## SciPost Submission Page

# Interacting Edge States of Fermionic Symmetry-Protected Topological Phases in Two Dimensions

### by Joseph Sullivan, Meng Cheng

### Submission summary

As Contributors: | Meng Cheng |

Arxiv Link: | https://arxiv.org/abs/1904.08953v3 |

Date submitted: | 2019-08-23 |

Submitted by: | Cheng, Meng |

Submitted to: | SciPost Physics |

Discipline: | Physics |

Subject area: | Condensed Matter Physics - Theory |

Approach: | Theoretical |

### Abstract

Recently, it has been found that there exist symmetry-protected topological phases of fermions, which have no realizations in non-interacting fermionic systems or bosonic models. We study the edge states of such an intrinsically interacting fermionic SPT phase in two spatial dimensions, protected by $\mathbb{Z}_4\times\mathbb{Z}_2^T$ symmetry. We model the edge Hilbert space by replacing the internal $\mathbb{Z}_4$ symmetry with a spatial translation symmetry, and design an exactly solvable Hamiltonian for the edge model. We show that at low-energy the edge can be described by a two-component Luttinger liquid, with nontrivial symmetry transformations that can only be realized in strongly interacting systems. We further demonstrate the symmetry-protected gaplessness under various perturbations, and the bulk-edge correspondence in the theory.

###### Current status:

### Submission & Refereeing History

## Reports on this Submission

### Anonymous Report 2 on 2019-11-4 Invited Report

### Report

The question studied by the authors, namely possible phases of edge mode of a topological phase due to interactions on the edge, is interesting and timely. The example that they work out is valuable and is written out clearly (Sections 3-5). Unfortunately, the introduction (Sections 1-2) is not well written, with many terms being left undefined ($Z_2^T$, FSPT, to state two examples), and many assumptions on prior knowledge of the reader being made. I propose that the authors make an effort to clarify the presentation in these sections. Once this is done, I recommend the paper for publication.

### Anonymous Report 1 on 2019-10-25 Invited Report

### Strengths

Interesting new result.

### Weaknesses

Poor presentation and incomplete references.

### Report

Summary of the manuscript:

***********************

The subject matter of this manuscript is the study of

fermionic symmetry-protected topological (FSPT) phase of maters in two-dimensional space. The idea of this manuscript, as I understood it, is the following.

Consider first an open Kitaev chain, i.e., a one-dimensional superconductor

that supports one Majorana zero mode at its left end and another one

at its right end. Impose symmetry under complex conjugation

so as to realize a fermionic symmetry protected topological (FSPT) phase

in the symmetry class BDI.

Second, consider a pair of such Kitaev chains, stacked on top of each others.

There are two Majorana zero modes on the left and two Majorana zero modes

on the right. The left (right) pair of Majorana zero modes combines into

a fermionic zero mode [the zero mode of a Schrieffer-Su-Heeger (SSH) chain]

localized on the left (right) side of the SSH chain.

Third, stack infinitely many such SSH chains. Because of the symmetry under

complex conjugation, there are infinitely many fermionic modes

localized at the left end of each SSH chains.

No one-body fermionic hopping between these fermionic zero modes

is allowed by the BDI symmetry.

The idea of the authors is to write down a quartic interaction

for the fermionic modes localized on the left sides of the SSH chains

that is invariant under (i) the representation of

the operation of complex conjugation for the original Majorana operators

and (ii) any translation along the stacking direction.

This fermionic model is an effectively one-dimensional lattice model

that respects fermionic parity but does not conserve the fermion number,

as it depends on a parameter $t\in\mathbb{R}$ compatible with the

conservation of the fermion number and a parameter $\Delta\in\mathbb{R}$

that breaks the U(1) global gauge symmetry down to $\mathbb{Z}_{2}$.

This effective one-dimensional fermionic Hamiltonian is solved exactly

by applying Jordan-Wigner transformations. It is found that there are

gapful phases separated by gapless phases in the phase diagram.

In the gapped phases, the representation of the operation of

complex conjugation for the original Majorana operators is always broken.

Moreover, the gapped phase with $t>0$ differs from the gapped phase with

$t<0$ in that translation by one lattice spacing

is also broken but not the product of the two broken symmetries

when $t<0$. The gapful phases thus always break spontaneously at least one of

the two protecting symmetries. These gapful phases are separated by

quantum critical lines in parameter space. By the bulk-edge correspondence,

these quantum critical lines are thus the signature of

fermionic symmetry protected topological phases in two-dimensional space.

Finally, a stability analysis of the boundary theory is performed at the

level of Abelian bosonization.

Criticism:

-------

This manuscript is interesting and I believe that the results are solid.

