Classifying quantum phases using Matrix Product States and PEPS

Abstract

We give a classification of gapped quantum phases of one-dimensional systems in the framework of Matrix Product States (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states, and both in the absence and presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, this is, the equivalence classes of its projective representations, a result first derived in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labelled by both the permutation action of the former and the cohomology class of the latter. Using Projected Entangled Pair States (PEPS), we subsequently extend our framework to the classification of two-dimensional phases in the neighborhood of a number of important cases, in particular systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus, we can focus on to the classification of isometric forms.

v2: 21 pages, 6 figures. Significantly rewritten and extended. Now contains a classification of phases both without and with symmetries, for systems with both unique and degenerate ground states. v3: 24 pages. Improved according to referees' suggestions (most notably added examples and improved definition of phases under symmetries). Accepted at Phys. Rev. B

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• Islands of Chiral Solitons in Integer Spin Kitaev Chains
• Universal resources for quantum computing
• Tensor Network Computations That Capture Strict Variationality, Volume Law Behavior, and the Efficient Representation of Neural Network States
• Local order parameters for symmetry fractionalization
• Fermionic criticality is shaped by Fermi surface topology: a case study of the Tomonaga-Luttinger liquid
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states
• Matrix Product Density Operators: when do they have a local parent Hamiltonian?
• Twice Hidden String Order and Competing Phases in the Spin-1/2 Kitaev-Gamma Ladder
• Transition of a ${\mathbb{Z}}_3$ topologically ordered phase to trivial and critical phases
• Quantum Error-Detection at Low Energies
• Category of SET orders
• Charting the space of ground states with tensor networks
• Symmetry protected topological phases beyond groups: The q-deformed bilinear-biquadratic spin chain
• Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension
• Theory of quantum kagome ice and vison zero modes
• Quantum simulation of lattice gauge theories via deterministic duality transformations assisted by measurements
• Pivoting through the chiral-clock family
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• Anomaly inflow, dualities, and quantum simulation of abelian lattice gauge theories induced by measurements
• Robust symmetry-protected metrology with the Haldane phase
• 4D beyond-cohomology topological phase protected by $C_2$ symmetry and its boundary theories
• The Gauge Theory of Measurement-Based Quantum Computation
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• Geometric Approach to 3D Interfaces at Strong Coupling
• Matrix product approximations to multipoint functions in two-dimensional conformal field theory
• Classical topological paramagnetism
• Locality optimization for parent Hamiltonians of Tensor Networks
• Generalized Phase-Space Techniques to Explore Quantum Phase Transitions in Critical Quantum Spin Systems
• Classification of measurement-based quantum wire in stabilizer PEPS
• Quantum computation by teleportation and symmetry
• Performance of the rigorous renormalization group for first order phase transitions and topological phases
• Composite symmetry protected topological order and effective models
• Bulk Entanglement Spectrum in Gapped Spin Ladders
• Matrix product approximations to conformal field theories
• Higher Berry curvature from matrix product states
• Anomalies and symmetry fractionalization in reflection-symmetric topological order
• Combination of Tensor Network States and Green's function Monte Carlo
• Phase transitions of a 2D deformed-AKLT model
• Fermionic tensor networks for higher order topological insulators from charge pumping
• Dual-isometric Projected Entangled Pair States
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Projected Entangled Pair States with continuous virtual symmetries
• Projected Entangled Pair States with flexible geometry
• Observation of average topological phase in disordered Rydberg atom array
• Detection of gapped phases of a 1D spin chain with onsite and spatial symmetries
• Weak Hopf non-invertible symmetry-protected topological spin liquid and lattice realization of (1+1)D symmetry topological field theory
• Topological phases of unitary dynamics: Classification in Clifford category
• Matrix-product unitaries: Beyond quantum cellular automata
• Self-$G$-ality in 1+1 dimensions
• Entanglement renormalization of thermofield double states
• Variational Tensor Network Operator
• Edge mode locality in perturbed symmetry protected topological order
• Exact Quantum Algorithms for Quantum Phase Recognition: Renormalization Group and Error Correction
• Symmetry protected topological phases in spin-1 ladders and their phase transitions
• The resource theory of tensor networks
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Area laws and efficient descriptions of quantum many-body states
• Unveiling Correlated Two-dimensional Topological Insulators through Fermionic Tensor Network States -- Classification, Edge Theories and Variational Wavefunctions
• Gauge theory and mixed state criticality
• Bulk and boundary entanglement transitions in the projective gauge-Higgs model
• Field Tensor Network States
• Exact correlations in topological quantum chains
• Entanglement of a alternating bipartition in spin chains: Relation with classical integrable models
• Symmetry-protected topological phases, conformal criticalities, and duality in exactly solvable SO($n$) spin chains
• Fusion of one-dimensional gapped phases and their domain walls
• Role of anomalous symmetry in 0-$\pi$ qubits
• Quantum antiferromagnetic Heisenberg half-odd integer spin model as the entanglement Hamiltonian of the integer spin Affleck-Kennedy-Lieb-Tasaki states
