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arXiv:1812.02721 · Physical review. B./Physical review. B

Classifying local fractal subsystem symmetry-protected topological phases

Trithep Devakul

Published Jun 14, 2019• 29 citations
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Abstract

We study symmetry-protected topological (SPT) phases of matter in $2\mathrm{D}$ protected by symmetries acting on fractal subsystems of a certain type. Despite the total symmetry group of such systems being subextensively large, we show that only a small number of phases are actually realizable by local Hamiltonians. Which phases are possible depends crucially on the spatial structure of the symmetries, and we show that in many cases, no nontrivial SPT phases are possible at all. In cases where nontrivial SPT phases do exist, we give an exhaustive enumeration of them in terms of their locality.

DOI
10.1103/physrevb.99.235131

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We study symmetry-protected topological (SPT) phases of matter in 2D2\mathrm{D}2D protected by symmetries acting on fractal subsystems of a certain type. Despite the total symmetry group of such systems being subextensively large, we show that only a small number of phases are actually realizable by local Hamiltonians. Which phases are possible depends crucially on the spatial structure of the symmetries, and we show that in many cases, no nontrivial SPT phases are possible at all. In cases where nontrivial SPT phases do exist, we give an exhaustive enumeration of them in terms of their locality.
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Abstract We refine our notion of a topoogical phase of matter to account for symmetry protection. We describe SPT (symmetry protected topological or trivial) phases and give some history of this idea. We focus on the case of on-site symmetries. The case of one-dimensional SPTs with on-site symmetry is discussed. For understanding two-dimensional SPTs we focus on the case of Z2 symmetry and argue that there are two possible Z2 paramagnetic phases, i.e., gapped phases where the Z2 symmetry is unbroken. These two phases are in one-to-one correspondence with the two possible Z2 loop gases. We argue that with any discrete group symmetry, we generally classify SPT phases with the possible 3-cocyles. We briefly discuss the idea of symmetry defects and explain that such particles are the excitations of the corresponding twisted Kitaev quantum double model of the same cocyle. We briefly discuss the edge modes of SPT phases. Finally we argue that the SPT phases can be considered as the “ungauging” of the corresponding Dijkgraaf-Witten model.
Abstract Classifying and understanding phases of matter is an important task in condensed matter physics. The class of ‘conventional’ symmetry-broken phases is well understood in terms of Landau’s theory. A paradigmatic example is the Z2-symmetric Ising model with symmetric (paramagnetic) and symmetry-broken (ferromagnet) phases, which can be distinguished by measuring the magnetization as a local order parameter. Topological phases of matter, however, are less understood and no complete classification is known. These lecture notes discuss schemes that provide an understanding and characterization of certain topological phases of matter in models with local Hamiltonians. The main focus is on symmetry-protected topological (SPT) phases in one-dimensional bosonic systems. On the basis of the entanglement properties of such systems, a motivation is given for the matrix product state (MPS) representation of ground states. Using the MPS framework, it is demonstrated how SPT phases can be classified using projective representations of the symmetries.
One problem of paramount importance in physics over various fields such as elementary particle physics, condensed matter physics, and statistical mechanics has been to make classification of phases of matter, including nontrivial structures of the vacuum. The triumphant history of theory on the phase classification and the phase transition can be traced back to Landau, which is based on the spontaneous breaking of symmetry. It is known, however, that other kinds of phases exist that cannot be characterized by conventional arguments of symmetry breaking, namely, topological phases. Topological phases are gapped phases at zero temperature without symmetry breaking. Examples of the topological phases include a quantum phase in a fractional quantum Hall system, where the ground state exhibits degeneracy, and an anyon, a quasi-particle with nontrivial statistical nature, neither boson nor fermion, emerges as a low-energy excitation. In contemporary physics of the 21st century, it has been revealed that, among various topological states, there exists a nontrivial ground state that does not have ground state degeneracy and yet can be distinguished from a trivial state. This nontrivial ground state, i.e., a topological insulator, has been recognized first in free electron systems [1–3] and later extended to more general quantum many-body systems with interactions including bosonic (spin) systems. Now, such a generalized situation is called a symmetry protected topological (SPT) phase [4–7]. From the point of a modern view of the SPT phase, nontrivial states, such as the Haldane state proposed in antiferromagnetic quantum spin systems in the 1980’s [8] and the ground state in the Affleck-Kennedy-Lieb-Tasaki model [9], belong to the SPT phase. A remarkable feature of the SPT phase is that, if the system has a surface, gapless surface states appear and they are stable against disturbances which do not break the symmetry. One way to understand these surface states is the quantum anomaly on the surface. That is, as long as the surface is concerned, the quantum anomaly arises, but the whole system has no quantum anomaly, and the symmetry is preserved due to cancellation from the bulk contribution. This canceling mechanism of the quantum anomaly has been long known as the anomaly inflow in elementary particle physics [10]. In fact, the classification of the SPT phases can be identified as the classification of the quantum anomaly through the bulk-boundary correspondence. This type of the quantum anomaly closely related to the SPT phase can provide us with a useful theoretical device to constrain phases that can or cannot be realized under the ’t Hooft anomaly matching condition [11–13]. It is quite difficult to scrutinize nonperturbatively interacting systems, but the ’t Hooft anomaly matching condition can be applied to those systems, and possible phases can be constrained. Nowadays, our understanding of the SPT phases has been so developed that applications of the anomaly matching condition can cover systems with finite groups and/or extending objects [14–16]. In this way, the concept of the topological phases has been essential by now in a wide range of physics, particularly in elementary particle physics, condensed matter physics, and statistical mechanics, as mentioned in the beginning. This special section is a collection of papers related to the topological aspects of physics including the topological phases. We would like to express our sincere gratitude to all the authors who made excellent contributions.
Symmetry-protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For noninteracting fermion systems with time reversal ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}$), charge conjugation ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{C}$), and/or U(1)U(1)U(1) ($\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}}^{\ensuremath{-}1}=\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$ or $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{T}}^{\ensuremath{-}1}=\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$. As a result, the SPT phases of those fermion systems with different symmetry groups have different classifications. In this paper, we use Kitaev's K-theory approach to classify the gapped free-fermion phases for those possible symmetry groups. In particular, we can view the U(1)U(1)U(1) as a spin rotation. We find that superconductors with the Sz{S}_{z}Sz​ spin-rotation symmetry are classified by ZZZ in even dimensions, while superconductors with the time reversal plus the Sz{S}_{z}Sz​ spin-rotation symmetries are classified by ZZZ in odd dimensions. We show that all 10 classes of gapped free-fermion phases can be realized by electron systems with certain symmetries. We also point out that, to properly describe the symmetry of a fermionic system, we need to specify its full symmetry group that includes the fermion number parity transformation ${(\ensuremath{-})}^{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}}$. The full symmetry group is actually a projective symmetry group.
We describe recent progress in our understanding of the interplay between interactions, symmetry, and topology in states of quantum matter. We focus on a minimal generalization of the celebrated topological band insulators (TBIs) to interacting many-particle systems known as symmetry-protected topological (SPT) phases. As with the TBIs, these states have a bulk gap and no exotic excitations but have nontrivial surface states that are protected by symmetry. We describe the various possible phases and their properties in three-dimensional systems with realistic symmetries. We develop many key ideas for the theory of these states using simple examples. The emphasis is on physical rather than mathematical properties. We survey insights obtained from the study of SPT phases for a number of other theoretical problems.