Symmetry-Protected Topological Interfaces and Entanglement Sequences

TL;DR

This paper explores how gapped interfaces in 2D Abelian topological phases host sequences of 1D symmetry-protected topological states, linking entanglement entropy corrections to these states and their symmetry properties.

Contribution

It introduces the Boundary Topological Entanglement Sequence, connecting entanglement entropy corrections to classes of 1D SPT states at topological interfaces.

Findings

Interfaces support a sequence of 1D SPT states.

Entanglement entropy correction signals SPT emergence.

Domain walls host parafermion-like excitations.

Abstract

Gapped interfaces (and boundaries) of two-dimensional (2D) Abelian topological phases are shown to support a remarkably rich sequence of 1D symmetry-protected topological (SPT) states. We show that such interfaces can provide a physical interpretation for the corrections to the topological entanglement entropy of a 2D state with Abelian topological order found in Ref.~\onlinecite{cano-2015}. The topological entanglement entropy decomposes as $\gamma = \gamma_a + \gamma_s$, where $\gamma_a > 0$ only depends on universal topological properties of the 2D state, while a correction $\gamma_s > 0$ signals the emergence of the 1D SPT state that is produced by interactions along the entanglement cut and provides a direct measure of the stabilizing symmetry of the resulting SPT state. A correspondence is established between the possible values of $\gamma_s$ associated with a given interface - which is named Boundary Topological Entanglement Sequence - and classes of 1D SPT states. We show that symmetry-preserving domain walls along such 1D interfaces (or boundaries) generally host localized parafermion-like excitations that are stable to local symmetry-preserving perturbations.

Equations

$$\gamma ={\gamma }_a+{\gamma }_s,$$

$$\mathcal{K}=\left(\begin{array}{cc}{K}_L& 0\\ 0& -K_R\end{array}\right),$$

$${\mathcal{L}}_{1D}=\frac{1}{4\pi }{\partial }_t{\Phi }_I{\mathcal{K}}_{IJ}{\partial }_x{\Phi }^J-\frac{1}{4\pi }{\partial }_x{\Phi }^I{V}_{IJ}{\partial }_x{\Phi }^J-H_{int}[\Phi ],$$

$$\left[{\partial }_x{\Phi }_I(x),{\Phi }_J(y)\right]=-2\pi i{\mathcal{K}}_{IJ}^{-1}\delta (x-y).$$

$$V_{\ell }=\exp \left(i{\ell }^T\Phi \right),$$

$$\begin{array}{rl}{H}_{int}[\Phi ]& =\sum _{i=1}^r{a}_i\cos \left({\Lambda }_i^T\mathcal{K}\Phi \right)\\ & =\sum _{i=1}^r{a}_i\cos \left({\Lambda }_{L,i}^T{K}_L{\varphi }_L-{\Lambda }_{R,i}^T{K}_R{\varphi }_R\right),\end{array}$$

$${\Lambda }_i^T\mathcal{K}{\Lambda }_j=0,\forall i,j=1,...,r.$$

$${\mathcal{M}}_L{K}_L{\mathcal{M}}_L^T-{\mathcal{M}}_R{K}_R{\mathcal{M}}_R^T=0,$$

$$\mathcal{M}_{L/R}=\left(\begin{array}[]{c}\Lambda^{T}_{L/R,1}\\ \Lambda^{T}_{L/R,2}\\ \ldots\\ \Lambda^{T}_{L/R,r}\end{array}\right)\,$$

$${\mathcal{S}}_j^{{\theta }_j}=e^{i\frac{{\theta }_j}{2\pi }{\int }_{\mathcal{I}}dx{\Lambda }_i^T\mathcal{K}{\partial }_x\Phi },j=1,...,r,$$

$$\begin{array}{rl}& {\mathcal{S}}_j^{{\theta }_j}{\Psi }_{\ell }({\mathcal{S}}_j^{{\theta }_j}{)}^{-1}=e^{i{\theta }_j{\ell }^T\mathcal{K}{\Lambda }_j}{\Psi }_{\ell },\\ & {\Psi }_{\ell }=e^{i{\ell }^T\mathcal{K}\Phi }=e^{i\left({\ell }_L^T{K}_L{\varphi }_L+{\ell }_R^T{K}_R{\varphi }_R\right)}.\end{array}$$

