Metallic and Deconfined Quantum Criticality in Dirac Systems

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Metallic and Deconfined Quantum Criticality in Dirac Systems

Zi Hong Liu, Matthias Vojta, Fakher F. Assaad, and Lukas Janssen

Phys. Rev. Lett. 128, 087201 – Published 23 February 2022

Abstract

Motivated by the physics of spin-orbital liquids, we study a model of interacting Dirac fermions on a bilayer honeycomb lattice at half filling, featuring an explicit global $SO (3) \times U (1)$ symmetry. Using large-scale auxiliary-field quantum Monte Carlo (QMC) simulations, we locate two zero-temperature phase transitions as function of increasing interaction strength. First, we observe a continuous transition from the weakly interacting semimetal to a different semimetallic phase in which the SO(3) symmetry is spontaneously broken and where two out of three Dirac cones acquire a mass gap. The associated quantum critical point can be understood in terms of a Gross-Neveu-SO(3) theory. Second, we subsequently observe a transition toward an insulating phase in which the SO(3) symmetry is restored and the U(1) symmetry is spontaneously broken. While strongly first order at the mean-field level, the QMC data are consistent with a direct and continuous transition. It is thus a candidate for a new type of deconfined quantum critical point that features gapless fermionic degrees of freedom.

Received 3 September 2021
Accepted 25 January 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.087201

Physics Subject Headings (PhySH)

Critical phenomena Quantum phase transitions

Honeycomb lattice Strongly correlated systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Zi Hong Liu ¹, Matthias Vojta², Fakher F. Assaad¹, and Lukas Janssen ²

¹Institut für Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, 97074 Würzburg, Germany
²Institut für Theoretische Physik and Würzburg-Dresden Cluster of Excellence ct.qmat, Technische Universität Dresden, 01062 Dresden, Germany

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Vol. 128, Iss. 8 — 25 February 2022

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Images

Figure 1
(a) Ground-state phase diagram of the model (1) as function of interaction strength $J$ , obtained from AFQMC simulations. The semimetal-to-semimetal transition at $J_{c 1}$ is continuous and can be understood in terms of a GN-SO(3) field theory. The semimetal-to-insulator transition at $J_{c 2}$ , while strongly first order at the mean-field level, appears continuous. (b) Variation of SO(3) and U(1) structure factors $S_{SO (3)} (k = Γ)$ and $S_{U (1)} (k = Γ)$ as function of $J$ for different lattice sizes. Extensive structure factors reflect spontaneous symmetry breaking.
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Figure 2
Mean-field results for the model (1), obtained on a $L = 60$ lattice. (a) SO(3) and U(1) order parameters $m_{α}$ and $V$ as a function of $J$ . (b)–(d) Single-particle density of states for representative fixed values of $J$ in the three phases.
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Figure 3
(a) Quasiparticle weight $Z (k = K)$ as a function of inverse system size, $1 / L$ , in the symmetric ( $J = 0.40$ ) and SO(3)-broken ( $J = 0.50$ ) phases, respectively. The polynomial fitting of the second curve yields $Z = 1.9 (1)$ for $L \to \infty$ . (b)–(d) Fermion spectral function $A (k, ω)$ along the path in momentum space shown in the inset of (a), shown for (b) the symmetric phase at $J = 0.30$ , (c) the SO(3)-broken phase at $J = 0.70$ , and (d) the U(1)-broken phase at $J = 1.10$ . Finite weight at the Dirac point is visible in (b) and (c), corresponding to semimetallic behavior, albeit with a reduced low-energy weight in the case of the SO(3)-broken semimetal (c). The low-energy weight at the $M$ point especially visible in (b) is an artifact of the magnetic flux and does not survive the thermodynamic limit (see Supplemental Material [45]).
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Figure 4
QMC characterization of GN-SO(3) transition at $J_{c 1}$ . (a) Correlation ratios of the U(1) and SO(3) order parameters. (b)–(d) Scaling collapse near $J_{c 1} = 0.461$ of (b) correlation ratio $R_{c}^{SO (3)}$ , (c) order parameter $m^{2}$ , and (d) fermion quasiparticle weight $Z$ , as a function of $j L^{1 / ν}$ , with $j = J - J_{c 1}$ and $ν$ the correlation-length exponent.
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Figure 5
QMC characterization of SO(3)-U(1) transition at $J_{c 2}$ . (a),(b) Correlation ratios as function of $J$ across transition. (c) Finite-size critical couplings $J_{c 2}^{SO (3) / U (1)} (L)$ as a function of $1 / L$ . While $J_{c 2}^{U (1)} (L)$ increases with system size, $J_{c 2}^{SO (3)} (L)$ stabilizes within our accuracy as $L \geq 9$ . By extrapolating $J_{c 2}^{U (1)} (L)$ using the power-law ansatz $J_{c 2} + a / L^{e}$ , we obtain the estimate $J_{c 2} = 1.0013 (18)$ . (d) First derivative of free energy $d F / d J$ near $J_{c 2}$ , showing no discontinuity within our accuracy.
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