Aurélien Grabsch, Satya N. Majumdar, Grégory Schehr, Christophe Texier
SciPost Phys. 4, 014 (2018) ·
published 24 March 2018
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We study a system of 1D noninteracting spinless fermions in a confining trap
at finite temperature. We first derive a useful and general relation for the
fluctuations of the occupation numbers valid for arbitrary confining trap, as
well as for both canonical and grand canonical ensembles. Using this relation,
we obtain compact expressions, in the case of the harmonic trap, for the
variance of certain observables of the form of sums of a function of the
fermions' positions, $\mathcal{L}=\sum_n h(x_n)$. Such observables are also
called linear statistics of the positions. As anticipated, we demonstrate
explicitly that these fluctuations do depend on the ensemble in the
thermodynamic limit, as opposed to averaged quantities, which are ensemble
independent. We have applied our general formalism to compute the fluctuations
of the number of fermions $\mathcal{N}_+$ on the positive axis at finite
temperature. Our analytical results are compared to numerical simulations. We
discuss the universality of the results with respect to the nature of the
confinement.
SciPost Phys. 4, 015 (2018) ·
published 27 March 2018
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We study a model in 1+2 dimensions composed of a spherical Fermi surface of
$N_f$ flavors of fermions coupled to a massless scalar. We present a framework
to non-perturbatively calculate general fermion $n$-point functions of this
theory in the limit $N_f\rightarrow0$ followed by $k_F\rightarrow\infty$ where
$k_F$ sets both the size and curvature of the Fermi surface. Using this
framework we calculate the zero-temperature fermion density-density correlation
function in real space and find an exponential decay of Friedel oscillations.
SciPost Phys. 4, 016 (2018) ·
published 27 March 2018
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At thermal equilibrium, the concept of effective central charge for massive
deformations of two-dimensional conformal field theories (CFT) is well
understood, and can be defined by comparing the partition function of the
massive model to that of a CFT. This temperature-dependent effective charge
interpolates monotonically between the central charge values corresponding to
the IR and UV fixed points at low and high temperatures, respectively. We
propose a non-equilibrium, time-dependent generalization of the effective
central charge for integrable models after a quantum quench, $c_{\rm eff}(t)$,
obtained by comparing the return amplitude to that of a CFT quench. We study
this proposal for a large mass quench of a free boson, where the charge is seen
to interpolate between $c_{\rm eff}=0$ at $t=0$, and $c_{\rm eff}\sim 1$ at
$t\to\infty$, as is expected. We use our effective charge to define an "Ising
to Tricritical Ising" quench protocol, where the charge evolves from $c_{\rm
eff}=1/2$ at $t=0$, to $c_{\rm eff}=7/10$ at $t\to\infty$, the corresponding
values of the first two unitary minimal CFT models. We then argue that the
inverse "Tricritical Ising to Ising" quench is impossible with our methods.
These conclusions can be generalized for quenches between any two adjacent
unitary minimal CFT models. We finally study a large mass quench into the
"staircase model" (sinh-Gordon with a particular complex coupling). At short
times after the quench, the effective central charge increases in a discrete
"staircase" structure, where the values of the charge at the steps can be
computed in terms of the central charges of unitary minimal CFT models. When
the initial state is a pure state, one always finds that $c_{\rm
eff}(t\to\infty)\geq c_{\rm eff}(t=0)$, though $c_{\rm eff}(t)$, generally
oscillates at finite times. We explore how this constraint may be related to RG
flow irreversibility.
SciPost Phys. 4, 017 (2018) ·
published 27 March 2018
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The time evolution of the entanglement entropy in non-equilibrium quantum
systems provides crucial information about the structure of the time-dependent
state. For quantum quench protocols, by combining a quasiparticle picture for
the entanglement spreading with the exact knowledge of the stationary state
provided by Bethe ansatz, it is possible to obtain an exact and analytic
description of the evolution of the entanglement entropy. Here we discuss the
application of these ideas to several integrable models. First we show that for
non-interacting systems, both bosonic and fermionic, the exact time-dependence
of the entanglement entropy can be derived by elementary techniques and without
solving the dynamics. We then provide exact results for interacting spin chains
that are carefully tested against numerical simulations. Finally, we apply this
method to integrable one-dimensional Bose gases (Lieb-Liniger model) both in
the attractive and repulsive regimes. We highlight a peculiar behaviour of the
entanglement entropy due to the absence of a maximum velocity of excitations.
SciPost Phys. 4, 018 (2018) ·
published 31 March 2018
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Superconducting Josephson vortices have direct analogues in ultracold-atom
physics as solitary-wave excitations of two-component superfluid Bose gases
with linear coupling. Here we numerically extend the zero-velocity Josephson
vortex solutions of the coupled Gross-Pitaevskii equations to non-zero
velocities, thus obtaining the full dispersion relation. The inertial mass of
the Josephson vortex obtained from the dispersion relation depends on the
strength of linear coupling and has a simple pole divergence at a critical
value where it changes sign while assuming large absolute values. Additional
low-velocity quasiparticles with negative inertial mass emerge at finite
momentum that are reminiscent of a dark soliton in one component with
counter-flow in the other. In the limit of small linear coupling we compare the
Josephson vortex solutions to sine-Gordon solitons and show that the
correspondence between them is asymptotic, but significant differences appear
at finite values of the coupling constant. Finally, for unequal and non-zero
self- and cross-component nonlinearities, we find a new solitary-wave
excitation branch. In its presence, both dark solitons and Josephson vortices
are dynamically stable while the new excitations are unstable.
