Order-by-Disorder and Quantum Coulomb Phase in Quantum Square Ice

Abstract

We reconstruct the equilibrium phase diagram of quantum square ice, realized by the transverse-field Ising model on the checkerboard lattice, using a combination of quantum Monte Carlo, degenerate perturbation theory and gauge mean-field theory. The extensive ground-state degeneracy of classical square ice is lifted by the transverse field, leading to two distinct order-by-disorder phases, a plaquette valence-bond solid for low field, and a canted Néel state for stronger fields. These two states appear via a highly non-linear effect of quantum fluctuations, and they can be identified with the phases of a lattice gauge theory (quantum link model) emerging as the effective Hamiltonian of the system within degenerate perturbation theory up to the 8th order. The plaquette valence-bond solid melts at a very low temperature, above which the system displays a thermally induced quantum Coulomb phase, supporting deconfined spinons. Introduction. Kinematically constrained systems represent a central theme of statistical physics and condensed matter, as they often represent the effective low-energy description of fundamental lattice many-body Hamiltonians. Prominent examples are to be found in models of frustrated magnetism (e.g. frustrated Ising models [1], quantum dimer models [2]) and of ice physics and its magnetic (spin-ice) analogs [3]. In such models the energy is typically minimized by an exponentially degenerate manifold of states satisfying a local constraint -the so-called ice rule. A fundamental insight is gained when recognizing that the ice rule can be cast in the form of a Gauss law for an emergent electric field -corresponding, in the case of spin models, to the orientation of one of the spin components. When these systems are endowed with quantum dynamics, their effective Hamiltonian describing quantum fluctuations within the constrained manifold takes the form of a quantum lattice gauge theory (LGT). A deconfined phase of the LGT -namely a phase in which the gauge field is not able to bind charges -corresponds to a novel phase supporting fractionalized excitations in the original spin model. Important examples thereof are represented by the deconfined phase of the Z 2 LGT in dimensions d = 2, 3 [4], corresponding to the so-called Z 2 spin liquid in the magnetic context; and the deconfined phase of the d = 3 compact lattice quantum electrodynamics (QED) [5], corresponding to the so-called U(1) (or Coulomb) spin liquid [6]. The former represents a strong candidate for the ground-state of frustrated S = 1/2 Heisenberg antiferromagnets (e.g. on the Kagomé lattice [7]) while the latter is expected to be realized as the ground state of 3d quantum spin ice [8].

Phase diagram of J1-J2 transverse field Ising model on the checkerboard lattice: a plaquette-operator approach

“…The existence of a canted RPS phase confirms the results of Monte-Carlo studies of Refs. 29 and 45 , as well as the result of quantum dimer model 28 .…”

“…A recently Monte-Carlo study of the TFIM on the isotropic J 2 = J 1 checkerboard lattice 45 , reports an RPS state, via an extrapolation to zero-temperature, that persists up to Γ ≃ 0.13 and a canted Néel state for 0.13 Γ 0.28 and finally a quantum paramagnet phase for higher fields (Γ 0.28). However, the presence of such a Néel phase is a very delicate issue, which requires more justifications.…”

“…According to Ref. 45 , the Néel phase does not come from direct simulation on the origianl Hamiltonian, rather it is an outcome of the simulation on the fourth order effective Hamiltonian that can be constructed from the extensive degenerate manifold at J 2 = J 1 . Moreover, the extrapolated (N → ∞) staggered magnetization is obtained to be m s ∼ 10 −2 , which is very small.…”

Quantum phase diagram of the two-dimensional transverse-field Ising model: Unconstrained tree tensor network and mapping analysis

“…In fact, it has been concluded that the exponential degeneracy of the classical ground state at the highly frustrated point, J 2 = J 1 , (known as square ice [26]) is lifted toward a unique quantum plaquette-VBS state that breaks translational symmetry of the lattice with two-fold degeneracy. It is a manifestation of order by disorder phenomena [23][24][25], which is induced by quantum fluctuations.…”

Generation of ice states through deep reinforcement learning

“…Therefore, it is possible to extend this framework to other physical models. For example, quantum Monte Carlo simulations of quantum spin ice models [39][40][41], toric code and related gauge models [42][43][44] have been difficult due to the physical constraints present in the model. The generality of the RL framework presented here can explore an enlarged state space and potentially discover new sampling schemes through the automatic exploration of the machine agent on a constrained model.…”