$${\varphi }_L\to {\varphi }_L+2\pi K_L^{-1}{\alpha }_L,$$

$${\varphi }_R\to {\varphi }_R+2\pi K_R^{-1}{\alpha }_R,$$

$${\psi }_L\to {\psi }_L{e}^{i2\pi {\ell }_L^T{\alpha }_L},$$

$${\psi }_R\to {\psi }_R{e}^{i2\pi {\ell }_R^T{\alpha }_R}.$$

$${\mathcal{M}}_L{\alpha }_L\in {\mathbb{Z}}^r,$$

$${\mathcal{M}}_R{\alpha }_R\in {\mathbb{Z}}^r.$$

$${\mathcal{D}}_{L/R}={\mathcal{U}}_{L/R}{\mathcal{M}}_{L/R}{\mathcal{V}}_{L/R},$$

$${\alpha }_{L/R}^{\prime }={\mathcal{D}}_{L/R}^{-1}{t}_{L/R}^{\prime }.$$

$${\mathcal{G}}_{L/R}={\mathbb{Z}}_{d_{1,L/R}}\times {\mathbb{Z}}_{d_{2,L/R}}\times \cdots \times {\mathbb{Z}}_{d_{r,L/R}}.$$

$$|{\mathcal{G}}_{L/R}|=|\text{det}({\mathcal{M}}_{L/R})|.$$

$$\begin{array}{rl}& S_{L/R}=\exp \left[i{\int }_{\mathcal{I}}dx{\left({\mathcal{M}}_{L/R}^{-1}{t}_{L/R}\right)}^T{\partial }_x{\varphi }_{L/R}\right],\end{array}$$

$$\begin{array}{rl}& S_{L/R}^{(p)}=\exp \left[i{\int }_{\mathcal{I}}dx{\left({\alpha }_{L/R}^{(p)}\right)}^T{\partial }_x{\varphi }_{L/R}\right],\\ & p=1,...,r,\end{array}$$

$${\alpha }_{L/R}^{(p)}={\mathcal{V}}_{L/R}{\mathcal{D}}_{L/R}^{-1}{e}_p$$

$$\gamma ={\gamma }_{a,L/R}+{\gamma }_{s,L/R},$$

$${\gamma }_{a,L/R}=\frac{1}{2}\log |\det K_{L/R}|$$

$${\gamma }_{s,L/R}=\log |{\mathcal{G}}_{L/R}|.$$

$$\frac{|\text{det}(K_L)|}{|\text{det}(K_R)|}=\frac{|{\mathcal{G}}_R{|}^2}{|{\mathcal{G}}_L{|}^2},$$

$${\Lambda }_i^{\text{f.t.}}=\left(\begin{array}{c}{e}_i\\ W^T{e}_i\end{array}\right)$$

$$K_L=K,K_R=GK{G}^T,$$

$$\begin{array}{rl}& {\left({\Lambda }_G\right)}_i^T=\left(e_i^{T}G,e_i^T\right),i=1,...,r.\end{array}$$

$${\mathcal{M}}_L=G,{\mathcal{M}}_R={\text{I}}_{r\times r},$$

Full text

1. I Introduction

2. II General Approach

3. II.1 Left/right discrete symmetries of the interactions in Eq. $\eqref{eq: gapping interaction}$

4. II.2 Symmetry and entanglement

5. II.3 Primitivity Condition for the Null Vectors

6. III Fully Transmitting Interface

7. IV Gauging Related Interfaces and Emergent Interface Symmetry

8. IV.1 Inhomogeneous gauging related interfaces

9. IV.2 Homogeneous interface with emergent symmetry

10. V Homogeneous Self-Gappable Interfaces

11. V.1 $K=\sigma_{x}$

12. V.2 $K=\sigma_{z}$

13. V.2.1 First type:

14. V.2.2 Second type:

15. V.3 $1$D SPTs as the boundary of $2$D phases

16. VI Heterogeneous Stable Interfaces

17. VII Homogeneous interface between two chiral phases

18. VII.1 $K=\textrm{I}_{2\times 2}$ interface

19. VIII General Properties of Boundary Topological Entanglement Sequences

20. VIII.1 Emergent symmetry and entanglement correction from lattice maps

21. VIII.2 Gauge-related topological phases share the same BTES

22. VIII.2.1 Proof that each TEE correction obtainable at the $\tilde{K}_{R/L}$ interface is obtainable at the $K_{R/L}$ interface

