arXiv subjects

Vincenzo Alba

Publications and source records attributed to Vincenzo Alba.

At least 73 records · Page 4Linked to original sources

arXiv2017

Out-of-equilibrium protocol for Rényi entropies via the Jarzynski equality

In recent years entanglement measures, such as the von Neumann and the Rényi entropies, provided a unique opportunity to access elusive feature of quantum many-body systems. However, extracting entanglement properties analytically, experimentally, or in numerical simulations can be a formidable task. Here, by combining the replica trick and the Jarzynski equality we devise a new effective out-of-equilibrium protocol for measuring the equilibrium Rényi entropies. The key idea is to perform a quench in the geometry of the replicas. The Rényi entropies are obtained as the exponential average of the work performed during the quench. We illustrate an application of the method in classical Monte Carlo simulations, although it could be useful in different contexts, such as in Quantum Monte Carlo, or experimentally in cold-atom systems. The method is most effective in the quasi-static regime, i.e., for a slow quench, where it allows to obtain the Rényi entropies in a single realization of the protocol. As a benchmark, we present results for the Rényi entropies in the Ising universality class in $1$$+$$1$ dimensions, which are found in perfect agreement with the well-known Conformal Field Theory (CFT) predictions.

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arXiv2017

Prethermalization at Low Temperature: the Scent of Long-Range Order

Non-equilibrium time evolution in isolated many-body quantum systems generally results in thermalization. However, the relaxation process can be very slow, and quasi-stationary non-thermal plateaux are often observed at intermediate times. The paradigmatic example is a quantum quench in an integrable model with weak integrability breaking; for a long time, the state can not escape the constraints imposed by the approximate integrability. We unveil a new mechanism of prethermalization, based on the presence of a symmetry of the pre-quench Hamiltonian, which is spontaneously broken at zero temperature and is explicitly broken by the post-quench Hamiltonian. The typical time scale of the phenomenon is proportional to the thermal correlation length of the initial state, which diverges as the temperature is lowered. We show that the prethermal quasi-stationary state can be approximated by a mixed state that violates cluster decomposition property. We consider two examples: the transverse-field Ising chain, where the full time evolution is computed analytically, and the (non integrable) ANNNI model, which is investigated numerically.

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arXiv2017

Entanglement and thermodynamics after a quantum quench in integrable systems

Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics, respectively. In the last decade the study of quantum quenches revealed that these two concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure initial state maintains the system globally at zero entropy, at long time after the quench local properties are captured by an appropriate statistical ensemble with non zero thermodynamic entropy, which can be interpreted as the entanglement accumulated during the dynamics. Therefore, understanding the post-quench entanglement evolution unveils how thermodynamics emerges in isolated quantum systems. An exact computation of the entanglement dynamics has been provided only for non-interacting systems, and it was believed to be unfeasible for genuinely interacting models. Conversely, here we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrability-based knowledge of the asymptotic state, leads to a complete analytical understanding of the entanglement dynamics in the space-time scaling limit. Our framework requires only knowledge about the steady state, and the velocities of the low-lying excitations around it. We provide a thorough check of our result focusing on the spin-1/2 Heisenberg XXZ chain, and considering quenches from several initial states. We compare our results with numerical simulations using both tDMRG and iTEBD, finding always perfect agreement.

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arXiv2017

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 $Δ\to1^+$ with a corrections exponent which is the same as that for the Rényi entropies.

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arXiv2016

Negativity spectrum of one-dimensional conformal field theories

The partial transpose $ρ_A^{T_2}$ of the reduced density matrix $ρ_A$ is the key object to quantify the entanglement in mixed states, in particular through the presence of negative eigenvalues in its spectrum. Here we derive analytically the distribution of the eigenvalues of $ρ_A^{T_2}$, that we dub negativity spectrum, in the ground sate of gapless one-dimensional systems described by a Conformal Field Theory (CFT), focusing on the case of two adjacent intervals. We show that the negativity spectrum is universal and depends only on the central charge of the CFT, similarly to the entanglement spectrum. The precise form of the negativity spectrum depends on whether the two intervals are in a pure or mixed state, and in both cases, a dependence on the sign of the eigenvalues is found. This dependence is weak for bulk eigenvalues, whereas it is strong at the spectrum edges. We also investigate the scaling of the smallest (negative) and largest (positive) eigenvalues of $ρ_A^{T_2}$. We check our results against DMRG simulations for the critical Ising and Heisenberg chains, and against exact results for the harmonic chain, finding good agreement for the spectrum, but showing that the smallest eigenvalue is affected by very large scaling corrections.

