Charting the space of ground states with tensor networks

Abstract

We employ matrix product states (MPS) and tensor networks to study topological properties of the space of ground states of gapped many-body systems. We focus on families of states in one spatial dimension, where each state can be represented as an injective MPS of finite bond dimension. Such states are short-range entangled ground states of gapped local Hamiltonians. To such parametrized families over $X$ we associate a gerbe, which generalizes the line bundle of ground states in zero-dimensional families (\emph{i.e.} in few-body quantum mechanics). The nontriviality of the gerbe is measured by a class in $H^3(X,\mathbb{Z})$, which is believed to classify one-dimensional parametrized systems. We show that when the gerbe is nontrivial, there is an obstruction to representing the family of ground states with an MPS tensor that is continuous everywhere on $X$. We illustrate our construction with two examples of nontrivial parametrized systems over $X=S^3$ and $X=\mathbb{R}{P}^2\times S^1$. Finally, we sketch using tensor network methods how the construction extends to higher dimensional parametrized systems, with an example of a two-dimensional parametrized system that gives rise to a nontrivial 2-gerbe over $X=S^4$.

v2, corrected error in appendix, minor revisions in main text

References in corpus (29)

• Entanglement spectrum of a topological phase in one dimension
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Classifying quantum phases using Matrix Product States and PEPS
• PEPS as ground states: degeneracy and topology
• Continuous Matrix Product States for Quantum Fields
• Lectures on Special Lagrangian Submanifolds
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications II
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications I
• Geometry of Deligne cohomology
• Geometry of Matrix Product States: metric, parallel transport and curvature
• A canonical form for Projected Entangled Pair States and applications
• Berry Phase in Quantum Field Theory: Diabolical Points and Boundary Phenomena
• Matrix product operator symmetries and intertwiners in string-nets with domain walls
• Higher-dimensional generalizations of the Berry curvature
• A generalization of the injectivity condition for Projected Entangled Pair States
• Quantum spin systems on infinite lattices
• Normal projected entangled pair states generating the same state
• Local Noether theorem for quantum lattice systems and topological invariants of gapped states
• Topological Goldstone phases of matter
• Projected Entangled Pair States: Fundamental analytical and numerical limitations
• On adiabatic cycles of quantum spin systems
• Homotopical Foundations of Parametrized Quantum Spin Systems
• Generalized Thouless Pumps in 1+1-dimensional Interacting Fermionic Systems
• On Topology of the Moduli Space of Gapped Hamiltonians for Topological Phases
• Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension
• Higher Berry Curvature from the Wave Function I: Schmidt Decomposition and Matrix Product States
• Classification and Construction of Topological Phases of Quantum Matter
• Many finite-dimensional lifting bundle gerbes are torsion

Cited by in corpus (5)

• Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions
• Charge pumps, boundary modes, and the necessity of unnecessary criticality
• Instanton Density Operator in Lattice QCD from Higher Category Theory
• Charge pumps, pivot Hamiltonians and symmetry-protected topological phases
• Equivariant Parameter Families of Spin Chains: A Discrete MPS Formulation