Random Matrix Theory and Entanglement in Quantum Spin Chains

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.

Entanglement in fermionic chains with finite-range coupling and broken symmetries

“…[19,20] Keating and Mezzadri established a relation between certain symmetries of one-dimensional quadratic fermionic Hamiltonians and classical compact Lie groups by means of entanglement entropy. This is analogous to the Altland-Zirnbauer classification [21].…”

“…In Sec. IV, we compute the determinant of the block Toeplitz matrix, first in the smooth case where the Its, Mezzadri, and Mo's formula (19) applies, and then by extending it in (23) to the case where discontinuities are present; later we use this result to obtain our general expression (26) for the asymptotic behavior of the entanglement entropy in a critical theory with long-range couplings and the above broken symmetries. In Secs.…”

Quantum phase transitions and quantum fidelity in free fermion graphs

“…It is worth noticing that the spin Hamiltonian obtained by the inverse Jordan-Wigner transformation of Eq. (19) is not the variable range extension of the XY model [15]. For example, the fully coordinated spin system, known as the Lipkin-Meshkov-Glick model [16], does not correspond to the fully coordinated fermionic graph discussed below.…”

“…In fact, in Refs. [15,17], it is shown how to express S l1,l2,..,lN in terms of the eigenvalues ν i of a matrix S N = (T N T † N ) 1/2 . There T N is the sub-block of order N , relative to the N sites belonging to the block, of a matrix (called T in Ref.…”

Entanglement and Density Matrix of a Block of Spins in AKLT Model

“…Entanglement properties also play an important role in condensed matter physics, such as phase transitions (Osterloh, et al 2002;Osborne & Nielsen 2002) and macroscopic properties of solids (Ghosh, et al 2003;Vedral 2004). Extensive research has been undertaken to understand quantum entanglement for spin chains, correlated electrons, interacting bosons as well as other models, see Amico, et al (2007), Audenaert, et al (2002), Fan & Korepin (2008), Katsura, et al (2007b), Fan, et al (2007), Arnesen, et al (2001), Korepin (2004), Verstraete, et al (2004a, b), Campos Venuti, et al (2006), Jin & Korepin (2004), Vedral (2004), Latorre, et al (2004aLatorre, et al ( , b, 2005, Orus (2005), Orus & Latorre (2004), Pachos & Plenio (2004), Plenio et al (2004), Fan & Lloyd (2005), Chen, et al (2004), Zanardi & Rasetti (1999), Popkov & Salerno (2004), Keating & Mezzadri (2004), Gu, et al (2003Gu, et al ( , 2004, , , Holzhey, et al (1994), Calabrese & Cardy (2004), Levin & Wen (2006), Kitaev & Preskill (2006), Ryu & Hatsugai (2006), Hirano & Hatsugai (2007) for reviews and references. Characteristic functions of quantum entanglement, such as von Neumann entropy and Renyi entropy, are obtained and discussed through studying reduced density matrices of subsystems (Fan, et al 2004;…”