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Interaction-induced nematic Dirac semimetal from quadratic band touching: A constrained-path quantum Monte Carlo study
by Zi Hong Liu, Hongyu Lu, Zi Yang Meng, Lukas Janssen
Submission summary
| Authors (as registered SciPost users): | ZIHONG LIU |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2507.15668v1 (pdf) |
| Date submitted: | Aug. 17, 2025, 11:48 p.m. |
| Submitted by: | ZIHONG LIU |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Electronic systems with quadratic band touchings, commonly found in two- and three-dimensional materials such as Bernal-stacked bilayer graphene, kagome metals, HgTe, and pyrochlore iridates, have attracted significant interest concerning the role of interactions in shaping their electronic properties. However, even in the simplest model of spinless fermions on a two-dimensional checkerboard lattice, the quantum phase diagram as a function of nearest-neighbor interaction remains under debate. We employ constrained-path quantum Monte Carlo simulations (CP-QMC) simulations to investigate the problem using a two-dimensional torus geometry. We cross-validate our results on small lattices by comparing them with density-matrix renormalization group calculations, finding quantitative agreement. In particular, we implement an improved optimization scheme within the CP-QMC simulations, enabling the identification of a bond-nematic Dirac semimetal phase that was found in tensor-network studies on cylindrical geometries, but remains inaccessible to Hartree-Fock mean-field methods. The CP-QMC approach makes it possible to establish the emergence of this phase in a geometry that preserves lattice rotational symmetry and permits extrapolation to the thermodynamic limit. Our results show that the quantum phase diagram of spinless fermions on the checkerboard lattice with nearest-neighbor repulsion features three interaction-induced phases at half filling: a quantum anomalous Hall insulator at weak coupling, a bond-nematic Dirac semimetal at intermediate coupling, and a site-nematic insulator at strong coupling.
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The main result of the paper is the stabilization, on relatively large two-dimensional clusters, of the bond-nematic Dirac phase, which does not appear already at the mean-field level, but was obtained within DMRG on cylindrical geometries.
Since this paper does not predict a new phase, but rather provides more solid numerical evidence of the bond-nematic phase's existence, the numerical evidence must be robust. In this respect, my primary concern is with the approach used by the authors, namely, performing multiple self-consistency runs initialized from different mean-field trial states, and then selecting the one with the lowest energy as the final result. This approach seems in contradiction with the fact that the constrained-path Monte Carlo (CPMC) mixed estimator for the energy is not an upper bound to the exact energy, that is, CPMC is not a variational approach where the lowest energy corresponds to the best approximation of the true ground state (see for instance Carlos, Gubernatis, Ortis, Zhang, PRB 59, 12788 (1999)). This point requires clarification for the validity of the paper itself.
Once this important point is fixed, there are a few more questions that should be addressed by the authors, as listed below:
-) In the conclusions and also in the introduction, the authors refer to a certain number of experimental realizations of quadratic band touching systems. I would suggest expanding this discussion a bit, discussing whether any material is characterized by a checkerboard lattice, and to what extent the results obtained on the checkerboard lattice can also be applied to other lattice geometries.
-) In the Hartree-Fock decoupling of Eq. 4, there are no terms like $\langle c^{\dagger}_{r,1} c_{r+\delta,2}\rangle c^{\dagger}_{r+\delta,2} c_{r,1}$ and $\langle c^{\dagger}_{r+\delta,2} c_{r,1}\rangle c^{\dagger}_{r,1} c_{r+\delta,2}$ that renormalize the electron hopping and might be relevant in principle. Did the authors try to include them in their mean-field analysis, or do they have some argument about their non-relevance in the mean-field calculations?
-) I think it would be beneficial for the reader to include, in Sec. IIIB, some more details about the Hubbard-Stratonovich decomposition in the $S_z$ channel.
-) At the end of Sec. IVA, the authors state that the transition from the BNDS to the SNI phase shows a jump in $\sqrt{\cal{J}_{\textrm{QAH}}}$, while I think that they refer to a jump in $\Delta_{\textrm{bond}}$.
-) The entanglement entropy $S_E$ is shown in a figure, but it is not defined in the text.
Finally, I would like to mention a couple of typos/inaccuracies that I encountered in the text:
-) There is one additional $V$ inside the square bracket of Eq. 4.
-) Toward the end of page 8, there is written "agree with DMRG to within".
-) In the introduction, where the authors say "Upon the inclusion of interactions, however, the ground state may break some of the symmetries", it is unclear if they refer to the checkerboard lattice or to the Bernal-stacked honeycomb layer.
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