Vincenzo Alba

Entanglement entropy of excited states

We study the entanglement entropy of a block of contiguous spins in excited states of spin chains. We consider the XY model in a transverse field and the XXZ Heisenberg spin chain. For the latter, we developed a numerical application of the algebraic Bethe ansatz. We find two main classes of states with logarithmic and extensive behavior in the dimension of the block, characterized by the properties of excitations of the state. This behavior can be related to the locality properties of the Hamiltonian having a given state as the ground state. We also provide several details of the finite size scaling.

Entanglement entropy of two disjoint blocks in critical Ising models

We study the scaling of the Renyi and entanglement entropy of two disjoint blocks of critical Ising models as function of their sizes and separations. We present analytic results based on conformal field theory that are quantitatively checked in numerical simulations of both the quantum spin chain and the classical two-dimensional Ising model. Theoretical results match the ones obtained from numerical simulations only after taking properly into account the corrections induced by the finite length of the blocks to their leading scaling behavior.

Entanglement entropy of two disjoint intervals in c = 1 theories

We study the scaling of the Renyi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge c = 1. We provide the analytic conformal field theory result for the second order Renyi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin–Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two-dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin chain with tree tensor network techniques that allowed us to obtain the reduced density matrices of disjoint blocks of the spin chain and to check the correctness of the predictions for Renyi and entanglement entropies from conformal field theory. In order to match these predictions, we have extrapolated the numerical results by properly taking into account the corrections induced by the finite length of the blocks on the leading scaling behavior.

Entanglement negativity and conformal field theory: a Monte Carlo study

We investigate the behavior of the moments of the partially transposed reduced density matrix in critical quantum spin chains. Given subsystem A as the union of two blocks, this is the (matrix) transpose of ρA with respect to the degrees of freedom of one of the two blocks. This is also the main ingredient for constructing the logarithmic negativity. We provide a new numerical scheme for efficiently calculating all the moments of using classical Monte Carlo simulations. In particular we study several combinations of the moments which are scale invariant at a critical point. Their behavior is fully characterized in both the critical Ising and the anisotropic Heisenberg XXZ chains. For two adjacent blocks we find, in both models, full agreement with recent conformal field theory (CFT) calculations. For disjoint blocks, in the Ising chain finite size corrections are nonnegligible. We demonstrate that their exponent is the same as that governing the unusual scaling corrections of the mutual information between the two blocks. Monte Carlo data fully match the theoretical CFT prediction only in the asymptotic limit of infinite intervals. Oppositely, in the Heisenberg chain scaling corrections are smaller and, already at finite (moderately large) block sizes, Monte Carlo data are in excellent agreement with the asymptotic CFT result.

Entanglement spreading after a geometric quench in quantum spin chains

We investigate the entanglement spreading in the anisotropic spin-1/2 Heisenberg (XXZ) chain after a geometric quench. This corresponds to a sudden change of the geometry of the chain or, in the equivalent language of interacting fermions confined in a box trap, to a sudden increase of the trap size. The entanglement dynamics after the quench is associated with the ballistic propagation of a magnetization wavefront. At the free fermion point (XX chain), the von-Neumann entropy S_A exhibits several intriguing dynamical regimes. Specifically, at short times a logarithmic increase is observed, similar to local quenches. This is accurately described by an analytic formula that we derive from heuristic arguments. At intermediate times partial revivals of the short-time dynamics are superposed with a power-law increase S_A t^\alpha, with \alpha<1. Finally, at very long times a steady state develops with constant entanglement entropy. As expected, since the model is integrable, we find that the steady state is non thermal, although it exhibits extensive entanglement entropy. We also investigate the entanglement dynamics after the quench from a finite to the infinite chain (sudden expansion). While at long times the entanglement vanishes, we demonstrate that its relaxation dynamics exhibits a number of scaling properties. Finally, we discuss the short-time entanglement dynamics in the XXZ chain in the gapless phase. The same formula that describes the time dependence for the XX chain remains valid in the whole gapless phase.

Entanglement negativity via the replica trick: A quantum Monte Carlo approach

Motivated by recent developments in conformal field theory (CFT), we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. These quantities can serve as witnesses of criticality. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. These are more pronounced in the presence of boundaries. For large chains we find perfect agreement with CFT. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. Remarkably, the finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.

Negativity spectrum in 1D gapped phases of matter

We investigate the spectrum of the partial transpose (negativity spectrum) of two adjacent regions in gapped one-dimensional models. We show that, in the limit of large regions, the negativity spectrum is entirely reconstructed from the entanglement spectrum of the bipartite system. We exploit this result in the XXZ spin chain, for which the entanglement spectrum is known by means of the corner transfer matrix. We find that the negativity spectrum levels are equally spaced, the spacing being half that in the entanglement spectrum. Moreover, the degeneracy of the spectrum is described by elegant combinatorial formulas, which are related to the counting of integer partitions. We also derive the asymptotic distribution of the negativity spectrum. We provide exact results for the logarithmic negativity and for the moments of the partial transpose. They exhibit unusual scaling corrections in the limit

Entanglement spectrum degeneracy and the Cardy formula in 1+1 dimensional conformal field theories

\Delta\to1^+

Magnetic and glassy transitions in the square-lattice XY model with random phase shifts

with a corrections exponent which is the same as that for the Renyi entropies.

Rényi entropies of generic thermodynamic macrostates in integrable systems

We investigate the effect of a global degeneracy in the distribution of entanglement spectrum in conformal field theories in one spatial dimension. We relate the recently found universal expression for the entanglement hamiltonian to the distribution of the entanglement spectrum. The main tool to establish this connection is the Cardy formula. It turns out that the Affleck-Ludwig non-integer degeneracy, appearing because of the boundary conditions induced at the entangling surface, can be directly read from the entanglement spectrum distribution. We also clarify the effect of the non-integer degeneracy on the spectrum of the partial transpose, which is the central object for quantifying the entanglement in mixed states. We show that the exact knowledge of the entanglement spectrum in some integrable spin-chains provides strong analytical evidences corroborating our results.