Bifurcation in entanglement renormalization group flow of a gapped spin model

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.

13 pages + supplementary Mathematica file containing calculation. (v2) Added a section on an algebro-geometric criterion for bifurcation

References in corpus (10)

• A class of quantum many-body states that can be efficiently simulated
• Local stabilizer codes in three dimensions without string logical operators
• Topological Quantum Distillation
• Tensor renormalization group approach to 2D classical lattice models
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Entanglement renormalization and topological order
• Scaling of entanglement entropy in the (branching) multi-scale entanglement renormalization ansatz
• Entanglement does not generally decrease under renormalization

Cited by in corpus (54)

• Quantum memories at finite temperature
• Fracton Models on General Three-Dimensional Manifolds
• Fracton topological order via coupled layers
• Fractal symmetries: Ungauging the cubic code
• Fracton Topological Order from Nearest-Neighbor Two-Spin Interactions and Dualities
• Hadamard-free circuits expose the structure of the Clifford group
• X-Cube Fracton Model on Generic Lattices: Phases and Geometric Order
• Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems
• Fractonic Matter in Symmetry-Enriched U(1) Gauge Theory
• Spurious topological entanglement entropy from subsystem symmetries
• Topological Defect Networks for Fractons of all Types
• Entanglement Entropy From Tensor Network States for Stabilizer Codes
• Fast State Transfer and Entanglement Renormalization Using Long-Range Interactions
• Renormalization group constructions of topological quantum liquids and beyond
• Deciphering the nonlocal entanglement entropy of fracton topological orders
• Algebraic Methods for Quantum Codes on Lattices
• Sorting topological stabilizer models in three dimensions
• Gauging fractons: immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies
• Non-Abelian Hybrid Fracton Orders
• Topological phases from higher gauge symmetry in 3+1D
• Algorithms for entanglement renormalization: boundaries, impurities and interfaces
• Gauging permutation symmetries as a route to non-Abelian fractons
• Bifurcating entanglement-renormalization group flows of fracton stabilizer models
• Higher-Form Subsystem Symmetry Breaking: Subdimensional Criticality and Fracton Phase Transitions
• Compactifying fracton stabilizer models
• Foliated fracton order in the Majorana checkerboard model
• Mixed s-sourcery: Building many-body states using bubbles of Nothing
• Hybrid Fracton Phases: Parent Orders for Liquid and Non-Liquid Quantum Phases
• The Group Structure of Quantum Cellular Automata
• Type-II fractons from coupled spin chains and layers
• Limits on the storage of quantum information in a volume of space
• Engineering 3D Floquet codes by rewinding
• Boundary theory of the X-cube model in the continuum
• Planar p-String Condensation: Chiral Fracton Phases from Fractional Quantum Hall Layers and Beyond
• A degeneracy bound for homogeneous topological order
• Bifurcating subsystem symmetric entanglement renormalization in two dimensions
• Non-Liquid Cellular States
• Designer non-Abelian fractons from topological layers
• Error-correcting codes for fermionic quantum simulation
• Anyon condensation web and multipartite entanglement in 2D modulated gauge theories
• Fractalizing quantum codes
• Topological defect network representations of fracton stabilizer codes
• Emergent fermionic gauge theory and foliated fracton order in the Chamon model
• A unified diagrammatic approach to topological fixed point models
• Hierarchy of Entanglement Renormalization and Long-Range Entangled States
• Non-local finite-depth circuits for constructing SPT states and quantum cellular automata
• No Strings Attached: Boundaries and Defects in the Cubic Code
• Entanglement renormalization of thermofield double states
• Fractal Subsystem Symmetries, Anomalies, Boundaries, and Effective Field Theory
• Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders
• Adiabatic ground state preparation in an expanding lattice
• Entanglement renormalization of fractonic anisotropic ${\mathbb{Z}}_N$ Laplacian models
• Planon-modular fracton orders
• Fracton models from product codes