# Random Matrix Theory and Entanglement in Quantum Spin Chains

- 243 Downloads
- 71 Citations

## Abstract

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians — those that are related to quadratic forms of Fermi operators — between the first *N* spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as *N*→∞. This is shown to grow logarithmically with *N*. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlevé type. In some cases these solutions can be evaluated to all orders using recurrence relations.

## Keywords

Entropy Central Charge Compact Group Random Matrix Spin Chain## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A
**53**, 2046–2052 (1996)CrossRefGoogle Scholar - 2.Amico, L., Osterloh, A., Plastina, F., Fazio, R., Palma, G.M.: Dynamics of entanglement in one-dimensional spin systems. Phys. Rev. A
**69**, 022304 (2004); Aharonov, D.: Quantum to classical phase transitions in noisy quantum computer. Phys. Rev. A**62**, 062311 (2000); Osborne, T.J., Nielsen, M.A.: Entanglement in a simple quantum phase transition. Phys. Rev. A**66**, 032110 (2002); Osterloh, A., Amico, L., Falci, G., Fazio, R.: Scaling of entanglement close to a quantum phase transition. Nature**416**, 608–610 (2002)CrossRefGoogle Scholar - 3.Vidal, G., Latorre, J.I., Rico, E., Kitaev, A.: Entanglement in quantum critical phenomena. Phys. Rev. Lett.
**90**, 227902 (2003)CrossRefGoogle Scholar - 4.Jin, B.Q., Korepin, V.E.: Entanglement, Toeplitz determinants and Fisher-Hartwig conjecture. J. Stat. Phys.
**116**, 79–95 (2004)CrossRefMathSciNetGoogle Scholar - 5.Korepin, V.E.: Universality of entropy scaling in 1D gap-less models. Phys. Rev. Lett.
**92**, 096402 (2004)CrossRefGoogle Scholar - 6.Calabrese, P., Cardy, J.: Entanglement Entropy and Quantum Field Theory. J. Stat. Mech. Theor. Exp. P06002 (2004)Google Scholar
- 7.Fisher, M.E., Hartwig, R.E.: Toeplitz determinants, some applications, theorems and conjectures. Adv. Chem. Phys.
**15**, 333–353 (1968)Google Scholar - 8.Basor, E.L., Ehrhardt, T.: Asymptotic formulas for the determinants of symmetric Toeplitz plus Hankel matrices. In:
*Toeplitz matrices and singular integral equations (Pobershau, 2001), Oper. Theory Adv. Appl.***135**, Basel: Birkhäuser 2002, pp. 61–90Google Scholar - 9.Forrester, P.J., Frankel, N.E.: Applications and generalizations of Fisher-Hartwig asymptotics. J. Math. Phys.
**45**, 2003–2028 (2004)CrossRefMathSciNetGoogle Scholar - 10.Lieb, E., Schultz, T., Mattis, D.: Two soluble models of an antiferromagnetic chain. Ann. Phys.
**16**, 407–466 (1961)zbMATHGoogle Scholar - 11.Barouch, E., McCoy, M.: Statistical mechanics of the XY model II. Spin-correlation functions. Phys. Rev. A
**3**, 786–804 (1971)Google Scholar - 12.Basor, E.L., Morrison, K.E.: The Fisher-Hartwig conjecture and Toeplitz eigenvalues. Lin. Alg. App.
**202**, 129–142 (2004)CrossRefzbMATHGoogle Scholar - 13.Heine, H.:
*Kugelfunktionen*, Berlin, 1878 and 1881. Reprinted by Würzburg: Physica Verlag, 1961Google Scholar - 14.Szegő, G.:
*Orthogonal Polynomials*. New York: AMS, 1959Google Scholar - 15.Szegő, G.:
*On certain hermitian forms associated with the Fourier series of a positive function*. Lund: Festkrift Marcel Riesz, 1952, pp. 222–238Google Scholar - 16.Basor, E.L.: Asymptotic formulas for Toeplitz determinants. Trans. Amer. Math. Soc.
**239**, 33–65 (1978)zbMATHGoogle Scholar - 17.Böttcher, A., Silbermann, B.: Toeplitz operators and determinants generated by symbols with one Fisher-Hartwig singularity. Math. Nachr.
**127**, 95–124 (1986)Google Scholar - 18.Böttcher, A., Silbermann, B.:
*Analysis of Toeplitz Operators*. Berlin: Springer-Verlag, 1990Google Scholar - 19.Forrester, P.J., Witte, N.S.: Discrete Painlevé equations, orthogonal polynomial on the unit circle and
*N*-recurrences for averages over U(*N*) – P_{ VI}*τ*-functions. http://arxiv.org/abs/math-ph/0308036, 2003 - 20.Forrester, P.J., Witte, N.S.: Application of the
*τ*-function theory of Painlevé equations to random matrices: P_{VI}, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J.**174**, 29–114 (2004)MathSciNetGoogle Scholar - 21.Weyl, H.:
*The Classical Groups*. Princeton, NJ: Princeton University Press, 1946Google Scholar