23. VIII.2.2 Proof that each TEE correction obtainable at the $K_{R/L}$ interface is obtainable at the $\tilde{K}_{R/L}$ interface

24. VIII.3 Proof that $K$ and $p\,K$ ($p\in\mathbb{Z}^{*}$) share the same BTES

25. VIII.4 Comments on using entanglement sequences for classification

26. IX Conclusions

27. A Derivation of TEE correction

28. A.1 Proof that when $\begin{pmatrix}\mathcal{M}_{R}&\mathcal{M}_{L}\end{pmatrix}$ is primitive, $|\det v_{R/L}|=|\det\mathcal{M}_{R/L}|$

29. A.2 Proof that $|\det S|=\sqrt{|\det K_{L}\det K_{R}|}$ when $\begin{pmatrix}\mathcal{M}_{R}&\mathcal{M}_{L}\end{pmatrix}$ is primitive

30. B Commutation relations between vertex operators

31. C Generalization of the $\mathbb{Z}_{n}\times\mathbb{Z}_{n}$ SPT states discussed in Sec. V.1

I Introduction

Over the past decade, entanglement has played an increasingly important role in characterizing quantum phases of matter.Wen-book ; TQC-Review Two of the most successful diagnostics are the entanglement entropykitaevpreskill ; levinwen and the entanglement spectrum.lihaldane ; pollmann-2010 ; cirac-2011 ; regnault-2011 For a gapped state $|\Psi\rangle$ in two spatial dimensions (2D), the bi-partite entanglement entropy between two spatial regions $A$ and $B$ obeys the scaling law, $S_{\rm ent}\equiv-{\rm Tr}_{A}(\rho_{A}\log\rho_{A})=\alpha L-\gamma_{a}$, where the reduced density matrix $\rho_{A}={\rm Tr}_{B}|\Psi\rangle\langle\Psi|.$ $S_{\rm ent}$ scales with the linear size $L$ of the boundary of region $A$ (“area law” scaling) and generally contains a sub-leading, constant correction $\gamma_{a}$ known as the topological entanglement entropy (TEE), in the limit where $L>>\xi$, where $\xi$ is the correlation length.kitaevpreskill ; levinwen The seminal work of Levin-Wen and Kitaev-Preskill showed that $\gamma_{a}$ is an important quantum characteristic of a ground state that is determined solely by the topological properties of $|\Psi\rangle$.

When $|\Psi\rangle$ has a vanishingly small correlation length, the entanglement entropy probes the quasi-one-dimensional (1D) gapped state living along the entanglement cut,qikatsuraludwig ; lundgrenentanglement ; chen-fradkin-2013 ; furukawa-kim-2013 i.e., the entanglement is only sensitive to the physics near the interface between regions $A$ and $B$. Such an interface between (possibly even distinct) 2D gapped states with Abelian topological order (living in regions $A$ and $B$) has been the subject of much recent theoretical work.kapustin-2011 ; Levin-2013 ; Barkeshli-2013-a ; Lindner-2012 ; Clarke-2013 ; Cheng-2012 ; Vaezi-2013 ; Barkeshli-2013-b ; Mong-2014 ; Plamadeala2013 ; Cano-2014 ; Cano-2015-a ; Khan-2014 ; Khan-2016 ; Santos-2017 ; Lu-2014 It was shown in Ref. cano-2015, that interactions along the entanglement cut can produce a positive correction to the topological entanglement entropy (even in situations where $\gamma_{a}=0$), i.e., $\gamma_{a}$ increases such that the total entropy decreases. In such cases, these interactions constrain the motion of certain quasiparticles across the interface and, thereby, reduce the total entanglement between regions $A$ and $B.$ In Ref. cano-2015, , the TEE was extracted from the infinite $L$ limit of a single bipartition calculation, in contrast to the procedures used in Refs. kitaevpreskill, ; levinwen, that use linear combinations of entropies.

