Volume-law Scaling Research Articles

Exact rainbow tensor networks for the colorful Motzkin and Fredkin spin chains

We present bulk tensor networks that exactly represent the ground states of a continuous family of one-dimensional frustration-free Hamiltonians. These states, which are known as area-deformed Motzkin and Fredkin states, exhibit a novel quantum phase transition. By tuning a single parameter, they go from a phase obeying an area law to a highly entangled rainbow phase, where the half-chain entropy scales with the volume. Using the representation of these ground states as superpositions of random walks, we introduce tensor networks for these ground states where local and global rules of the walker are baked into bulk tensors, thereby providing an efficient description of the ground states (some of which satisfy a volume law scaling of entanglement entropy).

Volume Law and Quantum Criticality in the Entanglement Entropy of Excited Eigenstates of the Quantum Ising Model.

Much has been learned about universal properties of entanglement entropies in ground states of quantum many-body lattice systems. Here we unveil universal properties of the average bipartite entanglement entropy of eigenstates of the paradigmatic quantum Ising model in one dimension. The leading term exhibits a volume-law scaling that we argue is universal for translationally invariant quadratic models. The subleading term is constant at the critical field for the quantum phase transition and vanishes otherwise (in the thermodynamic limit); i.e., the critical field can be identified from subleading corrections to the average (over all eigenstates) entanglement entropy.

Many-body localization-delocalization transition in the quantum Sherrington-Kirkpatrick model

We analyze many-body localization (MBL) to delocalization transition in Sherrington-Kirkpatrick (SK) model of Ising spin glass (SG) in the presence of a transverse field $\Gamma$. Based on energy resolved analysis, which is of relevance for a closed quantum system, we show that the quantum SK model has many-body mobility edges separating MBL phase which is non-ergodic and non-thermal from the delocalized phase which is ergodic and thermal. The range of the delocalized regime increases with increase in the strength of $\Gamma$ and eventually for $\Gamma$ larger than $\Gamma_{CP}$ the entire many-body spectrum is delocalized. We show that the Renyi entropy is almost independent of the system size in the MBL phase, hinting towards an area law in this infinite range model while the delocalized phase shows volume law scaling of Renyi entropy. We further obtain spin glass transition curve in energy density $\epsilon$-$\Gamma$ plane from the collapse of eigenstate spin susceptibility. We demonstrate that in most of the parameter regime SG transition occurs close to the MBL transition indicating that the SG phase is non-ergodic and non-thermal while the paramagnetic phase is delocalized and thermal.

Entanglement entropy in the spinless free fermion model and its application to the graph isomorphism problem

In this paper, the entanglement entropy is investigated in the ground state of a spinless free fermion Hamiltonian where its hopping matrix is given by the adjacency matrix of the graph. The bipartite entanglement entropy is calculated from the eigenvalues of the correlation matrix. The entanglement entropy and mutual information are calculated for some scalable graphs including a complete graph and three sets of strongly regular graphs (SRGs). In addition, the volume law scaling has been shown for these graphs. Then, we use the entanglement entropy as a tool for studying the graph isomorphism problem in some non-isomorphic pairs of graphs in order to distinguish them from each other. Two graphs are isomorphic when they are related to each other by relabeling the graph vertices. We study the graph isomorphism in non-isomorphic SRGs, distance-regular graphs and pseudo distance-regular graphs. Most pairs of non-isomorphic SRGs can be distinguished from each other by using fermionic entanglement entropy.

Entanglement entropy of free fermions on directed graphs

In this paper, we study the general spinless quadratic fermion Hamiltonian with interaction matrices given by the symmetric and antisymmetric parts of the adjacency matrix of a directed graph. The correlation matrix and entanglement entropy are provided for the ground state of the Hamiltonian, analytically. We also show that a volume law scaling holds for some scalable sets of nonsymmetric association-scheme graphs. The scaling of the entanglement entropy is then used as a tool for studying the graph isomorphism problem, in particular to distinguish some nonisomorphic pairs of directed strongly regular graphs.