• Characterization of photoexcited states in the half-filled one-dimensional extended Hubbard model assisted by machine learning
• Spectral properties for a family of two-dimensional quantum antiferromagnets
• Categorical descriptions of one-dimensional gapped phases with Abelian onsite symmetries
• Classifying parafermionic gapped phases using matrix product states
• Quantum teleportation implies symmetry-protected topological order
• Symmetry, topology, duality, chirality, and criticality in a spin-1/2 XXZ ladder with a four-spin interaction
• Symmetry-protected trivial phases and quantum phase transitions in an anisotropic antiferromagnetic spin-1 biquadratic model
• Fermionic symmetry-protected topological state in strained graphene
• Building Symmetry Enriched Topological Phases from a Bipartite Lattice Construction and Anyon Condensation
• Perturbative 2-body Parent Hamiltonians for Projected Entangled Pair States
• Gauging quantum states with non-anomalous matrix product operator symmetries
• Simulating chiral spin liquids with projected entangled-pair states
• Classes of topological qubits from low-dimensional quantum spin systems
• Unwinding of a one-dimensional topological superconductor
• Spectral Gap Optimization for Enhanced Adiabatic State Preparation
• Many-body quantum catalysts for transforming between phases of matter
• Models of interacting bosons with exact ground states: a unified approach
• Tensor Renormalization Group for interacting quantum fields
• A New Framework for Quantum Phases in Open Systems: Steady State of Imaginary-Time Lindbladian Evolution
• Fractional domain wall statistics in spin chains with anomalous symmetries
• Entanglement renormalization for chiral topological phases
• Complete characterization of spin chains with two Ising symmetries
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Quantum Computing with sine-Gordon Qubits
• Internal structure of gauge-invariant Projected Entangled Pair States
• Classification of the phases of 1D spin chains with commuting Hamiltonians
• Symmetries and local transformations of translationally invariant Matrix Product States
• Phases of Matrix Product States with Symmetric Quantum Circuits and Symmetric Measurements with Feedforward
• Systematic construction of stabilizer codes via gauging abelian boundary symmetries
• Parity-dependent phase diagrams in spin-cluster two-leg ladders
• Classification of matrix product ground states corresponding to one dimensional chains of two state sites of nearest neighbor interactions
• Adiabatic ground state preparation in an expanding lattice
• Enriched classification of parafermionic gapped phases with time reversal symmetry
• One-dimensional symmetric phases protected by frieze symmetries
• Variational Tensor Wavefunctions for the Interacting Quantum Spin Hall Phase
• Counter-intuitive yet efficient regimes for measurement based quantum computation on symmetry protected spin chains
• Redundant string symmetry-based error correction: Demonstrations on quantum devices
• Topological Dipole Insulator
• Detection of symmetry-protected topological phases in one dimension with multiscale entanglement renormalization
• Boundary anomaly detection in two-dimensional subsystem symmetry-protected topological phases
• A Classifying Space for Phases of Matrix Product States
• A Simple and General Equation for Matrix Product Unitary Generation
• Disentangling anomaly-free symmetries of quantum spin chains
• Subsystem Symmetry-Protected Topological Phases from Subsystem SymTFT of 2-Foliated Exotic Tensor Gauge Theory
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry
• Computing local properties in the trivial phase
• Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology
• Les Houches Lecture Notes on Tensor Networks
• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation
• Instability of the Haldane Phase: Roles of Charge Fluctuations and Hund's Coupling
• Impurity screening by defects in (1+1)$d$ quantum critical systems
• Topological phases of the compass ladder model
• Instanton effects in lattice models of bosonic symmetry-protected topological states
• Characterizing the parent Hamiltonian for a complete set of orthogonal wave functions: An inverse quantum problem
• Symmetries and field tensor network states
• C^1-Classification of gapped parent Hamiltonians of quantum spin chains with local symmetry
• SPT indices emerging from translation invariance in two dimensional quantum spin systems
• Spin chains, defects, and quantum wires for the quantum-double edge
• Model-Independent Quantum Phases Classifier
• Variational methods for characterizing matrix product operator symmetries
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Gauging Quantum Phases: A Matrix Product State Approach
• Parent Lindbladians for Matrix Product Density Operators
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks
• Entanglement renormalization and symmetry fractionalization
• Computational Characterization of Symmetry-Protected Topological Phases in Open Quantum Systems
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• Condensation Completion and Defects in 2+1D Topological Orders
• Interrelations among frustration-free models via Witten's conjugation
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• On ground state phases of quantum spin systems
• Geometric Approaches to Quantum Fields and Strings at Strong Couplings
• Construction of Entangled Many-body States via the Higgs Mechanism
• Matrix product state classification of 1D multipole symmetry protected topological phases
• Measurement-Based Quantum Computation Using the Spin-1 XXZ Model with Uniaxial Anisotropy
• A Subsystem Ginzburg-Landau and SPT Orders Co-existing on a Graph
• Tensorization of neural networks for improved privacy and interpretability
• Tensor network states for the description of quantum many-body systems
• Learning complexity gradually in quantum machine learning models
• Holographic duality between bulk topological order and boundary mixed-state order