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Exactly solvable models for topological phases

Tarantino, Nicolas Alessandro

Published Aug 1, 2017• 0 citations
OpenAlex

Abstract

Topological systems are characterized by some collection of features which remain unchanged under deformations of the Hamiltonian which leave the band gap open. The earliest examples of these were free fermion systems, allowing us to study the band structure to determine if a candidate material supports topological features. However, we can also ask the reversed question, i.e. Given a band gap, what topological features can be engineered? This classification problem proved to have numerous answers depending on which extra assumptions we allow, producing many candidate phases. While free fermion topological features could be classified by their band structures (culminating in the 10-fold way), strongly interacting systems defied this approach, and so classification outstripped the construction of even the most elementary Hamiltonians, leaving us with a number of phases which could exist, but do not have a single strongly interacting representative. The purpose of this thesis is to resolve this in certain cases by constructing commuting projector models (CPM), a class of exactly solvable models, for two types of topological phases, known as symmetry enriched topological (SET) order and fermionic symmetry protected topological (SPT) phases respectively. After introducing the background and history of commuting projector models, we will move on to the details of how these Hamiltonians are built. In the first case, we construct a CPM for a SET, showing how to encode the necessary group cohomology data into a lattice model. In the second, we construct a CPM for a fermionic SPT, and find that we must include a combinatorial representation of a spin structure to make the model consistent. While these two projects were independent, they are linked thematically by a technique known as decoration, where extra data is encoded onto simple models to generate exotic phases. | 90 pages

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The interplay between symmetry and topological properties plays a very important role in modern physics. In the past decade, the concept of symmetry-enriched topological (SET) phases was proposed and their classifications have been systematically studied for bosonic systems. Very recently, the concept of SET phases has been generalized into fermionic systems and their corresponding classification schemes are also proposed. Nevertheless, how to realize all these fermionic SET (fSET) phases in lattice models remains to be a difficult open problem. In this paper, we first construct exactly solvable models for nonanomalous nonchiral 2 + 1 D fSET phases, namely, the symmetry-enriched fermionic string-net models, which are described by commuting-projector Hamiltonians whose ground states are the fixed-point wave functions of each fSET phase. Mathematically, we provide a partial definition to G -graded super fusion category (when the total symmetry G f is a direct product of physical symmetry G and fermion-parity symmetry Z 2 f ), which is the input data of a symmetry-enriched fermionic string-net model. Next, we construct exactly solvable models for nonchiral 2 + 1 D fSET phases with ’t Hooft anomaly, especially the H 3 ( G , Z 2 ) fermionic ’t Hooft anomaly which is different from the well-known bosonic H 4 [ G , U ( 1 ) T ] anomaly. In our construction, this H 3 ( G , Z 2 ) fermionic ’t Hooft anomaly is characterized by a violation of fermion-parity conservation in some of the surface F moves (a type of renormalization moves for the ground state wave functions of surface fSET phases), and also by a new fermionic obstruction Θ in the surface pentagon equation. We demonstrate this construction in a concrete example that the surface topological order is a Z 4 gauge theory embedded into a fermion system and the total symmetry G f = Z 2 f × Z 2 × Z 4 . We further conjecture that the H 2 ( G , Z 2 ) fermionic ’t Hooft anomaly is characterized by a possible violation of nonterminating q -type (called σ -type) string in the fusion rules of surface models.

