# Gauging the Kitaev chain

### Submission summary

 As Contributors: Umberto Borla · Sergej Moroz Arxiv Link: https://arxiv.org/abs/2010.00607v2 (pdf) Date submitted: 2021-02-05 20:57 Submitted by: Borla, Umberto Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

We gauge the fermion parity symmetry of the Kitaev chain. While the bulk of the model becomes an Ising chain of gauge-invariant spins in a tilted field, near the boundaries the global fermion parity symmetry survives gauging, leading to local gauge-invariant Majorana operators. In the absence of vortices, the Higgs phase exhibits fermionic symmetry-protected topological (SPT) order distinct from the Kitaev chain. Moreover, the deconfined phase can be stable even in the presence of vortices. We also undertake a comprehensive study of a gently gauged model which interpolates between the ordinary and gauged Kitaev chains. This showcases rich quantum criticality and illuminates the topological nature of the Higgs phase. Even in the absence of superconducting terms, gauging leads to an SPT phase which is intrinsically gapless due to an emergent anomaly.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission 2010.00607v2 on 5 February 2021

## Reports on this Submission

### Strengths

1- Gauging the Kitaev chain in the presence of boundary.

2- Determining the gauge structure of the fermion number in the bulk and at the boundary.

3- Determining the phase structure of the Kitaev chain as a function of the coupling strengths.

### Weaknesses

1) Leaving a (possibly relevant) paper out.

2) Leaving certain quantities vague.

3) Gentle gauging of the Kitaev chain.

### Report

In this paper, the authors study gauging of the fermion parity symmetry in the Kitaev chain. The paper presents a detailed study of the gauged fermion parity, and arrives at new results concerning the phases, vorticity and boundary effects. Before recommending the paper for publication, I suggest the authors to discuss the points raised below.

### Requested changes

1) The authors have referred to gauge theory works in detail. But, still, it would be complementary to include a brief discussion of

Krauss LM, Wilczek F. Discrete gauge symmetry in continuum theories. Phys Rev Lett. 1989 Mar 13;62(11):1221-1223. doi: 10.1103/PhysRevLett.62.1221

This paper may help developing a more detailed (or slightly alternative) view of the transition from bulk to the boundary. (In gauge theories, boundary seldom appears, and this is one of the factors that make the present paper interesting.)

2) In Eq. (14), the coupling constant h needs be explained in terms of its origin. That “h” term appears as a gauge-fixing term. Is h a gauge-fixing parameter? What is then its physical relevance (other than causing vortices in Sec. 3.3 and Sec. 3.7)? Unless the emergent parameter h is clarified part of the analysis appears to result from parameter choices. For instance, authors take h=0 in Sec. 3.5 and Sec. 3.6 where the Higgs, deconfined and SPT phases are discussed. It would make the paper comprehensible and complete to include a discussion of h.

3) At the beginning of Sec. 3.4 the bosonic nature of the bulk needs be expanded somewhat. For example, what is the dynamical use/role of that “emergent fermion”?

4) In Fig. 3, the gauged (at the left?) and non-gauged (at the right?) Kitaev chains can be made more tractable.

5) In Sec. 4.1, Eq. (35), the coupling K multiplies the Gauss’ law. This means that Gauss’ law is imposed at high K, where at low K conservation laws are not guaranteed. Also role of K in the sense of effective field theory is not clear. It looks like small K means loss of Gauss’ law more than non-gauging (the authors propose). This point needs be clarified. (Is the anomaly in Sec. 5.2 connected to these features?) Speaking in general, the gentle gauging seems to involve somewhat ad hoc Hamiltonian in Eq.(35). The authors might want to reconsider this part in terms of discussions and analyses.

6) Fig. 5 can be extended from h=0 to larger h values to reveal the dependence of entropy on this gauge parameter.

7) I suggest the authors to have a full check of their equations and arguments. Things may be as simple as not defining hermiticity of γ ̃ in Eq. (2).

• validity: high
• significance: high
• originality: high
• clarity: good
• formatting: good
• grammar: good

### Report

The authors carefully work out the consequences of gauging the fermion-parity symmetry of the Kitaev chain, including the effect of including a "magnetic"-symmetry breaking (i.e. vortex creating term) term. They show how it maps onto an effective Ising chain, with the vortex term becoming the longitudinal magnetic field. They extend their results to a model with an enhanced U(1)-fermion number symmetry, explaining the role of an anomaly.

I recommend publication in SciPost. While this is basic stuff, that is a virtue: the field will benefit from such a clear presentation and thorough treatment. Moreover, the authors provide an engaging discussion of how to physically interpret the different phases arising, including the appearance of a fermionic SPT phase. I am happy that the authors took the time.

I don't have any major suggestions. Perhaps the authors should mention that their models are all free-fermionic except when $h\ne 0$. Perhaps also the authors should stress a little more why the deconfined phase is stable only for $\mu<0$ (antiferromagnetic coupling in the effective Ising picture). The simplest reason I know is that translation symmetry plays the role of the Ising $\mathbb{Z}_2$ symmetry, which makes the longitudinal magnetic field irrelevant.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -