Finitely correlated states on quantum spin chains

Springer Science and Business Media LLC - Tập 144 - Trang 443-490 - 1992
M. Fannes1,2, B. Nachtergaele3,4, R. F. Werner5
1Inst. Theor. Fysica, Universiteit Leuven, Leuven, Belgium
2Bevoegdverklaard Navorser N.F.W.O., Belgium
3Depto de Física, Universidad de Chile, Santiago de Chile
4Onderzoeker I.I.K.W., Belgium
5Dublin Institute for Advanced Studies, Dublin 4, Ireland

Tóm tắt

We study a construction that yields a class of translation invariant states on quantum spin chains, characterized by the property that the correlations across any bond can be modeled on a finite-dimensional vector space. These states can be considered as generalized valence bond states, and they are dense in the set of all translation invariant states. We develop a complete theory of the ergodic decomposition of such states, including the decomposition into periodic “Néel ordered” states. The ergodic components have exponential decay of correlations. All states considered can be obtained as “local functions” of states of a special kind, so-called “purely generated states,” which are shown to be ground states for suitably chosen finite range VBS interactions. We show that all these generalized VBS models have a spectral gap. Our theory does not require symmetry of the state with respect to a local gauge group. In particular we illustrate our results with a one-parameter family of examples which are not isotropic except for one special case. This isotropic model coincides with the one-dimensional antiferromagnet, recently studied by Affleck, Kennedy, Lieb, and Tasaki.

Tài liệu tham khảo

Accardi, L.: Topics in quantum probability. Phys. Rep.77, 169–192 (1981)

Accardi, L., Frigerio, A.: Markovian Cocycles. Proc. R. Ir. Acad.83A(2), 251–263 (1983)

Affleck, I.: Large-n limit ofSU(n) quantum “spin” chains. Phys. Rev. Lett.54, 966–969 (1985)

Affleck, I., Lieb, E. H.: A proof of part of Haldane's conjecture on quantum spin chains. Lett. Math. Phys.12, 57–69 (1986)

Affleck, I., Kennedy, T., Lieb, E. H., Tasaki, H.: Valence bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys.115, 477–528 (1988)

Affleck, I., Lieb, E. H., Kennedy, T., Tasaki, H.: Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett.59, 799–802 (1987)

Anderson, P.: The resonating valence bond state in La2CuO4 and superconductivity. Science235, 1196–1198 (1987)

Araki, H.: Gibbs states of a one-dimensional quantum lattice. Commun. Math. Phys.14, 120–157 (1969)

Arovas, D. P., Auerbach, A., haldane, F. D. M.: Extended heisenberg models of antiferromagnetism: Analogies to the fractional quantum Hall effect. Phys. Rev. Lett.60, 531–534 (1988)

Arveson, W.: Subalgebras ofC *-algebras. Acta Math.123, 141–224 (1969)

Babujian, H. M.: Exact solution of the one-dimensional isotropic heisenberg chain with arbitrary spins. Phys. Lett.90A, 479–482 (1982)

Batchelor, M. T., Barber, M. N.: Spin-s quantum chains and Temperley-Lieb algebras. J. Phys.A23, L15-L21 (1990)

Batchelor, M. T., Mezincescu, L., Nepmechie, R., Rittenberg, V.:q-deformations of theO(3)symmetric spin 1 chain. J. Phys.A23, L141-L144 (1990)

Bethe, H.: Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys.71, 205–226 (1931)

Blackwell, D.: The entropy of functions of finite Markov chains. Transactions of the first Prague conference on information theory, statistical decision functions and random processes. Publishing House of the Czechoslovak Academy of Sciences, Prague 1957

Bratteli, O., Robinson, D. W.: Operator algebras and quantum statistical mechanics, 2 vols. Berlin, Heidelberg, New York: Springer 1979 and 1981

van den Broeck, P. M.: Exact value of the ground state energy of the linear antiferromagnetic Heisenberg chain with nearest and next-nearest neighbor interactions. Phys. Lett.77A, 261–262 (1980)

