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arXiv:1412.0732 · Physical Review Letters

Tensor Network Renormalization

Glen Evenbly, Guifré Vidal

Published Oct 29, 2015• 362 citations
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Abstract

We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.

DOI
10.1103/physrevlett.115.180405

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We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.
We introduce a tensor renormalization group scheme for coarse graining a two-dimensional tensor network that can be successfully applied to both classical and quantum systems on and off criticality. The key innovation in our scheme is to deform a 2D tensor network into small loops and then optimize the tensors on each loop. In this way, we remove short-range entanglement at each iteration step and significantly improve the accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical Ising model and a frustrated 2D quantum model.
We discuss in detail algorithms for implementing tensor network renormalization (TNR) for the study of classical statistical and quantum many-body systems. First, we recall established techniques for how the partition function of a 2D2D2D classical many-body system or the Euclidean path integral of a 1D1D1D quantum system can be represented as a network of tensors, before describing how TNR can be implemented to efficiently contract the network via a sequence of coarse-graining transformations. The efficacy of the TNR approach is then benchmarked for the 2D2D2D classical statistical and 1D1D1D quantum Ising models; in particular the ability of TNR to maintain a high level of accuracy over sustained coarse-graining transformations, even at a critical point, is demonstrated.
We introduce a new coarse-graining algorithm, tensor network skeletonization, for the numerical computation of tensor networks. This approach utilizes a structure-preserving skeletonization procedure to remove short-range entanglements effectively at every scale. This approach is first presented in the setting of a two-dimensional (2D) statistical Ising model and is then extended to higher-dimensional tensor networks and disordered systems. When applied to the Euclidean path integral formulation, this approach also gives rise to new efficient representations of the ground states for 1D and 2D quantum Ising models.
We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors Tinv{T}_{\text{inv}}Tinv​ plus the symmetry group Gsym{G}_{\text{sym}}Gsym​ of the tensors (i.e., the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional (2D) statistical Ising model, 2D statistical loop-gas model, and 1+1D1+1\text{D}1+1D quantum spin-1/2 and spin-1 models. In particular, using such a (Gsym,Tinv)({G}_{\text{sym}},{T}_{\text{inv}})(Gsym​,Tinv​) characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry-protected topological phase. The (Gsym,Tinv)({G}_{\text{sym}},{T}_{\text{inv}})(Gsym​,Tinv​) characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.
The study of the ground states of local Hamiltonians in quantum many-body and quantum field theoretic systems is a source of many research problems of great complexity. Almost in its entirety, this thesis deals with constructions related to the field of tensor networks, which are efficient parametrizations of mathematical objects from quantum many-body theory (such as quantum states and operators) that exploit our knowledge of their entanglement structures in order to obtain improvements in our computational abilities. In the past decade, the new research program of continuous tensor networks has arisen with the goal of reproducing, in the setting of quantum field theory, the success that tensor network techniques have enjoyed on the lattice. First, we present a number of results from research performed on non-interacting examples of the continuous Multiscale Entanglement Renormalization Ansatz, or cMERA, a continuous tensor network that provides a variational wavefunctional for a quantum field theoretic ground state. Initially, we study the correlations and entanglement structure of cMERA states, providing evidence that such states can be interpreted as UV regularized versions of the physical states they approximate. After that, we study what modifications of the formalism are necessary in order to consider systems with gauge symmetry, or in the presence of boundaries and defects. We obtain prescriptions for the treatment of these cases which, we expect, may also hold in the interacting setting. We then abandon briefly the area of continuous tensor networks to present a piece of research within a different though related research program: that of the study of manifestations of universal emergent behaviour in critical lattice systems. In particular we bring attention to the existence of an approximate representation of the Virasoro algebra supported on the eigenvectors of the reduced density matrices of critical lattice systems. We call this the entanglement algebra, and study its accuracy in the particular case of the Ising model. Lastly, we present a proposal for a continuous version of the Tensor Network Renormalization (TNR) algorithm, which we dub continuous TNR, or cTNR. cTNR operates on UV regularized classical partition functions or quantum Euclidean path integrals, and generates a real-space renormalization group flow via continuous coarse-graining and rescaling operations. We show that a UV regularized version of the free boson path integral can be made a fixed point of such a renormalization flow, in a way that allows for recovery of the conformal data associated to the free boson conformal field theory.
On the lattice, a renormalization group (RG) flow for two-dimensional partition functions expressed as a tensor network can be obtained using the tensor network renormalization (TNR) algorithm [G. Evenbly, G. Vidal, Phys. Rev. Lett. 115 (18), 180405 (2015)]. In this work we explain how to extend TNR to field theories in the continuum. First, a short-distance length scale 1/Λ1/Λ1/Λ is introduced in the continuum partition function by smearing the fields. The resulting object is still defined in the continuum but has no fluctuations at distances shorter than 1/Λ1/Λ1/Λ. An infinitesimal coarse-graining step is then generated by the combined action of a rescalingrescalingrescaling operator LLL and a disentanglingdisentanglingdisentangling operator KKK that implements a quasi-local field redefinition. As demonstrated for a free boson in two dimensions, continuous TNR exactly preserves translation and rotation symmetries and can generate a proper RG flow. Moreover, from a critical fixed point of this RG flow one can then extract the conformal data of the underlying conformal field theory.