Quantum Entanglement in Neural Network States

Machine learning, one of today's most rapidly growing interdisciplinary fields, promises an unprecedented perspective for solving intricate quantum many-body problems. Understanding the physical aspects of the representative artificial neural-network states is recently becoming highly desirable in the applications of machine learning techniques to quantum many-body physics. Here, we study the quantum entanglement properties of neural-network states, with a focus on the restricted-Boltzmann-machine (RBM) architecture. We prove that the entanglement of all short-range RBM states satisfies an area law for arbitrary dimensions and bipartition geometry. For long-range RBM states we show by using an exact construction that such states could exhibit volume-law entanglement, implying a notable capability of RBM in representing efficiently quantum states with massive entanglement. We further examine generic RBM states with random weight parameters. We find that their averaged entanglement entropy obeys volume-law scaling and meantime strongly deviates from the Page-entropy of the completely random pure states. We show that their entanglement spectrum has no universal part associated with random matrix theory and bears a Poisson-type level statistics. Using reinforcement learning, we demonstrate that RBM is capable of finding the ground state (with power-law entanglement) of a model Hamiltonian with long-range interaction. In addition, we show, through a concrete example of the one-dimensional symmetry-protected topological cluster states, that the RBM representation may also be used as a tool to analytically compute the entanglement spectrum. Our results uncover the unparalleled power of artificial neural networks in representing quantum many-body states, which paves a novel way to bridge computer science based machine learning techniques to outstanding quantum condensed matter physics problems.

Entanglement entropy in a periodically driven quantum Ising ring

We numerically study the dynamics of entanglement entropy, induced by an oscillating time periodic driving of the transverse field, h(t), of a one-dimensional quantum Ising chain. We consider several realizations of h(t), and we find a number of results in analogy with entanglement entropy dynamics induced by a sudden quantum quench. After short-time relaxation, the dynamics of entanglement entropy synchronises with h(t), displaying an oscillatory behaviour at the frequency of the driving. Synchronisation in the dynamics of entanglement entropy, is spoiled by the appearance of quasi-revivals which fade out in the thermodynamic limit, and which we interpret using a quasi-particle picture adapted to periodic drivings. Taking the time-average of the entanglement entropy in the synchronised regime, we find that it obeys a volume law scaling with the subsystem's size. Such result is reminiscent of a thermal state or of a Generalised Gibbs ensemble of a quenched Ising chain, although the system does not heat up towards infinite temperature as a consequence of the integrability of the model.

Many-Body Localization and Quantum Nonergodicity in a Model with a Single-Particle Mobility Edge.

We investigate many-body localization in the presence of a single-particle mobility edge. By considering an interacting deterministic model with an incommensurate potential in one dimension we find that the single-particle mobility edge in the noninteracting system leads to a many-body mobility edge in the corresponding interacting system for certain parameter regimes. Using exact diagonalization, we probe the mobility edge via energy resolved entanglement entropy (EE) and study the energy resolved applicability (or failure) of the eigenstate thermalization hypothesis (ETH). Our numerical results indicate that the transition separating area and volume law scaling of the EE does not coincide with the nonthermal to thermal transition. Consequently, there exists an extended nonergodic phase for an intermediate energy window where the many-body eigenstates violate the ETH while manifesting volume law EE scaling. We also establish that the model possesses an infinite temperature many-body localization transition despite the existence of a single-particle mobility edge. We propose a practical scheme to test our predictions in atomic optical lattice experiments which can directly probe the effects of the mobility edge.

Eigenstate thermalization hypothesis and integrability in quantum spin chains

We investigate the eigenstate thermalization hypothesis (ETH) in integrable models, focusing on the $\text{spin}\ensuremath{-}\frac{1}{2}$ isotropic Heisenberg $(XXX)$ chain. We provide numerical evidence that the 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 ${L}^{\ensuremath{-}1/2}$, with $L$ the size of the chain. This is in contrast with the exponential decay that is found in generic nonintegrable 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 midspectrum eigenstates exhibit extensive entanglement entropy (i.e., volume-law scaling).