Chang, K., Affleck, I., Hayden, G. W., Soos, Z. G.: A study of the bilinear-biquadratic spin 1 antiferromagnetic chain using the valence-bond basis. J. Phys.C1, 153–167 (1989)

Caspers, W. J.: Exact ground states for a class of linear antiferromagnetic spin systems. Physica115A, 275–280 (1982)

Caspers, W. J., Magnus, W.: Exact ground states for a class of linear quantum spin systems. Physica119A, 291–294 (1983)

Caspers, W. J., Magnus, W.: Some exact excited states in a linear antiferromagnetic spin system. Phys. Lett.88A, 103–105 (1982)

Chayes, J., Chayes, L., Kivelson, S.: Valence bond ground states in a frustrated two-dimensional spin 1/2 Heisenberg antiferromagnet. Commun. Math. Phys.123, 53–83 (1989)

Choi, M. D.: A Schwarz inequality for positive linear maps onC *-algebras. Illinois J. Math.18, 565–574 (1974)

Choi, M. D., Effros, E.: Injectivity and operator spaces. J. Funct. Anal.24, 156–209 (1977)

Dyson, F. J., Lieb, E. H., Simon, B.: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys.18, 335–383 (1978)

Edmonds, A. R.: Angular momentum in quantum mechanics. Princeton, NJ: Princeton University Press 1957

Effros, E.: “Aspects of non-commutative order”. In: Araki, H., Kadison, R. V. (eds).C *-algebras and applications to physics. Lect. Notes Math. vol.650, Berlin Heidelberg New York: Springer 1978

Fannes, M., Nachtergaele, B., Slegers, L.: Functions of Markov processes and algebraic measures. Rev. Math. Phys. (in press)

Fannes, M., Nachtergaele, B., Werner, R. F.: Exact ground states of quantum spin chains. Europhys. Lett.10, 633–637 (1989)

Fannes, M., Nachtergaele, B., Werner, R. F.: Valence bond states on quantum spin chains as ground states with spectral gap. J. Phys. A, Math. Gen.24, L185-L190 (1991)

Fannes, M., Nachtergaele, B., Werner, R. F.: Entropy estimates for finitely correlated states. Preprint KUL-TF-91/09

Fannes, M., Nachtergaele, B., Werner, R. F.: Ground states of VBS models on Cayley trees. J. Stat. Phys. (in press)

Fannes, M., Nachtergaele, B., Werner, R. F.: Finitely correlated pure states and their symmetries. In preparation

Fannes, M., Verbeure, A.: On solvable models in classical lattice systems. Commun. Math. Phys.96, 115–124 (1984)

Freitag, W.-D., Müller-Hartmann, E.: Complete analysis of two-spin correlations of valence bond solid chains for all integer spins. Z. Physik B83, 381–390 (1991)

Groh, U.: The peripheral point spectrum of Schwarz operators onC *-algebras. Math. Z.176, 311–318 (1981)

Hagiwara, M., Katsuma, K., Affleck, I., Halperin, B. I., Renard, J. P.: Observation ofS=1/2 degrees of freedom in anS=1 linear chain Heisenberg antiferromagnet. Phys. Rev. Lett.25, 3181–3184 (1990)

Haldane, F. D. M.: Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with theO(3) nonlinear sigma model. Phys. Lett.93A, 464–468 (1983)

Hammermesh, M.: Group theory and its applications to physical problems. Reading, Mass.: Addison-Wesley 1962

Hudson, R. L., Moody, G. R.: Locally normal symmetric states and an analogue of de Finetti's theorem. Z. Wahrsch. Verw. Gebiete33, 343–351 (1976)

Hulthén, L.: Arkiv. Math. Astron. Fysik26A, (1938)

