These are lecture notes for a short course given at the Les Houches Summer School on ``Integrability in Atomic and Condensed Matter Physics'',
in summer 2018.
Here, I pedagogically discuss recent ...advances in the study of the entanglement spreading during the non-equilibrium dynamics
of isolated integrable quantum systems.
I first introduce the idea that the stationary thermodynamic entropy is the entanglement accumulated during the non-equilibrium dynamics
and then join such an idea with the quasiparticle picture for the entanglement spreading to provide quantitive predictions for the time evolution of the entanglement
entropy in arbitrary integrable models, regardless of the interaction strength.
A
bstract
We develop a systematic approach to compute the subsystem trace distances and relative entropies for subsystem reduced density matrices associated to excited states in different symmetry ...sectors of a 1+1 dimensional conformal field theory having an internal U(1) symmetry. We provide analytic expressions for the charged moments corresponding to the resolution of both relative entropies and distances for general integer
n
. For the relative entropies, these formulas are manageable and the analytic continuation to
n
= 1 can be worked out in most of the cases. Conversely, for the distances the corresponding charged moments become soon untreatable as
n
increases. A remarkable result is that relative entropies and distances are the same for all symmetry sectors, i.e. they satisfy entanglement equipartition, like the entropies. Moreover, we exploit the OPE expansion of composite twist fields, to provide very general results when the subsystem is a single interval much smaller than the total system. We focus on the massless compact boson and our results are tested against exact numerical calculations in the XX spin chain.
A
bstract
We consider the form factor bootstrap approach of integrable field theories to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement ...entropies. The bootstrap equations are determined in an intuitive way and their solution is presented for the massive Ising field theory and for the genuinely interacting sinh-Gordon model, both possessing a ℤ
2
symmetry. The solutions are carefully cross-checked by performing various limits and by the application of the ∆-theorem. The issue of symmetry resolution for discrete symmetries is also discussed. We show that entanglement equipartition is generically expected and we identify the first subleading term (in the UV cutoff) breaking it. We also present the complete computation of the symmetry resolved von Neumann entropy for an interval in the ground state of the paramagnetic phase of the Ising model. In particular, we compute the universal functions entering in the charged and symmetry resolved entanglement.
A
bstract
We consider the problem of the decomposition of the Rényi entanglement entropies in theories with a non-abelian symmetry by doing a thorough analysis of Wess-Zumino-Witten (WZW) models. We ...first consider SU(2)
k
as a case study and then generalise to an arbitrary non-abelian Lie group. We find that at leading order in the subsystem size
L
the entanglement is equally distributed among the different sectors labelled by the irreducible representation of the associated algebra. We also identify the leading term that breaks this equipartition: it does not depend on
L
but only on the dimension of the representation. Moreover, a log log
L
contribution to the Rényi entropies exhibits a universal prefactor equal to half the dimension of the Lie group.
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.
A
bstract
We present a thorough analysis of the entanglement entropies related to different symmetry sectors of free quantum field theories (QFT) with an internal U(1) symmetry. We provide explicit ...analytic computations for the charged moments of Dirac and complex scalar fields in two spacetime dimensions, both in the massive and massless cases, using two different approaches. The first one is based on the replica trick, the computation of the partition function on Riemann surfaces with the insertion of a flux
α
, and the introduction of properly modified twist fields, whose two-point function directly gives the scaling limit of the charged moments. With the second method, the diagonalisation in replica space maps the problem to the computation of a partition function on a cut plane, that can be written exactly in terms of the solutions of non-linear differential equations of the Painlevé V type. Within this approach, we also derive an asymptotic expansion for the short and long distance behaviour of the charged moments. Finally, the Fourier transform provides the desired symmetry resolved entropies: at the leading order, they satisfy entanglement equipartition and we identify the subleading terms that break it. Our analytical findings are tested against exact numerical calculations in lattice models.
Symmetry and symmetry breaking are two pillars of modern quantum physics. Still, quantifying how much a symmetry is broken is an issue that has received little attention. In extended quantum systems, ...this problem is intrinsically bound to the subsystem of interest. Hence, in this work, we borrow methods from the theory of entanglement in many-body quantum systems to introduce a subsystem measure of symmetry breaking that we dub entanglement asymmetry. As a prototypical illustration, we study the entanglement asymmetry in a quantum quench of a spin chain in which an initially broken global U(1) symmetry is restored dynamically. We adapt the quasiparticle picture for entanglement evolution to the analytic determination of the entanglement asymmetry. We find, expectedly, that larger is the subsystem, slower is the restoration, but also the counterintuitive result that more the symmetry is initially broken, faster it is restored, a sort of quantum Mpemba effect, a phenomenon that we show to occur in a large variety of systems.
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