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arXiv:1310.4507 · Physical Review B

Bifurcation in entanglement renormalization group flow of a gapped spin model

Jeongwan Haah

Published Feb 18, 2014• 66 citations
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Abstract

We study entanglement renormalization group transformations for the ground states of a spin model, called cubic code model ${H}_{A}$ in three dimensions, in order to understand long-range entanglement structure. The cubic code model has degenerate and locally indistinguishable ground states under periodic boundary conditions. In the entanglement renormalization, one applies local unitary transformations on a state, called disentangling transformations, after which some of the spins are completely disentangled from the rest and then discarded. We find a disentangling unitary to establish equivalence of the ground state of ${H}_{A}$ on a lattice of lattice spacing $a$ to the tensor product of ground spaces of two independent Hamiltonians ${H}_{A}$ and ${H}_{B}$ on lattices of lattice spacing $2a$. We further find a disentangling unitary for the ground space of ${H}_{B}$ with the lattice spacing $a$ to show that it decomposes into two copies of itself on the lattice of the lattice spacing $2a$. The disentangling transformations yield a tensor network description for the ground state of the cubic code model. Using exact formulas for the degeneracy as a function of system size, we show that the two Hamiltonians ${H}_{A}$ and ${H}_{B}$ represent distinct phases of matter.

DOI
10.1103/physrevb.89.075119

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We study entanglement renormalization group transformations for the ground states of a spin model, called cubic code model HA{H}_{A}HA​ in three dimensions, in order to understand long-range entanglement structure. The cubic code model has degenerate and locally indistinguishable ground states under periodic boundary conditions. In the entanglement renormalization, one applies local unitary transformations on a state, called disentangling transformations, after which some of the spins are completely disentangled from the rest and then discarded. We find a disentangling unitary to establish equivalence of the ground state of HA{H}_{A}HA​ on a lattice of lattice spacing aaa to the tensor product of ground spaces of two independent Hamiltonians HA{H}_{A}HA​ and HB{H}_{B}HB​ on lattices of lattice spacing 2a2a2a. We further find a disentangling unitary for the ground space of HB{H}_{B}HB​ with the lattice spacing aaa to show that it decomposes into two copies of itself on the lattice of the lattice spacing 2a2a2a. The disentangling transformations yield a tensor network description for the ground state of the cubic code model. Using exact formulas for the degeneracy as a function of system size, we show that the two Hamiltonians HA{H}_{A}HA​ and HB{H}_{B}HB​ represent distinct phases of matter.
Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long-range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry-breaking orders in a generic tensor-product state.
As a quantum-informative window into quantum many-body physics, the concept and application of the entanglement renormalization group (ERG) have been playing a vital role in the study of novel quantum phases of matter, especially long-range entangled (LRE) states in topologically ordered systems. For instance, by recursively applying local unitaries as well as adding and removing qubits that form product states, the 2D toric code ground states, i.e., the fixed point of Z2{\mathbb{Z}}_{2}Z2​ topological order, are efficiently coarse-grained with respect to the system size. As a further improvement, the addition and removal of 2D toric codes in the ground states of the 3D X-cube model is shown to be indispensable and, remarkably, leads to well-defined fixed points of a large class of fracton orders that are non-liquid-like. Here, we present a substantially unified ERG framework in which general degrees of freedom are allowed to be recursively added or removed. Specifically, we establish an exotic hierarchy of ERG and LRE states in Pauli stabilizer codes, where the 2D toric code and 3D X-cube models are naturally included. In the hierarchy, LRE states like 3D X-cube and 3D toric code ground states can be added or removed in ERG processes of more complex LRE states. In this way, a large group of Pauli stabilizer codes are categorized into a series of state towers; within each tower, in addition to local unitaries including cnot (controlled-NOT) gates, lower LRE states of level-nnn are added or removed in the level-nnn ERG process of an upper LRE state of level (n+1)(n+1)(n+1), connecting LRE states of different levels and unveiling complex relations among LRE states. As future directions, we expect this hierarchy can be applied to more general LRE states, leading to a unified ERG scenario of LRE states and exact tensor-network representations in the form of more generalized branching multiscale entanglement renormalization ansatz.
The fields of entanglement theory and tensor networks have recently emerged as central tools for characterizing quantum phases of matter. Here we determine the entanglement structure of ground states of gapped symmetric quantum lattice models and use this to obtain the most efficient tensor network representation of those ground states. We do this by showing that degeneracies in the entanglement spectrum arise through a duality transformation of the original model to the unique dual model where the entire dual symmetry is spontaneously broken. Physically, this duality transformation amounts to a-potentially twisted-gauging of the unbroken symmetry in the original ground state. In general, the dual symmetries of the resulting models are generalized non-invertible symmetries that cannot be described by groups. This result has strong implications for the complexity of simulating many-body systems using variational tensor network methods. For every phase in the phase diagram, the dual representation of the ground state that completely breaks the symmetry minimizes both the entanglement entropy and the required number of variational parameters. We demonstrate the applicability of this idea by developing a generalized density matrix renormalization group algorithm that works on constrained Hilbert spaces and quantify the computational gains obtained over traditional tensor network methods in a perturbed Heisenberg model. Our work testifies to the usefulness of generalized non-invertible symmetries and their formal category theoretic description for the practical simulation of strongly correlated systems.
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.
Gapped fracton phases constitute a new class of quantum states of matter, which connects to topological orders but does not fit easily into existing paradigms. They host unconventional features such as sub-extensive and robust ground-state degeneracies as well as sensitivity to lattice geometry. We investigate the anisotropic ZN{\mathbb{Z}}_{N}ZN​ Laplacian model [Phys. Rev. B 107, 125121 (2023)], which can describe a family of fracton phases defined on arbitrary graphs. Focusing on representative geometries where the 3D lattices are extensions of 2D square, triangular, honeycomb, and the Kagome lattices into the third dimension, we study their ground-state degeneracies and mobility of excitations, and examine their entanglement renormalization group (ERG) flows. All models show bifurcating behaviors under ERG but have distinct ERG flows sensitive to both NNN and lattice geometry. In particular, we show that the anisotropic ZN{\mathbb{Z}}_{N}ZN​ Laplacian models defined on the extensions of triangular and honeycomb lattices are equivalent when NNN is coprime to 3. We also point out that, in contrast to previous expectations, the model defined on the extension of Kagome lattice is robust against local perturbations if and only if NNN is coprime to 6.
This dissertation contributes to the ongoing effort of understanding the origins and applications of real-space renormalization group methods in tensor network representations of classical and quantum many-body systems. First, we construct a matrix product operator ansatz to coarse-grain real-space transfer matrices of matrix product state descriptions of one-dimensional quantum spin chains. By treating the physical spin as an impurity, we unravel the virtual entanglement degrees of freedom of matrix product states into a layered structure to reveal an inherent renormalization group scale. Secondly, we rephrase tensor network renormalization for two-dimensional classical lattice models in a manifestly nonnegative way. The resulting real-space renormalization group flow preserves positivity and hence yields an interpretation in terms of Hamiltonian flows, reconciling modern real-space tensor network renormalization methods with traditional block-spin approaches. Thirdly, we study non-local symmetries in tensor networks by expressing two-dimensional classical partition functions in terms of strange correlators of judiciously chosen product states and string-net wave functions. We exhibit and exploit the emerging non-local symmetries of the partition function at criticality and highlight parallels between topological sectors and conformal primary fields in the shared framework of matrix product operator symmetries. Additionally, we provide a complementary perspective on real-space renormalization by recognizing known tensor network renormalization methods as the approximate truncation of an exactly coarse-grained strange correlator.