Ionescu-Tulcea, A., Ionescu-Tulcea, C.: Topics in the theory of lifting. Berlin, Heidelberg, New York: Springer 1970

Iske, P. L., Caspers, W. J.: Exact ground states of one-dimensional valence-bond-solid Hamiltonians. Mod. Phys. Lett.B1, 231–237 (1987)

Iske, P. L., Caspers, W. J.: Scaling properties of valence-bond-solids with a linear structure. Mod. Phys. Lett.B2, 1223–1233 (1988)

Jacobson, N.: Basic Algebra I. San Francisco: Freeman 1974

Kennedy, T., Lieb, E. H., Tasaki, H.: A two-dimensional isotropic quantum antiferromagnet with unique disordered ground state. J. Stat. Phys.53, 383–415 (1988)

Kennedy, T., Lieb, E. H., Shastry, B. S.: Existence of Néel order in some spin 1/2 Heisenberg antiferromagnets. J. Stat. Phys.53, 1019–1030 (1988)

Klein, D. J.: Exact ground states for a class of antiferromagnetic Heisenberg models with short range interactions. J. Phys.A15, 661–671 (1982)

Klümper, A.: The spectra ofq-state vertex models and related antiferromagnetic quantum spin chains. J. Phys.A23, 809–823 (1990).

Knabe, S.: Energy gaps and elementary excitations for Certain VBS-Quantum Antiferromagnets. J. Stat. Phys.52, 627–638 (1988)

Lieb, E., Schulz, T., Mattis, D.: Two soluble models of an antiferromagnetic chain. Ann. Phys.16, 407–466 (1961)

Lieb, E., Mattis, D.: Ordering Energy Levels of Interacting Spin Systems. J. Math. Phys.3, 749–751 (1962)

Majumdar, C. K., Ghosh, D. K.: On next nearest-neighbor interaction in linear chains, I, II. J. Math. Phys.10, 1388–1398, and 1399–1402 (1969)

Majumdar, C. K.: Antiferromagnetic model with known ground state. J. Phys.C3, 911–915 (1970)

Matsui, T.: Uniqueness of the translationally invariant ground state in quantum spin systems. Commun. Math. Phys.126, 453–467 (1990)

Nachtergaele, B.: Working with Quantum Markov States and their Classical Analogues. In: Accardi, L., von Waldenfelds, W. (eds.). Quantum probability and applications V, Leeture Notes in Mathematics, vol. 442. Berlin, Heidelberg, New York: Springer 1990

Schmitt, L., Wittstock, G.: Kernel representation of completely positive Hilbert-Schmidt operators on standard forms. Arch. Math.38, 453–458 (1982)

Spitzer, F.: Markov random fields and Gibbs ensembles. Am. Math. Monthly78, 142–154 (1971)

Stinespring, W. F.: Positive functions onC *-algebras. Proc. Am. Math. Soc.6, 211–216 (1955)

Størmer, E.: Positive maps ofC *-algebras. In: Hartkämper, A., Neumann, H. (eds.). Foundations of quantum mechanics and ordered linear spaces. Lecture Notes in Physics, vol. 29, Berlin, Heidelberg, New York: Springer 1974

Størmer, E.: Symmetric states of infinite tensor products ofC *-algebras. J. Funct. Anal.3, 48–68 (1969)

Takesaki, M.: Theory of operator algebras I. Berlin, Heidelberg, New York: Springer 1979

Takhtajan, L. A.: The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins. Phys. Lett.87A, 479–482 (1982)

Werner, R. F.: Quantum states with Einstein-Rosen-Podolsky correlations admitting a hidden-variable model. Phys. Rev. A40, 4277–4281 (1989)

Werner, R. F.: Remarks on a quantum state extension problem. Lett. Math. Phys.19, 319–326 (1990)

Thông tin
Thông tin xuất bản

Springer Science and Business Media LLC

Tập 144

443-490

Thông tin tác giả