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arXiv2016

The entanglement negativity in random spin chains

We investigate the logarithmic negativity in strongly-disordered spin chains in the random-singlet phase. We focus on the spin-1/2 random Heisenberg chain and the random XX chain. We find that for two arbitrary intervals the disorder-averaged negativity and the mutual information are proportional to the number of singlets shared between the two intervals. Using the strong-disorder renormalization group (SDRG), we prove that the negativity of two adjacent intervals grows logarithmically with the intervals length. In particular, the scaling behavior is the same as in conformal field theory, but with a different prefactor. For two disjoint intervals the negativity is given by a universal simple function of the cross ratio, reflecting scale invariance. As a function of the distance of the two intervals, the negativity decays algebraically in contrast with the exponential behavior in clean models. We confirm our predictions using a numerical implementation of the SDRG method. Finally, we also implement DMRG simulations for the negativity in open spin chains. The chains accessible in the presence of strong disorder are not sufficiently long to provide a reliable confirmation of the SDRG results.

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arXiv2016

The quench action approach in finite integrable spin chains

We consider the problem of constructing the stationary state following a quantum quench, using the exact overlaps for finite size integrable models. We focus on the isotropic Heisenberg spin chain with initial state Néel or Majumdar-Ghosh (dimer), although the proposed approach is valid for an arbitrary integrable model. We consider only eigenstates which do not contain zero-momentum strings because the latter are affected by fictitious singularities that are very difficult to take into account. We show that the fraction of eigenstates that do not contain zero-momentum strings is vanishing in the thermodynamic limit. Consequently, restricting to this part of the Hilbert space leads to vanishing expectation values of local observables. However, it is possible to reconstruct the asymptotic values by properly reweighting the expectations in the considered subspace, at the price of introducing finite-size corrections. We also develop a Monte Carlo sampling of the Hilbert space which allows us to study larger systems. We accurately reconstruct the expectation values of the conserved charged and the root distributions in the stationary state, which turn out to match the exact thermodynamic results. The proposed method can be implemented even in cases in which an analytic thermodynamic solution is not obtainable.

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arXiv2016

The classical mutual information in mean-field spin glass models

We investigate the classical Renyi entropy S_n and the associated mutual information I_n in the Sherrington-Kirkpatrick (S-K) model, which is the paradigm model of mean-field spin glasses. Using classical Monte Carlo simulations and analytical tools we investigate the S-K model on the n-sheets booklet. This is obtained by gluing together n independent copies of the model, and it is the main ingredient to construct the Renyi entanglement-related quantities. We find a glassy phase at low temperature, whereas at high temperature the model exhibits paramagnetic behavior, consistent with the regular S-K model. The temperature of the paramagnetic-glassy transition depends non-trivially on the geometry of the booklet. At high-temperatures we provide the exact solution of the model by exploiting the replica symmetry. This is the permutation symmetry among the fictitious replicas that are used to perform disorder averages (via the replica trick). In the glassy phase the replica symmetry has to be broken. Using a generalization of the Parisi solution, we provide analytical results for S_n and I_n, and for standard thermodynamic quantities. Both S_n and I_n exhibit a volume law in the whole phase diagram. We characterize the behavior of the corresponding densities S_n/N, I_n/N, in the thermodynamic limit. Interestingly, at the critical point the mutual information does not exhibit any crossing for different system sizes, in contrast with local spin models.

