# Hidden Symmetries, the Bianchi Classification and Geodesics of the Quantum Geometric Ground-State Manifolds

### Submission summary

 As Contributors: Diego Liska Preprint link: scipost_202009_00015v2 Date accepted: 2021-01-25 Date submitted: 2020-12-19 15:00 Submitted by: Liska, Diego Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

We study the Killing vectors of the quantum ground-state manifold of a parameter-dependent Hamiltonian. We find that the manifold may have symmetries that are not visible at the level of the Hamiltonian and that different quantum phases of matter exhibit different symmetries. We propose a Bianchi-based classification of the various ground-state manifolds using the Lie algebra of the Killing vector fields. Moreover, we explain how to exploit these symmetries to find geodesics and explore their behaviour when crossing critical lines. We briefly discuss the relation between geodesics, energy fluctuations and adiabatic preparation protocols. Our primary example is the anisotropic transverse-field Ising model. We also analyze the Ising limit and find analytic solutions to the geodesic equations for both cases.

Published as SciPost Phys. 10, 020 (2021)

### List of changes

- Added references [28], [29] and [30]
- New figures 2 and 4
- Corrected typos and misnames

### Submission & Refereeing History

Resubmission scipost_202009_00015v2 on 19 December 2020
Submission scipost_202009_00015v1 on 18 September 2020

## Reports on this Submission

### Report 2 by Tapobrata Sarkar on 2021-1-11 (Invited Report)

• Cite as: Tapobrata Sarkar, Report on arXiv:scipost_202009_00015v2, delivered 2021-01-11, doi: 10.21468/SciPost.Report.2390

### Report

I am okay with the response of the authors and am happy to recommend publication at this stage.

There are just a couple of issues that I raise, although this has little to do with the present version.

1) In response to one of my queries, the authors state that "We do not agree that with the statement that the curvature does not have singularities at $h = \pm 1$. In fact, some components of the Riemann and Ricci tensors do have this singular behaviour, see Appendix C of arXiv: 1305.0568 for explicit expressions."

I would not fully agree with this. Components of the Riemann and Ricci tensors are not scalars, and the divergence of such components is a basis dependent statement, so one cannot really draw too many conclusions from there (meaning that a change of basis might do away with these divergences, although what the new basis means might be another story). What I meant was that the Ricci scalar is non-divergent (but discontinuous nonetheless). This is a coordinate independent statement and in two dimensions is a complete characterisation of the singularity. I am still a little surprised that geodesics can end at $h = \pm 1$ because there is no true (scalar) curvature singularity there. Typically, geodesics end on proper singularities, but maybe this is a matter of finer debate.

2) In a similar spirit, when I said the metric does not depend on $h$, I meant that there exists a coordinate transformation due to which the explicit dependence on $h$ can be done away with. And since one is finally interested in scalars anyway, one metric is as good as another transformed one. In any case, I accept the authors' response here.

In summary, I recommend publication of the manuscript in the present form.

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

### Strengths

See initial report.

### Weaknesses

No major weakness in particular.

### Report

All my comments have been addressed, and I am happy to recommend this work for publication in SciPost Physics.

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