Random matrix theory and critical phenomena in quantum spin chains

Abstract

We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups U(N ), O(N ), and Sp(2N ). In particular we calculate critical exponents s, ν, and z, corresponding to the energy gap, correlation length, and dynamic exponent, respectively. We also compute the ground state correlators σ , and n i=1 σ z i g , all of which display quasi-long-range order with a critical exponent dependent upon system parameters. Our approach establishes universality of the exponents for the class of systems in question.

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 particular, they find that translational invariant Hamiltonians preserving fermionic number, i.e., when the correlation matrix reduces to a scalar Toeplitz matrix, are related with the U(N) group.…”

“…,J and lateral limits t − j ,t + j at the discontinuities. Then from the discontinuities we get a logarithmic contribution to D X (λ), whose asymptotic expansion reads (21) where now the dots represent finite contributions in the large |X| limit, that can be computed explicitly but are not going to be necessary for us. The crucial fact is that contrary to the linear term (and also the constant one) the logarithmic contribution only depends on the behavior of the symbol near its discontinuities, more concretely on its lateral limits.…”

Asymptotic Correlations in Gapped and Critical Topological Phases of 1D Quantum Systems

“…This relation implies that ω = −c. The correlators O α (1)O α (N + 1) with |α| ≤ 1 in isotropic models with general c and ω = −c were derived in references [52,53] using the same methods as this paper. Our results go further by studying a wider class of models, including critical phases with general (c, ω), as well as a wider class of observables: O α (1)O α (N + 1) for all α.…”

“…Following [80], in order to write down the dominant asymptotics, it is helpful to introduce the notion of FH-representations. Given a symbol t(z) written in canonical form (52), replace all β j bỹ β j = β j + n j such that j n j = 0. This new function is the FH-representation t(z; n 1 , .…”

“…Theorem 9 (Deift, Its, Krasovsky 2011). Consider a Toeplitz matrix generated by t(z) in the canonical form (52). Suppose β j ∈ Z for all j.…”