32 research outputs found
Exact mean-field solution of a spin chain with short-range and long-range interactions
We consider the transverse field Ising model with additional all-to-all
interactions between the spins. We show that a mean-field treatment of this
model becomes exact in the thermodynamic limit, despite the presence of 1D
short-range interactions. This is established by looking for eigenstates as
coherent states with an amplitude that varies through the Hilbert space. We
study then the thermodynamics of the model and identify the different phases.
Among its peculiar features, this 1D model possesses a second-order phase
transition at finite temperature and exhibits inverse melting.Comment: 27 page
Concavity analysis of the tangent method
The tangent method has recently been devised by Colomo and Sportiello
(arXiv:1605.01388 [math-ph]) as an efficient way to determine the shape of
arctic curves. Largely conjectural, it has been tested successfully in a
variety of models. However no proof and no general geometric insight have been
given so far, either to show its validity or to allow for an understanding of
why the method actually works. In this paper, we propose a universal framework
which accounts for the tangency part of the tangent method, whenever a
formulation in terms of directed lattice paths is available. Our analysis shows
that the key factor responsible for the tangency property is the concavity of
the entropy (also called the Lagrangean function) of long random lattice paths.
We extend the proof of the tangency to -deformed paths.Comment: published version, 22 page
Continuous Hamiltonian dynamics on digital quantum computers without discretization error
We introduce an algorithm to compute Hamiltonian dynamics on digital quantum
computers that requires only a finite circuit depth to reach an arbitrary
precision, i.e. achieves zero discretization error with finite depth. This
finite number of gates comes at the cost of an attenuation of the measured
expectation value by a known amplitude, requiring more shots per circuit. The
gate count for simulation up to time is with the
-norm of the Hamiltonian, without dependence on the precision desired on the
result, providing a significant improvement over previous algorithms. The only
dependence in the norm makes it particularly adapted to non-sparse
Hamiltonians. The algorithm generalizes to time-dependent Hamiltonians,
appearing for example in adiabatic state preparation. These properties make it
particularly suitable for present-day relatively noisy hardware that supports
only circuits with moderate depth.Comment: 5 page
On tidal capture of primordial black holes by neutron stars
The fraction of primordial black holes (PBHs) of masses g
in the total amount of dark matter may be constrained by considering their
capture by neutron stars (NSs), which leads to the rapid destruction of the
latter. The constraints depend crucially on the capture rate which, in turn, is
determined by the energy loss by a PBH passing through a NS. Two alternative
approaches to estimate the energy loss have been used in the literature: the
one based on the dynamical friction mechanism, and another on tidal
deformations of the NS by the PBH. The second mechanism was claimed to be more
efficient by several orders of magnitude due to the excitation of particular
oscillation modes reminiscent of the surface waves. We address this
disagreement by considering a simple analytically solvable model that consists
of a flat incompressible fluid in an external gravitational field. In this
model, we calculate the energy loss by a PBH traversing the fluid surface. We
find that the excitation of modes with the propagation velocity smaller than
that of PBH is suppressed, which implies that in a realistic situation of a
supersonic PBH the large contributions from the surface waves are absent and
the above two approaches lead to consistent expressions for the energy loss.Comment: 7 page
A systematic -expansion of form factor sums for dynamical correlations in the Lieb-Liniger model
We introduce a framework for calculating dynamical correlations in the
Lieb-Liniger model in arbitrary energy eigenstates and for all space and time,
that combines a Lehmann representation with a expansion. The
term of the expansion is of order and takes into account all particle-hole excitations over the averaging eigenstate.
Importantly, in contrast to a 'bare' expansion it is uniform in space and
time. The framework is based on a method for taking the thermodynamic limit of
sums of form factors that exhibit non integrable singularities. We expect our
framework to be applicable to any local operator.
We determine the first three terms of this expansion and obtain an explicit
expression for the density-density dynamical correlations and the dynamical
structure factor at order . We apply these to finite-temperature
equilibrium states and non-equilibrium steady states after quantum quenches. We
recover predictions of (nonlinear) Luttinger liquid theory and generalized
hydrodynamics in the appropriate limits, and are able to compute sub-leading
corrections to these.Comment: 78 pages; corresponds to published versio
Out-of-equilibrium dynamics of the XY spin chain from form factor expansion
We consider the XY spin chain with arbitrary time-dependent magnetic field
and anisotropy. We argue that a certain subclass of Gaussian states, called
Coherent Ensemble (CE) following [1], provides a natural and unified framework
for out-of-equilibrium physics in this model. We show that correlation
functions in the CE can be computed using form factor expansion and expressed
in terms of Fredholm determinants. In particular, we present exact
out-of-equilibrium expressions in the thermodynamic limit for the previously
unknown order parameter one-point function, dynamical two-point function and
equal-time three-point function.Comment: 44 page
Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions
We consider the problems of calculating the dynamical order parameter
two-point function at finite temperatures and the one-point function after a
quantum quench in the transverse field Ising chain. Both of these can be
expressed in terms of form factor sums in the basis of physical excitations of
the model. We develop a general framework for carrying out these sums based on
a decomposition of form factors into partial fractions, which leads to a
factorization of the multiple sums and permits them to be evaluated
asymptotically. This naturally leads to systematic low density expansions. At
late times these expansions can be summed to all orders by means of a
determinant representation. Our method has a natural generalization to
semi-local operators in interacting integrable models.Comment: 51 pages - published versio
Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy and Generalized Hydrodynamics
We consider fermions defined on a continuous one-dimensional interval and subject to weak repulsive two-body interactions. We show that it is possible to perturbatively construct an extensive number of mutually compatible conserved charges for any interaction potential. However, the contributions to the densities of these charges at second order and higher are generally nonlocal and become spatially localized only if the potential fulfils certain compatibility conditions. We prove that the only solutions to the first of these conditions are the Cheon-Shigehara potential (fermionic dual to the Lieb-Liniger model) and the Calogero-Sutherland potentials. We use our construction to show how generalized hydrodynamics emerges from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, and argue that generalized hydrodynamics in the weak interaction regime is robust under nonintegrable perturbations