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Topological systems are characterized by some collection of features which remain unchanged under deformations of the Hamiltonian which leave the band gap open. The earliest examples of these were free fermion systems, allowing us to study the band structure to determine if a candidate material supports topological features. However, we can also ask the reversed question, i.e. Given a band gap, what topological features can be engineered? This classification problem proved to have numerous answers depending on which extra assumptions we allow, producing many candidate phases. While free fermion topological features could be classified by their band structures (culminating in the 10-fold way), strongly interacting systems defied this approach, and so classification outstripped the construction of even the most elementary Hamiltonians, leaving us with a number of phases which could exist, but do not have a single strongly interacting representative. The purpose of this thesis is to resolve this in certain cases by constructing commuting projector models (CPM), a class of exactly solvable models, for two types of topological phases, known as symmetry enriched topological (SET) order and fermionic symmetry protected topological (SPT) phases respectively. After introducing the background and history of commuting projector models, we will move on to the details of how these Hamiltonians are built. In the first case, we construct a CPM for a SET, showing how to encode the necessary group cohomology data into a lattice model. In the second, we construct a CPM for a fermionic SPT, and find that we must include a combinatorial representation of a spin structure to make the model consistent. While these two projects were independent, they are linked thematically by a technique known as decoration, where extra data is encoded onto simple models to generate exotic phases. | 90 pages
In this thesis, we study gapped topological phases of matter in systems with strong inter-particle interaction. They are challenging to analyze theoretically, because interaction not only gives rise to a plethora of phases that are otherwise absent, but also renders methods used to analyze non-interacting systems inadequate. By now, people have had a relatively systematic understanding of topological orders in two spatial dimensions. However, less is known about the higher dimensional cases. In Chapter 2, we will explore three dimensional long-range entangled topological orders in the framework of Walker-Wang models, which are a class of exactly solvable models for three-dimensional topological phases that are not known previously to be able to capture these phases. We find that they can represent a class of twisted discrete gauge theories, which were discovered using a different formalism. Meanwhile, a systematic theory of bosonic symmetry protected topological (SPT) phases in all spatial dimensions have been developed based on group cohomology. A generalization of the theory to group supercohomology has been proposed to classify and characterize fermionic SPT phases in all dimensions. However, it can only handle cases where the symmetry group of the system is a product of discrete unitary symmetries. Furthermore, the classification is known to be incomplete for certain symmetries. In Chapter 3, we will construct an exactly solvable model for the two-dimensional time-reversal-invariant topological superconductors, which could be valuable as a first attempt to a systematic understanding of strongly interacting fermionic SPT phases with anti-unitary symmetries in terms of exactly solvable models. In Chapter 4, we will propose an alternative classification of fermionic SPT phases using the spin cobordism theory, which hopefully can capture all the phases missing in the supercohomology classification. We test this proposal in the case of fermionic SPT phases with Z2 symmetry, where Z2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimensions less than or equal to 3.
In this thesis we investigate topological phases of matter that have a global, unbroken symmetry group -- also known as symmetry enriched topological (SET) phases. We address three questions about these phases: (1) how can we build exactly solvable models that realize them? (2) how can we determine if their edge theories can be gapped without breaking the symmetry? and (3) how do we understand the phenomenon of decoupled charge and neutral modes which occurs in certain fractional quantum Hall states? ,More specifically, we address the first question by constructing exactly solvable models for a wide class of symmetry enriched topological (SET) phases, which we call symmetry-enriched string nets. The construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group G, and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2 symmetry which exchanges the 'e' and 'm' type anyons. We further illustrate our construction with a number of additional examples. ,For the second question, we focus on the edge theories of 2D SET phases with Z2 symmetry. The central problem we seek to solve is to determine which edge theories can be gapped without breaking the symmetry. Previous attempts to answer this question in special cases relied on constructing perturbations of a particular type to gap the edge. This method proves the edge can be gapped when the appropriate perturbations can be found, but is inconclusive if they cannot be found. We build on this previous work by deriving a necessary and sufficient algebraic condition for when the edge can be gapped. Our results apply to Z2 symmetry protected topological phases as well as Abelian Z2 SET phases,Finally, in the fourth chapter, we describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode --- i.e. an edge mode that does not carry electric charge. These models consist of two counter-propagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the nu = 2/3 state is the most prominent example.
