primary 81T10
- DOI: https://doi.org/10.1090/S1061-0022-2011-01148-3
- MathSciNet review: 2729941
Dedicated: To Ludwig Dmitrievich Faddeev on his 75th birthday