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State selection in frustrated magnets: order by disorder and order by singularity

Subhankar Khatua, Sarvesh Srinivasan, R. Ganesh

Published Nov 23, 2020• 0 citations
OpenAlex

Abstract

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-$S$ 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 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.

Related papers

State selection in frustrated magnets

Subhankar Khatua, Sarvesh Srinivasan, R. Ganesh

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.

arXiv:2011.11668•Physical review. B./Physical review. B•May 11, 2021•12 citations

Effective theories for quantum spin clusters: Geometric phases and state selection by singularity

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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 $XY$ antiferromagnetic bond. The dimer with two spin-$S$ spins provides the simplest example---it maps to a quantum particle on a ring $({S}^{1})$. 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 $S$ 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.

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Effective theories for quantum spin clusters: geometric phases and order by singularity

Subhankar Khatua, Diptiman Sen, R. Ganesh

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•Mar 4, 2019•0 citations

Floquet prethermal order by disorder

Hui-Ke Jin, Johannes Knolle

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

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Quantum order by disorder in the Kitaev model on a triangular lattice

George Jackeli, Adolfo Avella

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.

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Finite-Size Spectral Signatures of Order by Quantum Disorder: A Perspective from Anderson’s Tower of States

Subhankar Khatua, Griffin C. Howson, Michel J. P. Gingras +1 more

In frustrated magnetic systems with a subextensive number of classical ground states, quantum zero-point fluctuations can select a unique long-range ordered state, a celebrated phenomenon referred to as order by quantum disorder (ObQD). For frustrated spin-1/2 models, unbiased numerical methods able to expose ObQD are necessary. We show that ObQD can be identified from exact diagonalization calculations through an analysis akin to the Anderson tower of states associated with spontaneous symmetry breaking. By defining an effective quantum rotor model, we describe the competition between ObQD-induced localization of the rotor and its tunneling between symmetry-related ground states, identifying the crossover length scale from the finite-size regime where the rotor is delocalized to the infinite system-size limit where it becomes localized. This rotor model relates the characteristic splittings in the exact diagonalization energy spectrum to the ObQD selection energy scale, providing an estimate that can be compared to spin wave calculations. We demonstrate the general applicability of this approach in one-, two-, and three-dimensional frustrated spin models that exhibit ObQD.

arXiv:2509.10610•Physical Review Letters•Feb 12, 2026•1 citation
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.
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.
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.
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
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.
In frustrated magnetic systems with a subextensive number of classical ground states, quantum zero-point fluctuations can select a unique long-range ordered state, a celebrated phenomenon referred to as order by quantum disorder (ObQD). For frustrated spin-1/2 models, unbiased numerical methods able to expose ObQD are necessary. We show that ObQD can be identified from exact diagonalization calculations through an analysis akin to the Anderson tower of states associated with spontaneous symmetry breaking. By defining an effective quantum rotor model, we describe the competition between ObQD-induced localization of the rotor and its tunneling between symmetry-related ground states, identifying the crossover length scale from the finite-size regime where the rotor is delocalized to the infinite system-size limit where it becomes localized. This rotor model relates the characteristic splittings in the exact diagonalization energy spectrum to the ObQD selection energy scale, providing an estimate that can be compared to spin wave calculations. We demonstrate the general applicability of this approach in one-, two-, and three-dimensional frustrated spin models that exhibit ObQD.