Unfortunately, it is poorly written. I fear that

the first two sections in which the problem to be solved is motivated and

the solution is explained are inaccessible to the target audience, at least

it was inaccessible to this referee.

The bulk of the paper is accessible but written in a hurry, with little

attention to details and precision. Citations are incomplete and bias.

Detailed comments:

--------------

--------------

Section 1:

-------

The notation "\mathbb{Z}^{\mathsf{T}}_{2}'' is not explained.

I am guessing that the superscript is supposed to imply that the physical

interpretation of this group is that it is generated by reversal of time.

The sentence

"As a result, a boundary without symmetry breaking is either gapless,

or gapped with intrinsic topological order [4].''

only applies to a boundary whose dimensionality is larger than one.

"... namely intrinsically interacting fermionic SPT phases, ...''

should become

"... namely intrinsically interacting fermionic SPT (FSPT) phases, ...''

"... can always be thought as a superposition of ...''

should become

"... can always be thought of as a superposition of ...''

Section 2:

-------

Section 2 is very hard to follow. This section reviews the concepts at

work in this manuscript. However, it is written in such a compact way that

I could only guess what the authors had in mind from reading the Sections

in which calculations are presented.

I am not familiar with the statement

"The ground state wavefunction of a SPT phase can

always be thought as a superposition of

all domain-wall configurations.''

I did not understand the sentence

"Many nontrivial SPT phases can be constructed by decorating

SPT states in one dimension lower on domain walls.''

From the context, it seems that

"\mathbb{Z}^{\mathsf{T}}_{2}'' refers to

a reversal of time that squares to unity.

Why not explain this? According to the tenfold way,

the symmetry classes CII and DIII are also protected by

time-reversal symmetry, but one that squares to minus unity.

I am not sure how the authors would denote such a symmetry,

but the very existence of the symmetry classes CII and DIII

caused my confusion.

The sentence

"In this work we adopt a different approach, invoking the connection

between SPT phases with internal symmetry and those with crystalline

symmetry of the same group structure. This correspondence has been

formalized in Ref. [31], and verified in many examples.''

is written in a way to discourage any reader except perhaps authors

of Ref. 31. More pedagogy would be welcome in this paragraph.

The last paragraph of Section 2 is again written as if one needed to

belong to a secret brotherhood to understand what is meant.

Section 3:

-------

Could the authors clarify if the particle-hole transformation

defined in Eq. (3) is antiunitary. It should be since it originates

from complex conjugation, but it would be nice to remove any ambiguity

since particle-hole transformations of operators are often defined to

be unitary.

The range $i=1,...,2N$ should be declared in Eq. (2), not only in Fig.\ 1.

The definition of the "physical fermion'' deserves its numbered equation

together with the algebra that it obeys.

Why is the fermion operator that is constructed from a pair of Majorana

operators called "physical'' after Eq. (2)?

The sentence

"We are interested in symmetric phases of the BDI chain,

which is necessarily gapless.''

is confusing for two reasons.

First, an explanation would be needed. Naively, one might invoke

the Lieb-Schultz-Mattis (LSM) theorem,

but this theorem presumes a U(1) symmetry, which is broken in Eq. (4).

Second, I thought that a fair amount of space is also devoted to

the gaped phases in this manuscript.

It is not specified if $t$ and $\Delta$ are real valued or complex valued

in Eq. (4). Both options are possible to the best of my knowledge at this stage,

but I believe that the authors restrict themselves to $t$ and $\Delta$

real valued later on.

The statement that the pair of fermion operators defined in Eq. (9)

commute should come just after (9), not after (10).

Moreover, \qquad should be inserted between the definitions of

$c_n$ and $\tilde{c}_{n}$. [Same applies to Eq. (8).]

The range of $i$ in Eq. (10) is not the same as that of $i$ in Eq.\ (7).

The authors could have reserved the letter $j,n=1,...,N$ when distinguishing

even from odd sites as they did in Eqs. (8) and (9). The same comment

applies to Eqs.\ (11), (12).

Section 4:

-------

The main goal of this Section is to show that the gaped phases of the

boundary Hamiltonian break spontaneously some of its symmetries.

It would be helpful to announce what will be done in this Section

in the first paragraph of this Section.

I do not understand the sentence

"Since the model is supposed to describe an anomalous edge,

the ground state is either gapless,

or gapped with spontaneous breaking of the symmetries.''

The dichotomy between gaplessness or a gap with

spontaneous symmetry breaking (SSB) seems to originate from the

one-dimensional nature of the boundary, not from it supporting an

anomalous theory. As I said earlier, a reference explaining this dichotomy

would be nice if it exists (LSM theorem cannot be applied).

I do not understand this claim

"..., which are gapped whenever $\Delta\neq 0$.''

What happens when $t=0$, is there a gap?

