20 research outputs found
On relations between one-dimensional quantum and two-dimensional classical spin systems
We exploit mappings between quantum and classical systems in order to obtain
a class of two-dimensional classical systems with critical properties
equivalent to those of the class of one-dimensional quantum systems discussed
in a companion paper (J. Hutchinson, J. P. Keating, and F. Mezzadri,
arXiv:1503.05732). In particular, we use three approaches: the Trotter-Suzuki
mapping; the method of coherent states; and a calculation based on commuting
the quantum Hamiltonian with the transfer matrix of a classical system. This
enables us to establish universality of certain critical phenomena by extension
from the results in our previous article for the classical systems identified.Comment: 36 page
Comb entanglement in quantum spin chains
Bipartite entanglement in the ground state of a chain of quantum spins
can be quantified either by computing pairwise concurrence or by dividing the
chain into two complementary subsystems. In the latter case the smaller
subsystem is usually a single spin or a block of adjacent spins and the
entanglement differentiates between critical and non-critical regimes. Here we
extend this approach by considering a more general setting: our smaller
subsystem consists of a {\it comb} of spins, spaced sites apart.
Our results are thus not restricted to a simple `area law', but contain
non-local information, parameterized by the spacing . For the XX model we
calculate the von-Neumann entropy analytically when and
investigate its dependence on and . We find that an external magnetic
field induces an unexpected length scale for entanglement in this case.Comment: 6 pages, 4 figure
A new correlator in quantum spin chains
We propose a new correlator in one-dimensional quantum spin chains, the
Emptiness Formation Probability (EFP). This is a natural generalization
of the Emptiness Formation Probability (EFP), which is the probability that the
first spins of the chain are all aligned downwards. In the EFP we let
the spins in question be separated by sites. The usual EFP corresponds to
the special case when , and taking allows us to quantify non-local
correlations. We express the EFP for the anisotropic XY model in a
transverse magnetic field, a system with both critical and non-critical
regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find
that the magnetic field induces an interesting length scale.Comment: 6 pages, 1 figur
Nodal domain distributions for quantum maps
The statistics of the nodal lines and nodal domains of the eigenfunctions of
quantum billiards have recently been observed to be fingerprints of the
chaoticity of the underlying classical motion by Blum et al. (Phys. Rev. Lett.,
Vol. 88 (2002), 114101) and by Bogomolny and Schmit (Phys. Rev. Lett., Vol. 88
(2002), 114102). These statistics were shown to be computable from the random
wave model of the eigenfunctions. We here study the analogous problem for
chaotic maps whose phase space is the two-torus. We show that the distributions
of the numbers of nodal points and nodal domains of the eigenvectors of the
corresponding quantum maps can be computed straightforwardly and exactly using
random matrix theory. We compare the predictions with the results of numerical
computations involving quantum perturbed cat maps.Comment: 7 pages, 2 figures. Second version: minor correction
Entanglement in Quantum Spin Chains, Symmetry Classes of Random Matrices, and Conformal Field Theory
We compute the entropy of entanglement between the first spins and the
rest of the system in the ground states of a general class of quantum
spin-chains. We show that under certain conditions the entropy can be expressed
in terms of averages over ensembles of random matrices. These averages can be
evaluated, allowing us to prove that at critical points the entropy grows like
as , where and are determined explicitly. In an important class of systems,
is equal to one-third of the central charge of an associated Virasoro algebra.
Our expression for therefore provides an explicit formula for the
central charge.Comment: 4 page
Rate of convergence of linear functions on the unitary group
We study the rate of convergence to a normal random variable of the real and
imaginary parts of Tr(AU), where U is an N x N random unitary matrix and A is a
deterministic complex matrix. We show that the rate of convergence is O(N^{-2 +
b}), with 0 <= b < 1, depending only on the asymptotic behaviour of the
singular values of A; for example, if the singular values are non-degenerate,
different from zero and O(1) as N -> infinity, then b=0. The proof uses a
Berry-Esse'en inequality for linear combinations of eigenvalues of random
unitary, matrices, and so appropriate for strongly dependent random variables.Comment: 34 pages, 1 figure; corrected typos, added remark 3.3, added 3
reference
On the multiplicativity of quantum cat maps
The quantum mechanical propagators of the linear automorphisms of the
two-torus (cat maps) determine a projective unitary representation of the theta
group, known as Weil's representation. We prove that there exists an
appropriate choice of phases in the propagators that defines a proper
representation of the theta group. We also give explicit formulae for the
propagators in this representation.Comment: Revised version: proof of the main theorem simplified. 21 page
Quantum cat maps with spin 1/2
We derive a semiclassical trace formula for quantized chaotic transformations
of the torus coupled to a two-spinor precessing in a magnetic field. The trace
formula is applied to semiclassical correlation densities of the quantum map,
which, according to the conjecture of Bohigas, Giannoni and Schmit, are
expected to converge to those of the circular symplectic ensemble (CSE) of
random matrices. In particular, we show that the diagonal approximation of the
spectral form factor for small arguments agrees with the CSE prediction. The
results are confirmed by numerical investigations.Comment: 26 pages, 3 figure
Entanglement entropy in quantum spin chains with finite range interaction
We study the entropy of entanglement of the ground state in a wide family of
one-dimensional quantum spin chains whose interaction is of finite range and
translation invariant. Such systems can be thought of as generalizations of the
XY model. The chain is divided in two parts: one containing the first
consecutive L spins; the second the remaining ones. In this setting the entropy
of entanglement is the von Neumann entropy of either part. At the core of our
computation is the explicit evaluation of the leading order term as L tends to
infinity of the determinant of a block-Toeplitz matrix whose symbol belongs to
a general class of 2 x 2 matrix functions. The asymptotics of such determinant
is computed in terms of multi-dimensional theta-functions associated to a
hyperelliptic curve of genus g >= 1, which enter into the solution of a
Riemann-Hilbert problem. Phase transitions for thes systems are characterized
by the branch points of the hyperelliptic curve approaching the unit circle. In
these circumstances the entropy diverges logarithmically. We also recover, as
particular cases, the formulae for the entropy discovered by Jin and Korepin
(2004) for the XX model and Its, Jin and Korepin (2005,2006) for the XY model.Comment: 75 pages, 10 figures. Revised version with minor correction
On the spacing distribution of the Riemann zeros: corrections to the asymptotic result
It has been conjectured that the statistical properties of zeros of the
Riemann zeta function near z = 1/2 + \ui E tend, as , to the
distribution of eigenvalues of large random matrices from the Unitary Ensemble.
At finite numerical results show that the nearest-neighbour spacing
distribution presents deviations with respect to the conjectured asymptotic
form. We give here arguments indicating that to leading order these deviations
are the same as those of unitary random matrices of finite dimension , where is a well
defined constant.Comment: 9 pages, 3 figure