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arXiv:2412.20905

Parametrized topological phases in 1d and T-duality

Roman Geiko

Published Dec 30, 2024• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2412.20905

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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.
We analyze topological phase transitions and higher Berry curvature in one-dimensional quantum spin systems, using a framework that explicitly incorporates the symmetry group action on the parameter space. Based on a G G -compatible discretization of the parameter space, we incorporate both group cochains and parameter-space differentials, enabling the systematic construction of equivariant topological invariants. We derive a fixed-point formula for the higher Berry invariant in the case where the symmetry action has isolated fixed points. This reveals that the phase transition point between Haldane and trivial phases acts as a monopole-like defect where higher Berry curvature emanates. We further discuss hierarchical structures of topological defects in the parameter space, governed by symmetry reductions and compatibility with subgroup structures.
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.
Parameterized Families of Toric Code Phase: ememem-duality family and higher-order anyon pumping
Within the toric-code phase, we study parameterized families of topologically ordered states. We construct 111- and 222-parameter families of local Hamiltonians and confirm their non-triviality via topological pumping. For the 111-parameter family, we show that the ememem-exchange defect is pumped into the bond Hilbert space of a tensor-network representation. For the 222-parameter case, we construct a ``pump of a pump'' that transports an S1S^1S1-family of a system in one lower spatial dimension. Using similar methods, we also present a 111-parameter family with a higher-order anyon pump that produces corner-localized anyon modes. These constructions provide explicit lattice realizations and concrete diagnostics of family-level topology. We use recently developed boundary algebra methods to study the non-triviality of these families.
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.
A C∗C^*C∗-Algebraic Approach to Parametrized Quantum Spin Systems and Their Phases in One Spatial Dimension
This thesis investigates parametrized quantum spin systems in the thermodynamic limit from a C∗C^*C∗-algebraic point of view. Our main physical result is the construction of a phase invariant for one-dimensional quantum spin chains parametrized by a topological space XXX. This invariant is constructed using C∗C^*C∗-algebraic techniques and takes values in degree one Čech cohomology H1(X;PU(H))H^1(X;\mathbb{P}\mathrm{U}(\mathscr{H}))H1(X;PU(H)), where PU(H)\mathbb{P}\mathrm{U}(\mathscr{H})PU(H) is the projective unitary group of an infinite-dimensional Hilbert space H\mathscr{H}H, endowed with the strong operator topology. Using Dixmier-Douady theory [Bry93, DD63, HJJS08] one may equivalently view this as an element of H3(X;Z)H^3(X;\mathbb{Z})H3(X;Z). An exactly solvable model of a one-dimensional spin system parametrized by the 3-sphere X=S3X = \mathbb{S}^3X=S3 is presented and it is shown that its invariant is nontrivial [WQB+22]. We also prove several mathematical results on topological aspects of the pure state space P(A)\mathscr{P}(\mathfrak{A})P(A) of a C∗C^*C∗-algebra A\mathfrak{A}A. We prove that P(A)\mathscr{P}(\mathfrak{A})P(A), endowed with the weak* topology, has trivial fundamental group for every UHF algebra A\mathfrak{A}A [BHM+23]. We review the results of [SMQ+22] that show how the outputs of the GNS representation and Kadison transitivity theorem can be understood to depend continuously on their inputs. These results are expanded upon with the construction of a distinguished Čech class in H1(P(A);U(1))H^1(\mathscr{P}(\mathfrak{A});\mathrm{U}(1))H1(P(A);U(1)) that generalizes the principal U(1)\mathrm{U}(1)U(1)-bundle SH→PH\mathbb{S} \mathscr{H} \rightarrow \mathbb{P} \mathscr{H}SH→PH in a representation independent way, where SH\mathbb{S} \mathscr{H}SH and PH\mathbb{P} \mathscr{H}PH are the unit sphere and projective Hilbert space of H\mathscr{H}H. Finally, we prove a selection theorem in the vein of [SMQ+22] for the weak* topology on P(A)\mathscr{P}(\mathfrak{A})P(A).
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.