In spin chains with local unitary evolution preserving the magnetization \(S^{\rm z}\), the domain-wall state \(\left| \dots \uparrow \uparrow \uparrow \uparrow \uparrow \downarrow \downarrow ...\downarrow \downarrow \downarrow \dots \right>\) typically "melts". At large times, a non-trivial magnetization profile develops in an expanding region around the initial position of the domain-wall. For non-integrable dynamics the melting is diffusive, with entropy production within a melted region of size \(\sqrt{t}\). In contrast, when the evolution is integrable, ballistic transport dominates and results in a melted region growing linearly in time, with no extensive entropy production: the spin chain remains locally in states of zero entropy at any time. Here we show that, for the integrable spin-\(1/2\) XXZ chain, low-energy quantum fluctuations in the melted region give rise to an emergent Luttinger liquid which, remarkably, differs from the equilibrium one. The striking feature of this emergent Luttinger liquid is its quasi-particle charge (or Luttinger parameter \(K\)) which acquires a fractal dependence on the XXZ chain anisotropy parameter \(\Delta\).
We consider the non-equilibrium dynamics after a sudden quench of the magnetic field in the transverse field Ising chain starting from excited states of the pre-quench Hamiltonian. We prove that ...stationary values of local correlation functions can be described by the generalised Gibbs ensemble (GGE). Then we study the full time evolution of the transverse magnetisation by means of stationary phase methods. The equal time two-point longitudinal correlation function is analytically derived for a particular class of excited states for quenches within the ferromagnetic phase, and studied numerically in general. The full time dependence of the entanglement entropy of a block of spins is also obtained analytically for the same class of states and for arbitrary quenches.
It is widely believed that the stationary properties after a quantum quench in integrable systems can be described by a generalized Gibbs ensemble (GGE), even if all the analytical evidence is based ...on free theories in which the pre- and post-quench modes are linearly related. In contrast, we consider the experimentally relevant quench of the one-dimensional Bose gas from zero to infinite interaction, in which the relation between modes is nonlinear, and consequently Wick's theorem does not hold. We provide exact analytical results for the time evolution of the dynamical density-density correlation function at any time after the quench and we prove that its stationary value is described by a GGE in which Wick's theorem is restored.
We analyze the entanglement properties of the asymptotic steady state after a quench from free to hard-core bosons in one dimension. The Rényi and von Neumann entanglement entropies are found to be ...extensive, and the latter coincides with the thermodynamic entropy of the Generalized Gibbs Ensemble (GGE). Computing the spectrum of the two-point function, we provide exact analytical results both for the leading extensive parts and the subleading terms for the entropies as well as for the cumulants of the particle number fluctuations. We also compare the extensive part of the entanglement entropy with the thermodynamic ones, showing that the GGE entropy equal the entanglement one and it is the double of the diagonal entropy.
In this paper, we apply the form factor bootstrap approach to branch point twist fields in the \(q\)-state Potts model for \(q\leq 3\). For \(q=3\) this is an integrable interacting quantum field ...theory with an internal discrete \(\mathbb{Z}_3\) symmetry and therefore provides an ideal starting point for the investigation of the symmetry resolved entanglement entropies. However, more generally, for \(q\leq 3\) the standard Rényi and entanglement entropies are also accessible through the bootstrap programme. In our work we present form factor solutions both for the standard branch point twist field with \(q\leq 3\) and for the composite (or symmetry resolved) branch point twist field with \(q=3\). In both cases, the form factor equations are solved for two particles and the solutions are carefully checked via the \(\Delta\)-sum rule. Using our analytic predictions, we compute the leading finite-size corrections to the entanglement entropy and entanglement equipartition for a single interval in the ground state.
We study the Renyi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the ...reduced density matrix in the spin variables by taking properly into account the Jordan-Wigner string between the two blocks. From this we can evaluate any Renyi entropy of finite integer order. We study in details critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results in the gapped phase and after a quantum quench.
We present a general theory of the corrections to the asymptotic behaviour of the Renyi entropies which measure the entanglement of an interval A of length L with the rest of an infinite ...one-dimensional system, in the case when this is described by a conformal field theory of central charge c. These can be due to bulk irrelevant operators of scaling dimension x>2, in which case the leading corrections are of the expected form L^{-2(x-2)} for values of n close to 1. However for n>x/(x-2) corrections of the form L^{2-x-x/n} and L^{-2x/n} arise and dominate the conventional terms. We also point out that the last type of corrections can also occur with x less than 2. They arise from relevant operators induced by the conical space-time singularities necessary to describe the reduced density matrix. These agree with recent analytic and numerical results for quantum spin chains. We also compute the effect of marginally irrelevant bulk operators, which give a correction (log L)^{-2}, with a universal amplitude. We present analogous results for the case when the interval lies at the end of a semi-infinite system.
We review the conformal field theory approach to entanglement entropy. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization ...to different situations such as finite size, systems with boundaries, and the case of several disjoint intervals. We discuss the behaviour away from the critical point and the spectrum of the reduced density matrix. Quantum quenches, as paradigms of non-equilibrium situations, are also considered.
We consider the non-equilibrium dynamics of a gas of impenetrable bosons released from a harmonic trapping potential to a circle. The many body dynamics is solved analytically and the time dependence ...of all the physically relevant correlations is described. We prove that, for large times and in the thermodynamic limit, the reduced density matrix of any subsystem converges to a generalized Gibbs ensemble as a consequence of the integrability of the model. We discuss the approach to the stationary behavior at late times. We also describe the time-dependence of the entanglement entropy which attains a very simple form in the stationary state.
We study the non-equilibrium dynamics of a Tonks-Girardeau gas released from a parabolic trap to a circle. We present the exact analytic solution of the many body dynamics and prove that, for large ...times and in a properly defined thermodynamic limit, the reduced density matrix of any finite subsystem converges to a generalized Gibbs ensemble. The equilibration mechanism is expected to be the same for all one-dimensional systems.