Equation (17) seems to demand that $\Delta$ and $t$ are real valued.

Equation (18) needs some horizontal spaces for readability.

Equation (18) should have been presented after Eq.\ (7).

Equation (22) is incomplete as it is not specified explicitly

of which Pauli matrix $\boldsymbol{\tau}_{i}$

$|\uparrow\rangle_{i}$ is eigenstate with eigenvalue +1.

If the action of $T_r$ on

$|\uparrow\rangle_{i}$

had been given explicitly,

it would be obvious by inspection that eigenstates

(23) and (25) are not eigenstates of $T_r$.

Section 5:

-------

Section 5 deals with the line of critical points $\Delta=0$.

When $\Delta=0$, the boundary Hamiltonian is gapless.

This is a necessary condition for the

two-dimensional theory to realize a fermionic symmetry protected boundary.

To show that the $\Delta=0$ condition is sufficient for the

two-dimensional theory to realize a fermionic symmetry protected boundary,

the authors will prove that this line of quantum critical points is robust to

interactions that are allowed by the protecting symmetries.

To this end, they will describe the line of critical points $\Delta=0$

by a bosonic theory and use and a reasoning going back to a seminal paper by

Haldane to show that no interaction can gap this bosonic theory without

spontaneously breaking the protecting symmetries.

This bosonic theory is referred to as a "chiral bosonic theory''.

I will come back to this terminology below.

Equation (30) omits the sector with $\tilde{c}$ operators.

This seems to be a typo.

"... do form the right group structure faithfully.''

should be changed to

"... do represent faithfully the group

$\mathrm{Z}^{\mathsf{T}}_{2}\times\mathbb{Z}$.''

The references in the sentence

"Here we discuss an alternative

formulation using K matrix [39–42], ...''

are incomplete and bias.

The second paper after Ref.\ [39] to be cited in this context

should be that of Haldane 10.1103/PhysRevLett.74.2090,

since it is in this paper that the condition (54) was introduced

for the first time, to the best of my knowledge.

Haldane as Wen were only considering $K$ matrices with a net chirality

(their traces was non vanishing). Early manuscripts dealing with the stability

of the boundary theory using non-chiral bosonization techniques

($K$ matrices whose traces are vanishing) are those of

Levin and Stern, 10.1103/PhysRevLett.103.196803,

and Neupert et al., 10.1103/PhysRevB.84.165107.

The extension to all ten symmetry classes from the tenfold way was done

by Neupert et al in 10.1103/PhysRevB.90.205101.

I do not understand the sentence

"... to understand the perturbative stability.''.

My understanding of Haldane's 10.1103/PhysRevLett.74.2090

is that his approach to the stability of the edge states

was designed to be nonperturbative, as opposed to that used by

Kane, Fisher, and Polchinski in 10.1103/PhysRevLett.72.4129.

The references [42,43] in the sentence

"In an effort to gap out the theory

we can consider adding Higgs terms of the form [42, 43] ...''

are problematic for two reasons. First, Ref.\ [43] is the same

as Ref.\ [40]. Second both References are written after

Neupert et al., 10.1103/PhysRevB.84.165107

in which the strategy of "adding Higgs terms ''

is applied to study the effects of

strong interactions on the boundary of a topological insulator in

the symmetry class AII.

I have the same difficulty as above with the use of "perturbatively'' in

"Through a null vector analysis,

we have shown that perturbatively the gapless edge modes are

protected by the symmetry.''

I would like to comment about the use of the adjective "chiral''.

The authors correctly use the adjective chiral when describing

operators that obey either left-moving or right-moving equations of motion.

The authors state correctly that their K matrix is "non chiral''

as it is traceless. I am not sure it is wise to call Eq. (42)

a general chiral Luttinger liquid if one does not state explicitly that

the K matrix is not traceless. For this reason, I would omit chiral in

the sentence "... and the edge chiral boson theory following ...''

and in the first section of Section 6. To put it differently,

I can always recast a nonchiral bosonic theory in one dimension

in terms of pairs of chiral fields with opposite chiralities,

but a bosonic theory with an odd number

of chiral fields is necessarily chiral.

Section 6:

-------

The case $t=0$ was never discussed.

It is reiterated by the authors that the main result of this manuscript is

the construction of a FSPT phase in two-dimensional space that

cannot be realized by free fermions. In this context, I would like to point

out that a (nonchiral) FSPT phase that cannot be realized with

free fermions was constructed by

Neupert et al.\ 10.1103/PhysRevB.90.205101.

This FSPT phase displays a nonvanishing quantum Hall conductivity

with vanishing thermal Hall conductivity. I do not know if this was the first

example of a FSPT phase that has no free fermion counterpart, but

it is certainly one of the early examples.

### Requested changes

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