Nevertheless, the connection between the 1D state at the interface and the correction to the entanglement entropy in Ref. cano-2015, has remained elusive. The purpose of this work is to provide a physical interpretation for some of these 1D states by relating them to symmetry-protected topological phases (SPTs).Senthil-SPT-review SPTs, the spin-$1$ antiferromagnetic chainHaldane-1983-a ; Haldane-1983-b ; Affleck-1987 ; Affleck-1988 being the earliest known example, possess a gap to bulk excitations, and harbor symmetry-protected boundary modes. These states have been systematically studied using the tools of tensor networks,pollmann-2010 ; Schuch-2011 group cohomology,Chen-2013 ; Gu-Wen-2014 non-linear sigma models,Bi-2013 and effective field theories.Freedman2004 ; Fidkowski-2010 ; Fidkowski-2011 ; LevinStern2013 ; Neupert-2011 ; Santos-2011 ; Levin2011 ; LevinStern-2012 ; Ryu-2012 ; Levin-2012 ; Lu-2012 ; Vishwanath-2013 ; Metliski-2013-a ; Kapustin-2014 ; C-Wang-2014 ; J-Wang-2015 ; Ye-2016 One of the main results of this work is a general mechanism that generates 1D SPTs protected by Abelian discrete symmetries, that exist at the interface between 2D topological phases of matter. Importantly, the discrete symmetry that stabilizes the 1D phase acts solely on the local degrees of freedom along the interface, i.e., it does not affect the (possible) fractionalized quasiparticles of the bulk states. As a consequence, the properties of the 1D SPT phase are robust to local perturbations that respect the SPT symmetry.

In related work, it was shown in Ref. Zou-2016, that a non-zero correction to the TEE of topologically trivial bosonic $2$D states can be attributed to 1D SPTs living along the entanglement cut. In this respect, our work can be viewed as an extension of the relation between TEE corrections and 1D SPTs to a wide class of 2D Abelian topological phases, which include fermionic and bosonic topologically ordered states. Consequently, our results provide new insights into the connection between 1D SPTs and entanglement entropy of 2D gauge theories,casinihuertarosabal ; Ghosh2015 ; Fliss2017 and may provide useful theoretical guidance to the realization of 1D SPTs from 2D heterostructures.

Although our construction is more general, in this article we primarily focus on explicit examples of gapped interfaces that realize 1D bosonic SPTs protected by $\mathbb{Z}_{n}\times\mathbb{Z}_{n}$ symmetry, where the integer $n>1$. In this case, there are $n$ distinct classes of $\mathbb{Z}_{n}\times\mathbb{Z}_{n}$-protected 1D SPTs. These phases can be viewed as generalizationsGeraedts ; Santos-2015-a of the gapped, Haldane phase of the spin-1 chainHaldane-1983-a ; Haldane-1983-b ; Affleck-1987 ; Affleck-1988 for which $n=2$. Remarkably, we will show that these 1D SPTs can appear at interfaces between systems whose fundamental constituents are bosonic or fermionic, and that they can arise even when the bulk states are not topologically ordered with fractionalized excitations. As we demonstrate in our examples, the existence of these 1D SPT states can be inferred from the presence of symmetry-protected bound states that are localized at any domain wall between distinct classes of the 1D gapped phases.Santos-2015-a ; Geraedts Such domain walls can support parafermionic defectsLindner-2012 ; Clarke-2013 ; Cheng-2012 ; Vaezi-2013 ; Barkeshli-2013-b ; Mong-2014 ; Khan-2014 ; Khan-2016 ; Santos-2017 ; Fradkin-1980 whose properties may be useful for topological quantum computation architectures.TQC-Review

From this point of view, it is natural to link the correction to the entanglement entropy found in Ref. cano-2015, to the emergence of a 1D SPT. Given an entanglement cut that runs along such a gapped interface, the sub-leading constant in the entanglement entropy can be separated into a sum of two terms,

