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arXiv:2303.04252 · Physical review. B./Physical review. B

Discrete higher Berry phases and matrix product states

Shuhei Ohyama, Yuji Terashima, Ken Shiozaki

Published Jul 3, 2024• 11 citations
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Abstract

A one-parameter family of invertible states gives a topological transport phenomenon, similar to the Thouless pumping. As a natural generalization of this, we can consider a family of invertible states parametrized by some topological space $X$. This is called a higher pump. It is conjectured that a $(1+1)$-dimensional bosonic invertible state parametrized by $X$ is classified by ${\mathrm{H}}^{3}(X;\mathbb{Z})$. In this paper, we construct two higher pumping models parametrized by $X=\mathbb{R}{P}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ and $X=\mathrm{L}(3,1)\ifmmode\times\else\texttimes\fi{}{S}^{1}$ that corresponds to the torsion part of ${\mathrm{H}}^{3}(X;\mathbb{Z})$. As a consequence of the nontriviality as a family, we find that a quantum mechanical system with a nontrivial discrete Berry phase is pumped to the boundary of the $(1+1)$-dimensional system. We also study higher pump phenomena by using matrix product states, and construct a higher pump invariant which takes value in a torsion part of ${\mathrm{H}}^{3}(X;\mathbb{Z})$. This is a higher analog of the ordinary discrete Berry phase that takes value in the torsion part of ${\mathrm{H}}^{2}(X;\mathbb{Z})$. In order to define the higher pump invariant, we utilize the smooth Deligne cohomology and its integration theory. We confirm that the higher pump invariant of the model has a nontrivial value.

DOI
10.1103/physrevb.110.035114

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A one-parameter family of invertible states gives a topological transport phenomenon, similar to the Thouless pumping. As a natural generalization of this, we can consider a family of invertible states parametrized by some topological space XXX. This is called a higher pump. It is conjectured that a (1+1)(1+1)(1+1)-dimensional bosonic invertible state parametrized by XXX is classified by H3(X;Z){\mathrm{H}}^{3}(X;\mathbb{Z})H3(X;Z). In this paper, we construct two higher pumping models parametrized by X=RP2\ifmmode×\else\texttimes\fiS1X=\mathbb{R}{P}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}X=RP2\ifmmode×\else\texttimes\fiS1 and X=L(3,1)\ifmmode×\else\texttimes\fiS1X=\mathrm{L}(3,1)\ifmmode\times\else\texttimes\fi{}{S}^{1}X=L(3,1)\ifmmode×\else\texttimes\fiS1 that corresponds to the torsion part of H3(X;Z){\mathrm{H}}^{3}(X;\mathbb{Z})H3(X;Z). As a consequence of the nontriviality as a family, we find that a quantum mechanical system with a nontrivial discrete Berry phase is pumped to the boundary of the (1+1)(1+1)(1+1)-dimensional system. We also study higher pump phenomena by using matrix product states, and construct a higher pump invariant which takes value in a torsion part of H3(X;Z){\mathrm{H}}^{3}(X;\mathbb{Z})H3(X;Z). This is a higher analog of the ordinary discrete Berry phase that takes value in the torsion part of H2(X;Z){\mathrm{H}}^{2}(X;\mathbb{Z})H2(X;Z). In order to define the higher pump invariant, we utilize the smooth Deligne cohomology and its integration theory. We confirm that the higher pump invariant of the model has a nontrivial value.
We consider families of invertible many-body quantum states in ddd spatial dimensions that are parameterized over some parameter space XXX. The space of such families is expected to have topologically distinct sectors classified by the cohomology group Hd+2(X;Z){\mathrm{H}}^{d+2}(X;\mathbb{Z})Hd+2(X;Z). These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work [Phys. Rev. B 109, 115152 (2024)], we introduced a generalized inner product for three one-dimensional many-body quantum states (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in H3(X;Z){\mathrm{H}}^{3}(X;\mathbb{Z})H3(X;Z), can be extracted. In this paper, we introduce an inner product of four two-dimensional invertible quantum many-body states. We use it to measure the topological nontriviality of parameterized families of 2d invertible states. In particular, we define a topological invariant of such families that takes values in H4(X;Z){\mathrm{H}}^{4}(X;\mathbb{Z})H4(X;Z). Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of nontrivial parameterized families of 2d invertible states parameterized over RP4\mathbb{R}{P}^{4}RP4 and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.
We consider families of invertible many-body quantum states in ddd spatial dimensions that are parameterized over some parameter space XXX. The space of such families is expected to have topologically distinct sectors classified by the cohomology group Hd+2(X;Z)\mathrm{H}^{d+2}(X;\mathbb{Z})Hd+2(X;Z). These topological sectors are distinguished by a topological invariant built from a generalization of the Berry phase, called the higher Berry phase. In the previous work, we introduced a generalized inner product for three one-dimensional many-body quantum states, (``triple inner product''). The higher Berry phase for one-dimensional invertible states can be introduced through the triple inner product and furthermore the topological invariant, which takes its value in H3(X;Z)\mathrm{H}^{3}(X;\mathbb{Z})H3(X;Z), can be extracted. In this paper, we introduce an inner product of four two-dimensional invertible quantum many-body states. We use it to measure the topological nontriviality of parameterized families of 2d invertible states. In particular, we define a topological invariant of such families that takes values in H4(X;Z)\mathrm{H}^{4}(X;\mathbb{Z})H4(X;Z). Our formalism uses projected entangled pair states (PEPS). We also construct a specific example of non-trivial parameterized families of 2d invertible states parameterized over RP4\mathbb{R}P^4RP4 and demonstrate the use of our formula. Applications for symmetry-protected topological phases are also discussed.
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 ddd 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 ddd-dimensional parametrized system, and produce new (d+1)(d+1)(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.
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 ddd-dimensional lattice systems, for which we construct the closed (d+2)(d+2)(d+2)-form higher Berry curvature, which is a generalization of the well-known 2-form Berry curvature. Such (d+2)(d+2)(d+2)-form higher Berry curvature characterizes a flow of (d+1)(d+1)(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)\mathrm{U}(1)U(1) symmetry, which corresponds to a closed ddd-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=2d=2d=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.
For a parameterized family of invertible states (short-range-entangled states) in (1+1)(1+1)(1+1) dimensions, we discuss a generalization of the Berry phase. Using translationally invariant, infinite matrix product states (MPSs), we introduce a gerbe structure, a higher generalization of complex line bundles, as an underlying mathematical structure describing topological properties of a parameterized family of MPSs. Furthermore, we introduce a generalization of a quantum mechanical inner product, which we call the ``triple inner product,'' defined for three matrix product states. The triple inner product proves to extract a topological invariant, the Dixmier-Douady class over the parameter space.