We develop a method to calculate the bipartite entanglement entropy of quantum models, in the thermodynamic limit, using a Numerical Linked Cluster Expansion (NLCE) involving only rectangular clusters. It is based on exact diagonalization of all n × m rectangular clusters at the interface between entangled subsystems A and B. We use it to obtain the Renyi entanglement entropy of the two-dimensional transverse field Ising model, for arbitrary real Renyi index α. Extrapolating these results as a function of the order of the calculation, we obtain universal pieces of the entanglement entropy associated with lines and corners at the quantum critical point. They show NLCE to be one of the few methods capable of accurately calculating universal properties of arbitrary Renyi entropies at higher dimensional critical points.
In the presence of strong disorder and weak interactions, closed quantum systems can enter a many-body localized phase where the system does not conduct, does not equilibrate even for arbitrarily long times, and robustly violates quantum statistical mechanics. The starting point for such a many-body localized phase is usually taken to be an Anderson insulator where, in the limit of vanishing interactions, all degrees of freedom of the system are localized. Here, we instead consider a model where in the non-interacting limit, some degrees of freedom are localized while others remain delocalized. Such a system can be viewed as a model for a many-body localized system brought into contact with a small bath of a comparable number of degrees of freedom. We numerically and analytically study the effect of interactions on this system and find that generically, the entire system delocalizes. However, we find certain parameter regimes where results are consistent with localization of the entire system, an effect recently termed many-body proximity effect.
Tensor network algorithms have been remarkably successful solving a variety of problems in quantum many-body physics. However, algorithms to optimize two-dimensional tensor networks known as PEPS lack many of the aspects that make the seminal density matrix renormalization group (DMRG) algorithm so powerful for optimizing one-dimensional tensor networks known as matrix product states. We implement a framework for optimizing twodimensional PEPS tensor networks which includes all of steps that make DMRG so successful for optimizing one-dimension tensor networks. We present results for several 2D spin models and discuss possible extensions and applications.
Tensor networks impose a notion of geometry on the entanglement of a quantum system. In some cases, this geometry is found to reproduce key properties of holographic dualities, and subsequently much work has focused on using tensor networks as tractable models for holographic dualities. Conventionally, the structure of the network -and hence the geometry -is largely fixed a priori by the choice of tensor network ansatz. Here, we evade this restriction and describe an unbiased approach that allows us to extract the appropriate geometry from a given quantum state. We develop an algorithm that iteratively finds a unitary circuit that transforms a given quantum state into an unentangled product state. We then analyze the structure of the resulting unitary circuits. In the case of non-interacting, critical systems in one dimension, we recover signatures of scale invariance in the unitary network, and we show that appropriately defined geodesic paths between physical degrees of freedom exhibit known properties of a hyperbolic geometry.Tensor networks have proven to be a powerful and universal tool to describe quantum states. Originating as variational ansatz states for low-dimensional quantum systems, they have become a common language between condensed matter and quantum information theory. More recently, the realization that some key properties of holographic dualities [1][2][3][4][5] are reproduced in certain classes of tensor network states (TNS) [6,7] has led to new connections to quantum gravity. In particular, many questions about holographic dualities appear more tractable in TN models [8][9][10][11][12][13][14][15][16][17][18]. The study of the geometry of TN states underlies these developments. Here, the physical legs of the network represent the boundary of some emergent "holographic" space that is occupied by the TN. While in networks such as matrix-product states (MPS) [19][20][21] and projected entangled-pair states (PEPS) [22][23][24] this space just reflects the physical geometry, other networks -such as the multi-scale entanglement renormalization ansatz (MERA) [25,26] -can have non-trivial geometry in this space [7]. We will refer to this geometry as "entanglement geometry".In this paper, we investigate whether this entanglement geometry can be extracted from a given quantum state without pre-imposing a particular structure on the TN [27]. We first describe a greedy, iterative algorithm that, given a quantum state, finds a 2-local unitary circuit that transforms this state into an unentangled (product) state (see Fig. 1). Such circuits, composed from unitary operators acting on two sites (which are not necessarily spatially close to each other), can be viewed as a particular class of TNS where the tensors are the unitary operators that form the circuit.We then develop a framework for analyzing the geometry of these circuits. First, we introduce a locally computable notion of distance between two points in the circuit, thus inducing a geometry in the bulk. We then focus on a particular property of thi...
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