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arXiv:2301.11948 · Physical Review Letters

Pseudo-Goldstone Modes and Dynamical Gap Generation from Order by Thermal Disorder

Subhankar Khatua, Michel J. P. Gingras, Jeffrey G. Rau

Published Jun 26, 2023• 17 citations
arXivHTMLPDF

Abstract

Accidental ground state degeneracies-those not a consequence of global symmetries of the Hamiltonian-are inevitably lifted by fluctuations, often leading to long-range order, a phenomenon known as "order-by-disorder" (ObD). The detection and characterization of ObD in real materials currently lacks clear, qualitative signatures that distinguish ObD from conventional energetic selection. We show that for order by thermal disorder (ObTD) such a signature exists: a characteristic temperature dependence of the fluctuation-induced pseudo-Goldstone gap. We demonstrate this in a minimal two-dimensional model that exhibits ObTD, the ferromagnetic Heisenberg-compass model on a square lattice. Using spin-dynamics simulations and self-consistent mean-field calculations, we determine the pseudo-Goldstone gap, Δ, and show that at low temperatures it scales as the square root of temperature, sqrt[T]. We establish that a power-law temperature dependence of the gap is a general consequence of ObTD, showing that all key features of this physics can be captured in a simple model of a particle moving in an effective potential generated by the fluctuation-induced free energy.

