123 research outputs found
Fisher Hartwig determinants, conformal field theory and universality in generalised XX models
We discuss certain quadratic models of spinless fermions on a 1D lattice, and
their corresponding spin chains. These were studied by Keating and Mezzadri in
the context of their relation to the Haar measures of the classical compact
groups. We show how these models correspond to translation invariant models on
an infinite or semi-infinite chain, which in the simplest case reduce to the
familiar XX model. We give physical context to mathematical results for the
entanglement entropy, and calculate the spin-spin correlation functions using
the Fisher-Hartwig conjecture. These calculations rigorously demonstrate
universality in classes of these models. We show that these are in agreement
with field theoretic and renormalization group arguments that we provide
Random Matrix Theory and Entanglement in Quantum Spin Chains
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 -->
infinity . 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
Painlev\'e type. In some cases these solutions can be evaluated to all orders
using recurrence relations.Comment: 39 pages, 1 table, no figures. Revised version: minor correction
Universal parity effects in the entanglement entropy of XX chains with open boundary conditions
We consider the Renyi entanglement entropies in the one-dimensional XX
spin-chains with open boundary conditions in the presence of a magnetic field.
In the case of a semi-infinite system and a block starting from the boundary,
we derive rigorously the asymptotic behavior for large block sizes on the basis
of a recent mathematical theorem for the determinant of Toeplitz plus Hankel
matrices. We conjecture a generalized Fisher-Hartwig form for the corrections
to the asymptotic behavior of this determinant that allows the exact
characterization of the corrections to the scaling at order o(1/l) for any n.
By combining these results with conformal field theory arguments, we derive
exact expressions also in finite chains with open boundary conditions and in
the case when the block is detached from the boundary.Comment: 24 pages, 9 figure
The multicomponent 2D Toda hierarchy: Discrete flows and string equations
The multicomponent 2D Toda hierarchy is analyzed through a factorization
problem associated to an infinite-dimensional group. A new set of discrete
flows is considered and the corresponding Lax and Zakharov--Shabat equations
are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix
types are proposed and studied. Orlov--Schulman operators, string equations and
additional symmetries (discrete and continuous) are considered. The
continuous-discrete Lax equations are shown to be equivalent to a factorization
problem as well as to a set of string equations. A congruence method to derive
site independent equations is presented and used to derive equations in the
discrete multicomponent KP sector (and also for its modification) of the theory
as well as dispersive Whitham equations.Comment: 27 pages. In the revised paper we improved the presentatio
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