We construct exactly solved commuting projector Hamiltonian lattice models for all known (2+1)-dimensional (2+1D) fermionic symmetry protected topological phases (SPTs) with on-site unitary symmetry group ${G}_{f}=G\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{f}$, where GGG is finite and Z2f{\mathbb{Z}}_{2}^{f}Z2f​ is the fermion parity symmetry. In particular, our models transcend the class of group supercohomology models, which realize some, but not all, fermionic SPTs in 2+1D. A natural ingredient in our construction is a discrete form of the spin structure of the 2D spatial surface MMM on which our model is defined, namely a ``Kasteleyn'' orientation of a certain graph associated with the lattice. As a special case, our construction yields commuting projector models for all eight members of the Z8{\mathbb{Z}}_{8}Z8​ classification of 2D fermionic SPTs with G=Z2G={\mathbb{Z}}_{2}G=Z2​.
Using the decorated domain wall procedure, we construct finite-depth local unitaries that realize fermionic symmetry-protected topological (SPT) phases. This results in explicit full commuting projector Hamiltonians, where ``full'' implies the fact that the ground state as well as all excited states of these Hamiltonians realize the nontrivial SPT phase. We begin by constructing explicit examples of 1+1D phases protected by the symmetry group $G={\mathbb{Z}}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}$, which also has a free fermion realization in class BDI, and by the symmetry group $G={\mathbb{Z}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{4}^{F}$, which does not. We then turn to 2+1D, and construct the square roots of the Levin-Gu bosonic SPT phase, protected by ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{F}$ symmetry, in a concrete model of fermions and spins on the triangular lattice. Edge states and the anomalous symmetry action on them are explicitly derived. Although this phase has a free fermion representation as two copies of p+ipp+ipp+ip superconductors combined with their $p\ensuremath{-}ip$ counterparts with a different symmetry charge, the full set of commuting projectors is only realized in the strongly interacting version, which also implies that it admits a many-body localized realization.
The classification and lattice model construction of symmetry protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group Gf=Z2f×ω2GbG_f=\mathbb{Z}_2^f \times_{\omega_2} G_bGf​=Z2f​×ω2​​Gb​ which is a central extension of bosonic symmetry group GbG_bGb​ (may contain time reversal symmetry) by the fermion parity symmetry group Z2f={1,Pf}\mathbb{Z}_2^f = \{1,P_f\}Z2f​={1,Pf​}. Our construction is based on the concept of equivalence class of finite depth fermionic symmetric local unitary (FSLU) transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We will also discuss the systematical construction and classification of boundary anomalous SPT (ASPT) states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction-free and trivialization-free constructions naturally lead to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point/space group symmetry as well as continuum Lie group symmetry.
The interplay between symmetry and topological properties plays a very important role in modern physics. In the past decade, the concept of symmetry-enriched topological (SET) phases was proposed and their classifications have been systematically studied for bosonic systems. Very recently, the concept of SET phases has been generalized into fermionic systems and their corresponding classification schemes are also proposed. Nevertheless, how to realize all these fermionic SET (fSET) phases in lattice models remains to be a difficult open problem. In this paper, we first construct exactly solvable models for nonanomalous nonchiral 2 + 1 D fSET phases, namely, the symmetry-enriched fermionic string-net models, which are described by commuting-projector Hamiltonians whose ground states are the fixed-point wave functions of each fSET phase. Mathematically, we provide a partial definition to G -graded super fusion category (when the total symmetry G f is a direct product of physical symmetry G and fermion-parity symmetry Z 2 f ), which is the input data of a symmetry-enriched fermionic string-net model. Next, we construct exactly solvable models for nonchiral 2 + 1 D fSET phases with ’t Hooft anomaly, especially the H 3 ( G , Z 2 ) fermionic ’t Hooft anomaly which is different from the well-known bosonic H 4 [ G , U ( 1 ) T ] anomaly. In our construction, this H 3 ( G , Z 2 ) fermionic ’t Hooft anomaly is characterized by a violation of fermion-parity conservation in some of the surface F moves (a type of renormalization moves for the ground state wave functions of surface fSET phases), and also by a new fermionic obstruction Θ in the surface pentagon equation. We demonstrate this construction in a concrete example that the surface topological order is a Z 4 gauge theory embedded into a fermion system and the total symmetry G f = Z 2 f × Z 2 × Z 4 . We further conjecture that the H 2 ( G , Z 2 ) fermionic ’t Hooft anomaly is characterized by a possible violation of nonterminating q -type (called σ -type) string in the fusion rules of surface models.