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arXiv:2011.11668 · Physical review. B./Physical review. B

State selection in frustrated magnets

Subhankar Khatua, Sarvesh Srinivasan, R. Ganesh

Published May 11, 2021• 12 citations
arXivHTMLPDF

Abstract

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-$S$ 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.

DOI
10.1103/physrevb.103.174412

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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.
Magnets with frustration often show accidental degeneracies, characterised by a large classical ground state space (CGSS). Quantum fluctuations may select a particular element of this space via the well known `order by disorder' (ObD) paradigm. Recent studies by the current authors propose a distinct selection mechanism -- `order by singularity' (ObS) -- that can occur when the CGSS self-intersects. This effect has been demonstrated in clusters containing a few spins. In this article, we draw a qualitative distinction between these two selection paradigms. We argue that the 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. In ObD, the particle localizes at a minimum of a fluctuation-generated potential. In contrast, in ObS, the particle localizes at a self-intersection point due to bound state formation -- a consequence of quantum interference. 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 upon introducing an additional XY antiferromagnetic coupling. The CGSS simplifies dramatically, taking the form of a circle. Spin wave fluctuations generate a potential on this space. We discuss the conditions under which this potential gives rise to state selection by ObD. Apart from contrasting ObS and ObD, we discuss the possibility of ObS in macroscopic magnets.
Magnetic systems with frustration often have large classical degeneracy. We show that their low-energy physics can be understood as dynamics within the space of classical ground states. We demonstrate this mapping in a family of quantum spin clusters where every pair of spins is connected by an XYXYXY antiferromagnetic bond. The dimer with two spin-SSS spins provides the simplest example---it maps to a quantum particle on a ring (S1)({S}^{1})(S1). The trimer is more complex, equivalent to a particle that lives on two disjoint rings $({S}^{1}\ensuremath{\bigotimes}{\mathbb{Z}}_{2})$. It has an additional subtlety for half-integer SSS values, wherein both rings must be threaded by $\ensuremath{\pi}$ fluxes to obtain a satisfactory mapping. This is a consequence of the geometric phase incurred by spins. For both the dimer and the trimer, the validity of the effective theory can be seen from a path-integral-based derivation. This approach cannot be extended to the quadrumer which has a nonmanifold ground-state space, consisting of three tori that touch pairwise along lines. In order to understand the dynamics of a particle in this space, we develop a tight-binding model with this connectivity. Remarkably, this successfully reproduces the low-energy spectrum of the quadrumer. For half-integer spins, a geometric phase emerges which can be mapped to two $\ensuremath{\pi}$-flux tubes that reside in the space between the tori. The nonmanifold character of the space leads to a remarkable effect---the dynamics at low energies is not ergodic as the particle is localized around singular lines of the ground-state space. The low-energy spectrum consists of an extensive number of bound states formed around singularities. Physically, this manifests as an order-by-disorder-like preference for collinear ground states. However, unlike order-by-disorder, this ``order by singularity'' persists even in the classical limit. We discuss consequences for field theoretic studies of magnets.
Magnetic systems with frustration often have large classical degeneracy. We show that their low-energy physics can be understood as dynamics within the space of classical ground states. We demonstrate this mapping in a family of quantum spin clusters where every pair of spins is connected by an XYXYXY antiferromagnetic bond. The dimer with two spin-SSS spins provides the simplest example, it maps to a quantum particle on a ring (S1S^1S1). The trimer is more complex, equivalent to a particle that lives on two disjoint rings (S1⊗Z2S^1\otimes \mathbb{Z}_2S1⊗Z2​). It has an additional subtlety for half-integer SSS values, wherein both rings must be threaded by π\piπ-fluxes to obtain a satisfactory mapping. This is a consequence of the geometric phase incurred by spins. For both the dimer and the trimer, the effective theory can be seen from a path-integral-based derivation. This approach cannot be extended to the quadrumer which has a non-manifold ground state space, consisting of three tori that touch pairwise along lines. In order to understand the dynamics of a particle in this space, we develop a tight-binding model with this connectivity. Remarkably, this successfully reproduces the low-energy spectrum of the quadrumer. For half-integer spins, a geometric phase emerges which can be mapped to two π\piπ-flux tubes that reside in the space between the tori. The non-manifold character of the space leads to a remarkable effect - the dynamics at low energies is not ergodic as the particle is localized around singular lines of the ground state space. The low-energy spectrum consists of an extensive number of bound states formed around singularities. Physically, this manifests as an order-by-disorder-like preference for collinear ground states. However, unlike order-by-disorder, this `order by singularity' persists even in the classical limit. We discuss consequences for field theoretic studies of magnets.
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.
In selected lattice systems, it is impossible to satisfy all the antiferromagnetic interactions between the spins simultaneously. Due to this frustration the degeneracy of the ground state becomes very large, often scaling with the size of the system. An antiferromagnetic model of Ising spins with on the three-dimensional Swedenborgite lattice will similarly be frustrated and will behave like a spin liquid in all parameter regimes. \n\nFluctuations, which may be either thermal or quantum-mechanical in nature, between the states in the large (disordered) ground state manifold may be biased towards one of them. This causes the system to select a unique (ordered) ground state, in an effect called order-by-disorder. Our objective is now to investigate the effects of quantum fluctuations on the ground states of the Swedenborgite Ising model and to examine it for potential order-by-disorder effects. This will ultimately result in the construction of a tentative phase diagram of this system.
We identify and discuss the ground state of a quantum magnet on a triangular lattice with bond-dependent Ising-type spin couplings, that is, a triangular analog of the Kitaev honeycomb model. The classical ground-state manifold of the model is spanned by decoupled Ising-type chains, and its accidental degeneracy is due to the frustrated nature of the anisotropic spin couplings. We show how this subextensive degeneracy is lifted by a quantum order-by-disorder mechanism and study the quantum selection of the ground state by treating short-wavelength fluctuations within the linked cluster expansion and by using the complementary spin-wave theory. We find that quantum fluctuations couple next-nearest-neighbor chains through an emergent four-spin interaction, while nearest-neighbor chains remain decoupled. The remaining discrete degeneracy of the ground state is shown to be protected by a hidden symmetry of the model.