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arXiv2015

Overlap distributions for quantum quenches in the anisotropic Heisenberg chain

The dynamics after a quantum quench is determined by the weights of the initial state in the eigenspectrum of the final Hamiltonian, i.e., by the distribution of overlaps in the energy spectrum. We present an analysis of such overlap distributions for quenches of the anisotropy parameter in the one-dimensional anisotropic spin-1/2 Heisenberg model (XXZ chain). We provide an overview of the form of the overlap distribution for quenches from various initial anisotropies to various final ones, using numerical exact diagonalization. We show that if the system is prepared in the antiferromagnetic Néel state (infinite anisotropy) and released into a non-interacting setup (zero anisotropy, XX point) only a small fraction of the final eigenstates gives contributions to the post-quench dynamics, and that these eigenstates have identical overlap magnitudes. We derive expressions for the overlaps, and present the selection rules that determine the final eigenstates having nonzero overlap. We use these results to derive concise expressions for time-dependent quantities (Loschmidt echo, longitudinal and transverse correlators) after the quench. We use perturbative analyses to understand the overlap distribution for quenches from infinite to small nonzero anisotropies, and for quenches from large to zero anisotropy.

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arXiv2015

Simulating the Generalized Gibbs Ensemble (GGE): a Hilbert space Monte Carlo approach

By combining classical Monte Carlo and Bethe ansatz techniques we devise a numerical method to construct the Truncated Generalized Gibbs Ensemble (TGGE) for the spin-1/2 isotropic Heisenberg ($XXX$) chain. The key idea is to sample the Hilbert space of the model with the appropriate GGE probability measure. The method can be extended to other integrable systems, such as the Lieb-Liniger model. We benchmark the approach focusing on GGE expectation values of several local observables. As finite-size effects decay exponentially with system size, moderately large chains are sufficient to extract thermodynamic quantities. The Monte Carlo results are in agreement with both the Thermodynamic Bethe Ansatz (TBA) and the Quantum Transfer Matrix approach (QTM). Remarkably, it is possible to extract in a simple way the steady-state Bethe-Gaudin-Takahashi (BGT) roots distributions, which encode complete information about the GGE expectation values in the thermodynamic limit. Finally, it is straightforward to simulate extensions of the GGE, in which, besides the local integral of motion (local charges), one includes arbitrary functions of the BGT roots. As an example, we include in the GGE the first non-trivial quasi-local integral of motion.

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arXiv2015

Eigenstate thermalization hypothesis (ETH) and integrability in quantum spin chains

We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the spin-1/2 isotropic Heisenberg (XXX) chain. We provide numerical evidence that ETH holds for typical eigenstates (weak ETH scenario). Specifically, using a numerical implementation of state-of-the-art Bethe ansatz results, we study the finite-size scaling of the eigenstate-to-eigenstate fluctuations of the reduced density matrix. We find that fluctuations are normally distributed, and their standard deviation decays in the thermodynamic limit as L^{-1/2}, with L the size of the chain. This is in contrast with the exponential decay that is found in generic non-integrable systems. Based on our results, it is natural to expect that this scenario holds in other integrable spin models and for typical local observables. Finally, we investigate the entanglement properties of the excited states of the XXX chain. We numerically verify that typical mid-spectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).

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arXiv2015

Phase diagram of the J1-J2 Heisenberg model on the kagome lattice

We perform an extensive density matrix renormalization group (DMRG) study of the ground-state phase diagram of the spin-1/2 J_1-J_2 Heisenberg model on the kagome lattice. We focus on the region of the phase diagram around the kagome Heisenberg antiferromagnet, i.e., at J_2=0. We investigate the static spin structure factor, the magnetic correlation lengths, and the spin gaps. Our results are consistent with the absence of magnetic order in a narrow region around J_2\approx 0, although strong finite-size effects do not allow us to accurately determine the phase boundaries. This result is in agreement with the presence of an extended spin-liquid region, as it has been proposed recently. Outside the disordered region, we find that for ferromagnetic and antiferromagnetic J_2 the ground state displays signatures of the magnetic order of the \sqrt{3}\times\sqrt{3} and the q=0 type, respectively. Finally, we focus on the structure of the entanglement spectrum (ES) in the q=0 ordered phase. We discuss the importance of the choice of the bipartition on the finite-size structure of the ES.