[TABLE]

where $\gamma_{a}=\log\mathcal{D}$ (with ${\cal D}$ the so-called total quantum dimensionkitaevpreskill ; levinwen ) is non-zero whenever the bulk 2D system has deconfined anyonic excitations, i.e., non-trivial topological order, and we will show below that $\gamma_{s}$ is a non-negative correction that can be associated with the emergent SPT interface. For a given set of bulk states that can share a gapped interface, there are many possible discrete values of $\gamma_{s}$ that can arise; the value of $\gamma_{s}$ depends on the gapping interactions at the interface. We refer to the set of possible $\gamma_{s}$ as the boundary topological entanglement sequence (BTES) corresponding to a given interface. It has been shown that $\gamma_{a}$ depends upon the universal topological data of the 2D state, which for an Abelian topological phase is encoded in its K-matrix (see Sec. II) and results in $\gamma_{a}=\log{\sqrt{|\textrm{det}(K)|}}.$ The correction $\gamma_{s}$ depends on the properties of the local interactions that give rise to a particular gapped interface, however it still only depends on universal data because the allowed sets of gapping interactions are solely determined by the properties of the K-matrix through the Haldane null-vector criterion.Haldane-1995 We will give a simple algorithm for calculating the BTES for Abelian 2D topological phases, and show to what extent $\gamma_{s}$ can serve to classify the possible SPT states that are supported along 1D interfaces.

For most of the calculations in this article we will imagine a generic setup where two topological phases of matter $L$ and $R$ are connected laterally through a gapped interface. The phases $L$ and $R$ are Abelian topological phases that will be described using the K-matrix Chern-Simons formalism, read-1990 ; wen-1992 and they could be identical or different. The analysis of the possible $1$D gapped interfaces between $L$ and $R$ reveals five different scenarios.

Fully transmitting Interface: In Sec. III, we consider interfaces across which every quasiparticle can move. Thus, the bulk phases on either side of the interface necessarily have the same topological order. When a quasiparticle crosses the interface, it either retains its topological properties or changes into another type of quasiparticle. As we will discuss, a fully transmitting interface does not give rise to a SPT phase. This is revealed by the absence of an emergent symmetry that is preserved by the interactions that gap the interface, and has $\gamma_{s}=0$. 2. 2.

Gauging Related Interfaces: In Sec. IV, we first study heterogeneous interfaces that appear at the boundary between distinct Abelian topological phases whose total quantum dimensions satisfy $\frac{\mathcal{D}_{R}}{\mathcal{D}_{L}}=k\in\mathbb{Z}$, where the $R$ phase is obtained from gauging a discrete $\mathbb{Z}_{k}$ global symmetry of the $L$ phase. As such, the local backscattering process that gaps this heterogeneous interface has an emergent $\mathbb{Z}_{k}$ global symmetry associated with the local degrees of freedom of $L.$ In Sec. VIII, we show that bulk Abelian phases whose K-matrices are either proportional or related by a gauging mechanism share the same BTES and SPT interfaces. This permits the grouping of different 2D phases into equivalence classes associated to their BTES and SPT interfaces. In the latter half of Section IV we study a sandwich structure with two identical Abelian topological phases separated by an intermediate gauged phase. We show that as the intermediate gauged region is thinned down to zero width, there is a clear mechanism for the formation of an emergent symmetry leading to an SPT state and an entanglement correction. 3. 3.

Homogeneous Self-Gappable SPT Interface: In Sec. V, we discuss interfaces separating the same non-chiral 2D bulk phase for which the degrees of freedom living along the left ($L$) and right ($R$) sides of the interface can be gapped independently. A 1D SPT can be generated by backscattering processes that mix the local degrees of freedom on the two sides of the interface such that the local interactions are invariant under a symmetry group $\mathcal{G}_{L}\times\mathcal{G}_{R}$, acting on the $L$ and $R$ degrees of freedom with $\mathcal{G}_{L}=\mathcal{G}_{R}=\prod_{i}\,\mathbb{Z}_{d_{i}}$. This type of interface yields a correction to the entanglement entropy given by $\gamma_{s}=\sum_{i}\,\log d_{i}$. We will show that in this case there are (pairs of) symmetry-protected non-Abelian zero modes centered around the positions of the domain walls separating distinct 1D phases. 4. 4.

Heterogeneous Stable Interface: In Sec. VI, we consider interfaces between distinct $L$ and $R$ bulk phases that are related by the gauging mechanism studied in Sec. IV. We specialize to the case where neither bulk $L$ or $R$ phase has boundary degrees of freedom that can be gapped independently.Levin-2013 We refer to the associated interface as “stable without symmetry,” or stable for short. Although one cannot generate a trivial SPT phase in a heterogeneous stable interface, since one cannot gap the $L$ and