DOI
10.1103/physrevlett.130.266702

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Accidental ground state degeneracies-those not a consequence of global symmetries of the Hamiltonian-are inevitably lifted by fluctuations, often leading to long-range order, a phenomenon known as "order-by-disorder" (ObD). The detection and characterization of ObD in real materials currently lacks clear, qualitative signatures that distinguish ObD from conventional energetic selection. We show that for order by thermal disorder (ObTD) such a signature exists: a characteristic temperature dependence of the fluctuation-induced pseudo-Goldstone gap. We demonstrate this in a minimal two-dimensional model that exhibits ObTD, the ferromagnetic Heisenberg-compass model on a square lattice. Using spin-dynamics simulations and self-consistent mean-field calculations, we determine the pseudo-Goldstone gap, Δ, and show that at low temperatures it scales as the square root of temperature, sqrt[T]. We establish that a power-law temperature dependence of the gap is a general consequence of ObTD, showing that all key features of this physics can be captured in a simple model of a particle moving in an effective potential generated by the fluctuation-induced free energy.
Frustrated magnets can have accidental ground state degeneracies which may be lifted by various forms of disorder, for example in the form of thermal or quantum fluctuations. This order by disorder (ObD) paradigm is well established in equilibrium and here is generalized to Floquet many-body systems. Investigating a periodically driven XXZ-compass model on the square lattice, we show that in a prethermal regime, dynamical fluctuations induced by high-frequency drives select a discrete set of states out of a degenerate ground state manifold of the lowest order Floquet Hamiltonian chosen as initial states. Remarkably, prior to the ObD selection, an unusual fluctuating regime emerges leading to a prethermalization timescale scaling linearly with the drive frequency. We argue that prethermal ObD with its unusual approach to the selected states is a generic phenomenon of driven frustrated systems and confirm it in the paradigmatic J 1 − J 2 XX model. Published by the American Physical Society 2024
Abstract The order from quantum disorders (OFQD) phenomenon is well-known and ubiquitous in particle physics and frustrated magnetic systems. Typically, OFQD transfers a spurious Goldstone mode into a pseudo-Goldstone mode with a tiny gap. Here, we report an opposite phenomenon: OFQD transfers a spurious quadratic mode into a true linear Goldstone mode with a very small velocity (named slow-Goldstone mode). This new phenomenon is demonstrated in an interacting bosonic system subjected to an Abelian flux. We develop a new and systematic OFQD analysis to determine the true quantum ground state and the whole excitation spectrum. In the weak-coupling limit, the superfluid ground state has a 4-sublattice 90° coplanar spin structure, which supports 4 linear Goldstone modes with 3 different velocities. One of which is generated by the OFQD is much softer than the other 3 Goldstone modes, so it can be easily detected in the cold atom or photonic experiments. In the strong-coupling limit, the ferromagnetic Mott ground state with a true quadratic Goldstone mode. We speculate that there could be some topological phases intervening between the two symmetry broken states. These novel phenomena may be observed in the current cold-atom or photonic experiments subjected to an Abelian flux at the weak coupling limit where the heatings may be well under control. Possible connections to Coleman-Weinberg potential in particle physics, 1/N expansion of Sachdev-Ye-Kitaev models and zero temperature quantum black hole entropy are outlined.
This thesis presents novel theoretical results on two cold atomic systems with particular emphasis on the order-by-disorder mechanism that they exhibit. The latter selects one or a number of classically degenerate states of lowest energy as the true ground states on the basis of Helmholtz free energy contributions of fluctuations about the classical solutions. It has traditionally played an important role in solid-state systems but has so far not been conclusively observed. The first system considered is that of cold bosons hopping in the two-dimensional dice lattice in the presence of an artificial gauge field, tuned to provide half an elementary flux per plaquette. The single-particle band structure consists entirely of flat bands. Many-particle behaviour is captured by a Bose-Hubbard model with contact interactions. Mean-field analysis yields a large degeneracy of classical ground states which is lifted through Order by Disorder. A closer analysis yields a picture of free-energy-mediated interactions between domain walls separating distinct, classically degenerate regions. The second system is that of spin-2 species in a tightly confining spatial potential, with a generalised quadratic-Zeeman coupling. It is analysed by means of an exact mapping of the many-body Hamiltonian onto that of a five-dimensional rotor. The new Hamiltonian is in general non-Hermitian, but Hermitianising transforms may be found in a number of regimes. Analytical results for the ordinary quadratic Zeeman potential are presented and shown to yield qualitatively different behaviour from the mean-field analysis. In particular, there are no signs of a fluctuation-induced phase transition, predicted by mean-field theory. Motivated by this, an alternative potential, breaking less symmetry, is considered within the rotor framework and shown to display microscopic parallels of Order by Disorder.
Magnets with frustration often show accidental degeneracies, characterized by a large classical ground-state space (CGSS). Quantum fluctuations may ``select'' one of these ground states---a phenomenon labeled ``order by (quantum) disorder'' in literature. In this article, we examine the mechanism(s) by which such state selection takes place. We argue that a magnet, at low energies, maps to a particle moving on the CGSS. State selection corresponds to localization of the particle at a certain point on this space. We distinguish two mechanisms that can bring about localization. In the first, quantum fluctuations generate a potential on the CGSS space. If the potential has a deep enough minimum, then the particle localizes in its vicinity. We denote this as ``order by potential'' (ObP). In the second scenario, the particle localizes at a self-intersection point due to bound-state formation---a consequence of geometry and quantum interference. Following recent studies by the present authors, we denote this scenario as ``order by singularity'' (ObS). In either case, localization leads to an energy gap between the ground state(s) and higher-energy states. This pseudo-Goldstone gap behaves differently in the two mechanisms, scaling differently with the spin length. We place our discussion within the context of the one-dimensional spin-SSS Kitaev model. We map out its CGSS which grows systematically with increasing system size. It resembles a network where the number of nodes increases exponentially. In addition, the number of wires that cross at each node also grows exponentially. This self-intersecting structure leads to ObS, with the low-energy physics determined by a small subset of the CGSS, consisting of ``Cartesian'' states. A contrasting picture emerges when an additional XY antiferromagnetic coupling is introduced. The CGSS simplifies dramatically, taking the form of a circle. Spin wave fluctuations generate a potential on this space, giving rise to state selection by ObP under certain conditions. Apart from contrasting ObS and ObP, we discuss the possibility of ObS in macroscopic magnets.
In this study, an order-by-disorder mechanism has been proposed in a two-dimensional PXP model, where the extensive degeneracy of the classical ground-state manifold is due to strict occupation constraints instead of geometrical frustrations. By performing an unbias large-scale quantum Monte Carlo simulation, we find that local quantum fluctuations, which usually work against long-range ordering, lift the macroscopic classical degeneracy and give rise to a compressible ground state with charge-density-wave long-range order. A simple trial wave function has been proposed to capture the essence of the ground-state of the two-dimensional PXP model. The finite-temperature properties of this model have also been studied, and we find a thermal phase transition with a universality class of a two-dimensional Ising model.
We show that a weak periodic drive removes the accidental degeneracy in the ground state of the XY antiferromagnet on the triangular lattice in a uniform static magnetic field. The underlying mechanism involves adding a small periodically modulated component to the magnetic field, which influences finite-frequency modes to generate an effective potential for the accidental pseudo-Goldstone mode. This selection can be arranged to compete with the degeneracy lifting via the thermal order by disorder mechanism. This yields a nonequilibrium phase transition as the relative strength of the two mechanisms is tuned by varying temperature. This proposal may be amenable to experimental realization, in particular, as applying the field is noninvasive, and no complex bath engineering is required.