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

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.

Interesting new result.

Poor presentation and incomplete references.

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:
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--------------

Section 1:
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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.

See Report