The higher Berry curvature was introduced by Kapustin and Spodyneiko as an extension of the Berry curvature in quantum mechanical systems with finite degrees of freedom to quantum many-body systems in finite spatial dimensions. In this paper, we propose an alternative formulation of the higher Berry curvature using translationally invariant matrix product states. They are the ground states of a set of gapped Hamiltonians which are evolved adiabatically through a discretized parameter space. Because matrix product states transform under a projective representation, evaluating the Berry curvature on a closed loop through parameter space is not sufficient to fix all the gauge degrees of freedom. To obtain a gauge-invariant real quantity, the higher-dimensional Berry curvature is evaluated on small tetrahedra in parameter space. Our numerical calculations confirm that the higher Berry curvature varies continuously throughout an adiabatic evolution and becomes quantized over a closed 3-dimensional parameter space.
The higher Berry curvature was introduced by Kapustin and Spodyneiko as an extension of the Berry curvature in quantum mechanical systems with finite degrees of freedom to quantum many-body systems in finite spatial dimensions. In this paper, we propose an alternative formulation of the higher Berry curvature using translationally invariant matrix product states. They are the ground states of a set of gapped Hamiltonians which are evolved adiabatically through a discretized parameter space. Because matrix product states transform under a projective representation, evaluating the Berry curvature on a closed loop through parameter space is not sufficient to fix all the gauge degrees of freedom. To obtain a gauge-invariant real quantity, the higher-dimensional Berry curvature is evaluated on small tetrahedra in parameter space. Our numerical calculations confirm that the higher Berry curvature varies continuously throughout an adiabatic evolution and becomes quantized over a closed 3-dimensional parameter space.
Higher Berry curvature (HBC) is the proposed generalization of Berry curvature to infinitely extended systems. Heuristically, HBC captures the flow of local Berry curvature in a system. Here, we provide a simple formula for computing the HBC for extended d=1 systems at the level of wave functions using the Schmidt decomposition. We also find a corresponding formula for matrix product states and show that for translationally invariant matrix product states this gives rise to a quantized invariant. We demonstrate our approach with an exactly solvable model and numerical calculations for generic models using the infinite density matrix renormalization group algorithm.
This paper is concerned with the physics of parametrized gapped quantum many-body systems, which can be viewed as a generalization of conventional topological phases of matter. In such systems, rather than considering a single Hamiltonian, one considers a family of Hamiltonians that depend continuously on some parameters. After discussing the notion of phases of parametrized systems, we formulate a bulk-boundary correspondence for an important bulk quantity, the Kapustin-Spodyneiko higher Berry curvature, first in one spatial dimension and then in arbitrary dimension. This clarifies the physical interpretation of the higher Berry curvature, which in one spatial dimension is a flow of (ordinary) Berry curvature. In d dimensions, the higher Berry curvature is a flow of $(d\ensuremath{-}1)$-dimensional higher Berry curvature. Based on this, we discuss one-dimensional systems that pump Chern number to/from spatial boundaries, resulting in anomalous boundary modes featuring isolated Weyl points. In higher dimensions, there are pumps of the analogous quantized invariants obtained by integrating the higher Berry curvature. We also discuss the consequences for parametrized systems of Kitaev's proposal that invertible phases are classified by a generalized cohomology theory, and emphasize the role of the suspension isomorphism in generating new examples of parametrized systems from known invertible phases. Finally, we present a pair of general quantum pumping constructions, based on physical pictures introduced by Kitaev, which take as input a d-dimensional parametrized system, and produce new (d+1)-dimensional parametrized systems. These constructions are useful for generating examples, and we conjecture that one of the constructions realizes the suspension isomorphism in a generalized cohomology theory of invertible phases.
A family of finite-dimensional quantum systems with a nondegenerate ground state gives rise to a closed two-form on the parameter space, the curvature of the Berry connection. Its integral over a surface detects the presence of degeneracy points inside the volume enclosed by the surface. We seek generalizations of the Berry curvature to gapped many-body systems in D spatial dimensions which can detect gapless or degenerate points in the phase diagram of a system. Field theory predicts that in spatial dimension D the analog of the Berry curvature is a closed (D+2)-form on the parameter space (the Wess-Zumino-Witten form). We construct such closed forms for arbitrary families of gapped interacting lattice systems in all dimensions. We show that whenever the integral of the Wess-Zumino-Witten form over a (D+2)-dimensional surface in the parameter space is nonzero, there must be gapless edge modes for at least one value of the parameters. These edge modes arise even when the bulk system is in a trivial phase for all values of the parameters and are protected by the nontrivial topology of the phase diagram.
We propose a systematic wave-function-based approach to construct topological invariants for families of lattice systems that are short-range entangled using local parameter spaces. This construction is particularly suitable when given a family of tensor networks that can be viewed as the ground states of d-dimensional lattice systems, for which we construct the closed (d+2)-form higher Berry curvature, which is a generalization of the well-known 2-form Berry curvature. Such (d+2)-form higher Berry curvature characterizes a flow of (d+1)-form higher Berry curvature in the system. Our construction is equally suitable for constructing other higher pumps, such as the (higher) Thouless pump in the presence of a global onsite U(1) symmetry, which corresponds to a closed d-form. The cohomology classes of such higher differential forms are topological invariants and are expected to be quantized for short-range entangled states. We illustrate our construction with exactly solvable lattice models that are in nontrivial higher Berry classes in d=2.
In one spatial dimension, families of short-range entangled many-body quantum states, parameterized over some parameter space, can be topologically distinguished and classified by topological invariants built from the higher Berry phase---a many-body generalization of the Berry phase. Previous works identified the underlying mathematical structure (the gerbe structure) and introduced a multi-wave-function overlap, a generalization of the inner product in quantum mechanics, which allows for the extraction of the higher Berry phase and topological invariants. In this paper, building on these works, we introduce a connection, the higher Berry connection, for a family of parameterized matrix product states (MPS) over a parameter space. We demonstrate the use of our formula for simple nontrivial models.
In this work, we study the connection between two subjects: the space of conformal boundary conditions in boundary conformal field theories (BCFTs) and the space of gapped systems characterized by higher Berry phases. We explore this connection by analyzing multi-parameter spectral flow in Dirac fermion BCFTs with continuously parametrized conformal boundary conditions, which are introduced by coupling a CFT to a family of gapped systems. When the gapped systems belong to a nontrivial higher Berry class, the associated conformal boundary conditions induce a flow of the ordinary Berry curvature, resulting in a Chern number pump in the Fock space of the BCFT. This phenomenon is the BCFT analog of Berry curvature flow in one-dimensional parametrized gapped systems, where the flow occurs in real space. Building on this correspondence, we introduce the notions of higher Berry curvature and higher Berry invariants within the BCFT framework. Our results provide a new perspective for studying the topological properties of families of conformal boundary states and gapped ground states: if a family of gapped states belongs to a nontrivial higher Berry class, then the corresponding entanglement Hamiltonians exhibit a multi-parameter spectral flow that carries Berry curvature in the Fock space.