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arXiv:2304.05356 · Physical review. B./Physical review. B

Higher structures in matrix product states

Shuhei Ohyama, Shinsei Ryu

Published Mar 25, 2024• 16 citations
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Abstract

For a parameterized family of invertible states (short-range-entangled states) in $(1+1)$ 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.

DOI
10.1103/physrevb.109.115152

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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.
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.
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.
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.
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 $X=\mathbb{R}{P}^{2}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ and $X=\mathrm{L}(3,1)\ifmmode\times\else\texttimes\fi{}{S}^{1}$ that corresponds to the torsion part of 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 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 X we associate a gerbe, which generalizes the line bundle of ground states in zero-dimensional families ( in few-body quantum mechanics). The nontriviality of the gerbe is measured by a class in H^3(X, \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 X . We illustrate our construction with two examples of nontrivial parametrized systems over X=S^3 X = S 3 and X = \R P^2 × 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 X = S 4 .
There are families of physical systems that cannot be adiabatically evolved to the trivial system uniformly across the parameter space, even if each system in the family belongs to the trivial phase. The obstruction is measured by higher Berry class. We analyze families of topological systems in 1+1d using families of invertible TQFTs and families of RG fixed states of spin chains. We use the generalized matrix-product states to describe RG fixed points of all translation invariant pure splits states on spin chains. Families of such fixed points correspond to bundles of Hilbert-Schmidt operators. There exists a global MPS parametrization of the family if and only if the latter bundle is trivial. We propose a novel duality of parametrized topological phases which is an avatar of the T-duality in string theory. The duality relates families with different parameter spaces and different higher Berry classes. Mathematically, the T-duality is realized by gauging the circle action on the continuous trace algebra generated by parametrized matrix-product tensors.