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arXiv2014

Bound states and entanglement in the excited states of quantum spin chains

We investigate entanglement properties of the excited states of the spin-1/2 Heisenberg (XXX) chain with isotropic antiferromagnetic interactions, by exploiting the Bethe ansatz solution of the model. We consider eigenstates obtained from both real and complex solutions ("strings") of the Bethe equations. Physically, the former are states of interacting magnons, whereas the latter contain bound states of groups of particles. We first focus on the situation with few particles in the chain. Using exact results and semiclassical arguments, we derive an upper bound S_MAX for the entanglement entropy. This exhibits an intermediate behavior between logarithmic and extensive, and it is saturated for highly-entangled states. As a function of the eigenstate energy, the entanglement entropy is organized in bands. Their number depends on the number of blocks of contiguous Bethe-Takahashi quantum numbers. In presence of bound states a significant reduction in the entanglement entropy occurs, reflecting that a group of bound particles behaves effectively as a single particle. Interestingly, the associated entanglement spectrum shows edge-related levels. At finite particle density, the semiclassical bound S_MAX becomes inaccurate. For highly-entangled states S_A\propto L_c, with L_c the chord length, signaling the crossover to extensive entanglement. Finally, we consider eigenstates containing a single pair of bound particles. No significant entanglement reduction occurs, in contrast with the few-particle case.

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arXiv2014

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^α, with α<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.

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arXiv2013

Bethe ansatz description of edge-localization in the open-boundary XXZ spin chain

At large values of the anisotropy Δ, the open-boundary Heisenberg spin-1/2 chain has eigenstates displaying localization at the edges. We present a Bethe ansatz description of this `edge-locking' phenomenon in the entire Δ>1 region. We focus on the simplest spin sectors, namely the highly polarized sectors with only one or two overturned spins, i.e., one-particle and two-particle sectors. Edge-locking is associated with pure imaginary solutions of the Bethe equations, which are not commonly encountered in periodic chains. In the one-particle case, at large anisotropies there are two eigenstates with imaginary Bethe momenta, related to localization at the two edges. For any finite chain size, one of the two solutions become real as the anisotropy is lowered below a certain value. For two particles, a richer scenario is observed, with eigenstates having the possibility of both particles locked on the same or different edge, one locked and the other free, and both free either as single magnons or as bound composites corresponding to `string' solutions. For finite chains, some of the edge-locked spins get delocalized at certain values of the anisotropy (`exceptional points'), corresponding to imaginary solutions becoming real. We characterize these phenomena thoroughly by providing analytic expansions of the Bethe momenta for large chains, large anisotropy, and near the exceptional points. In the large-chain limit all the exceptional points coalesce at the isotropic point (Δ=1) and edge-locking becomes stable in the whole Δ>1 region.

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arXiv2013

Entanglement negativity via 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.

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arXiv2013

Entanglement spectrum of the two dimensional Bose-Hubbard model

We study the entanglement spectrum (ES) of the Bose-Hubbard model on the two dimensional square lattice at unit filling, both in the Mott insulating and in the superfluid phase. In the Mott phase, we demonstrate that the ES is dominated by the physics at the boundary between the two subsystems. On top of the boundary-local (perturbative) structure, the ES exhibits substructures arising from one-dimensional dispersions along the boundary. In the superfluid phase, the structure of the ES is qualitatively different, and reflects the spontaneously broken U(1) symmetry of the phase. We attribute the basic low-lying structure to a so-called "tower of states" (TOS) Hamiltonian of the model. We then discuss how these characteristic structures evolve across the superfluid to Mott insulator transition and their influence on the behavior of the entanglement entropies. Finally, we briefly outline the implications of the ES structure on the efficiency of matrix-product-state based algorithms in two dimensions.

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arXiv2013

Entanglement negativity and conformal field theory: a Monte Carlo study

We investigate the behavior of the moments of the partially transposed reduced density matrix ρ^{T_2}_A in critical quantum spin chains. Given subsystem A as union of two blocks, this is the (matrix) transposed of ρ_A with respect to the degrees of freedom of one of the two. This is also the main ingredient for constructing the logarithmic negativity. We provide a new numerical scheme for calculating efficiently all the moments of ρ_A^{T_2} 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 CFT calculations. For disjoint ones, in the Ising chain finite size corrections are non negligible. We demonstrate that their exponent is the same 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.

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