Simple Fermionic Model of Deconfined Phases and Phase Transitions

Abstract

Using quantum Monte Carlo simulations, we study a series of models of fermions coupled to quantum Ising spins on a square lattice with N flavors of fermions per site for N=1 , 2, and 3. The models have an extensive number of conserved quantities but are not integrable, and they have rather rich phase diagrams consisting of several exotic phases and phase transitions that lie beyond the Landau-Ginzburg paradigm. In particular, one of the prominent phases for N>1 corresponds to 2N gapless Dirac fermions coupled to an emergent Z2 gauge field in its deconfined phase. However, unlike a conventional Z2 gauge theory, we do not impose “Gauss’s Law” by hand; instead, it emerges because of spontaneous symmetry breaking. Correspondingly, unlike a conventional Z2 gauge theory in two spatial dimensions, our models have a finite-temperature phase transition associated with the melting of the order parameter that dynamically imposes the Gauss’s law constraint at zero temperature. By tuning a parameter, the deconfined phase undergoes a transition into a short-range entangled phase, which corresponds to Néel antiferromagnet or superconductor for N=2 and a valence-bond solid for N=3 . Furthermore, for N=3 , the valence-bond solid further undergoes a transition to a Néel phase consistent with the deconfined quantum critical phenomenon studied earlier in the context of quantum magnets.

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Plain Language Summary

A rich variety of phases beyond solid, liquid, and gas are found in nature, and these phases originate from simple rules that encode interactions among elementary constituents. Quantum mechanics provides a window to new phases in which constituents can be in several different patterns all at once, much like Schrödinger’s cat can be dead and alive simultaneously. Here, we introduce a simple model, free of the fermion sign problem, that harbors fascinating new phases and can be used to conduct unbiased quantum Monte Carlo simulations.

The basic rules of our model are simple: Fermions of up to three flavors tunnel between sites of a square lattice, and each tunneling event changes the sign of an Ising field (subject to a magnetic field) residing on the links of the lattice. One fundamental question is the following: Does our model host a phase in which strong quantum fluctuations do not allow the system to pick any one ordering pattern? We find that the answer to this question is indeed yes. There exists a regime in which there is no ordering, and the description is in terms of fluctuating strings whose ends are the fermions. Additionally, we find two symmetry-breaking phases that are separated by a continuous transition that lies beyond the Landau paradigm of phase transitions. The transition from the symmetry-unbroken phase to the symmetry-broken phase is again seemingly continuous.

We expect that our findings will motivate work to determine a larger theoretical framework encompassing our model.