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arXiv:1707.05770

Implicitly disentangled renormalization

Glen Evenbly

Published Jul 18, 2017• 1 citation
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Abstract

We propose a new implementation of real-space renormalization group (RG) transformations for quantum states on a lattice. Key to this approach is the removal of short-ranged entanglement, similar to Vidal's entanglement renormalization (ER), which allows a proper RG flow to be achieved. However, our proposal only uses operators that act locally within each block, such that the use of disentanglers acting across block boundaries is not required. By avoiding the use of disentanglers we argue many tensor network algorithms for studying quantum many-body systems can be significantly improved. The effectiveness of this RG approach is demonstrated through application to the ground state of a 1D system at criticality, which is shown to reach a scale-invariant fixed point.

DOI
10.48550/arxiv.1707.05770

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We propose a new implementation of real-space renormalization group (RG) transformations for quantum states on a lattice. Key to this approach is the removal of short-ranged entanglement, similar to Vidal's entanglement renormalization (ER), which allows a proper RG flow to be achieved. However, our proposal only uses operators that act locally within each block, such that the use of disentanglers acting across block boundaries is not required. By avoiding the use of disentanglers we argue many tensor network algorithms for studying quantum many-body systems can be significantly improved. The effectiveness of this RG approach is demonstrated through application to the ground state of a 1D system at criticality, which is shown to reach a scale-invariant fixed point.
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 propose a real-space renormalization group (RG) transformation for quantum systems on a D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining it into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the entanglement of a block.
We propose a real-space renormalization group method to explicitly decouple into independent components a many-body system that, as in the phenomenon of spin-charge separation, exhibits separation of degrees of freedom at low energies. Our approach produces a branching holographic description of such systems that opens the path to the efficient simulation of the most entangled phases of quantum matter, such as those whose ground state violates a boundary law for entanglement entropy. As in the coarse-graining transformation of Vidal [Phys. Rev. Lett. 99, 220405 (2007)], the key ingredient of this decoupling transformation is the concept of entanglement renormalization, or removal of short-range entanglement. We demonstrate the feasibility of the approach, both analytically and numerically, by decoupling in real space the ground state of a critical quantum spin chain into two. Generalized notions of renormalization group flow and of scale invariance are also put forward.
The ability of entanglement renormalization (ER) to generate a proper real-space renormalization group (RG) flow in extended quantum systems is analyzed in the setting of harmonic lattice systems in D =1 and 2 spatial dimensions. A conceptual overview of the steps involved in momentum-space RG is provided and contrasted against the equivalent steps in the real-space setting. The real-space RG flow, as generated by ER, is compared against the exact results from momentum-space RG, including an investigation of a critical fixed point and the effect of relevant and irrelevant perturbations.
We make Kadanoff's block idea into a reliable three-dimensional (3D) real-space renormalization group (RG) method. Kadanoff's idea, expressed in spin representation, offers a qualitative intuition for clarifying scaling behavior in criticality, but has difficulty as a quantitative tool due to uncontrolled approximations. A tensor-network reformulation equips the block idea with a measure of RG errors. In 3D, we propose an entanglement filtering scheme to enhance such a block-tensor map, with the lattice reflection symmetry exploited. When the proposed RG is applied to the cubic-lattice Ising model, the RG errors are reduced to about 2% by retaining more couplings. The estimated scaling dimensions of the two relevant fields have errors 0.4% and 0.1% in the best case, compared with the accepted values. The proposed RG is promising as a systematically improvable real-space RG method in 3D. The unique feature of our method is its ability to numerically obtain a 3D critical fixed point in a high-dimensional tensor space. A fixed-point tensor contains much more information than a handful of observables estimated in conventional techniques for analyzing critical systems.
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.