Tensor Network Renormalization

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 PACS numbers: 05.30.-d, 02.70.-c, 03.67.Mn, 75.10.Jm Understanding emergent phenomena in many-body systems remains one of the major challenges of modern physics. With sufficient knowledge of the microscopic degrees of freedom and their interactions, we can write the partition function of a classical system, namely, a weighted sum of all of its microscopic configurations, or the analogous Euclidean path integral of a quantum many-body system, where the weighted sum is now over all conceivable trajectories. These objects contain complete information on the collective properties of the many-body system. However, evaluating partition functions or Euclidean path integrals is generically very hard. Kadanoff's spin-blocking procedure [1] opened the path to nonperturbative approaches based on coarse graining a lattice [2,3]. More recently, Levin and Nave proposed the tensor renormalization group (TRG) [4], a versatile realspace coarse-graining transformation for 2D classical partition functions-or, equivalently, Euclidean path integrals of 1D quantum systems.The TRG is an extremely useful approach that has revolutionized how we coarse-grain lattice models [4][5][6][7][8][9]. However, this method fails to remove part of the shortrange correlations in the partition function and, as a result, the coarse-grained system still contains irrelevant microscopic information. Conceptually, this is in conflict with the very spirit of the renormalization group (RG) and results in a RG flow with the wrong structure of noncritical fixed points, as discussed in Ref. [7]. Computationally, the accumulation of short-range correlations over successive TRG coarse-graining transformation also has important consequences: as pointed out by Levin and Nave, it implies the breakdown of the TRG at criticality [4,10] (although universal information, such as critical exponents, can still be obtained from finite systems).An analogous problem, faced earlier in the context of ground state wave functions, was resolved with the introduction of entanglement renormalization techniques [18,19]. In this L...

Horizon as critical phenomenon

“…For other approaches of entanglement based renormalization group, see refs. [21][22][23][24]. In the present case, the disentangler e −zĤ is a non-unitary operator.…”

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

“…Instead, as we will see later in this section, our optimization changes the tensor network structures as in tensor network renormalization [23,24].…”

“…Essentially, we reformulate the conjectured relation between tensor networks and AdS/CFT from the viewpoint of Euclidean path-integrals. Indeed, the method called tensor network renormalization (TNR) [23,24] shows that an Euclidean path-integral computation of a ground state wave function can be regarded as a tensor network description of MERA. In this argument, one first discretizes the path-integral into a lattice version and rewrites it as a tensor network.…”

“…(2.9) is partly motivated by an interesting connection between the tensor network renormalization (TNR) [23,24] and our optimization procedure of Euclidean path-integral. This is because the number of tensors in TNR is an estimation of complexity and the Liouville action has a desired property in this sense, e.g.…”

“…[45,46]). The optimization of tensor network was introduced in [23,24], called tensor network renormalization. We are now considering a path-integral counterpart of the same optimization here.…”