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.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.

arXiv:1903.01499Physical review. B./Physical review. BOct 9, 201910 citations

The Geometric Phase in Quantum Physics

Arno Böhm

Springer eBooksJan 1, 199310 citations

The Geometric Phase in Quantum Systems

Arno Böhm, Ali Mostafazadeh, Hiroyasu Koizumi +2 more

Springer eBooksJan 1, 2003426 citations

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

Subhankar Khatua, Diptiman Sen, R. Ganesh

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 ( $S^1\otimes \mathbb{Z}_2$ ). It has an additional subtlety for half-integer $S$ values, wherein both rings must be threaded by $\pi$ -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 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 $\pi$ -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.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 ($S^1\otimes \mathbb{Z}_2$). It has an additional subtlety for half-integer $S$ values, wherein both rings must be threaded by $\pi$-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 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 $\pi$-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.

Mar 4, 20190 citations

Geometrical Phases in Quantum Mechanics

Joy Christian

Jan 1, 19910 citations

Quantum Adiabatic Approximation, Quantum Action, and Berry’s Phase

Ali Mostafazadeh

An alternative interpretation of the quantum adiabatic approximation is presented. This interpretation is based on the ideas originally advocated by David Bohm in his quest for establishing a hidden variable alternative to quantum mechanics. It indicates that the validity of the quantum adiabatic approximation is a sufficient condition for the separability of the quantum action function in the time variable. The implications of this interpretation for Berry’s adiabatic phase and its semi-classical limit are also discussed. Probably one of the best recognized applications of the quantum adiabatic approximation [1, 2] is in Berry’s derivation of the adiabatic geometrical phases [3]. Following Berry’s, by now, classical article on the adiabatic geometric phase [3], Hannay proposed a classical analogue of Berry’s phase [4] and Berry [5] and Anandan [6] explored the semiclassical limit of Berry’s phase and the classical analogue of the general, non-adiabatic geometric phase [7], respectively. The purpose of this note is to provide an alternative interpretation of the quantum adiabatic approximation which yields a natural approach to study the semi-classical limit of this approximation and consequently Berry’s phase.An alternative interpretation of the quantum adiabatic approximation is presented. This interpretation is based on the ideas originally advocated by David Bohm in his quest for establishing a hidden variable alternative to quantum mechanics. It indicates that the validity of the quantum adiabatic approximation is a sufficient condition for the separability of the quantum action function in the time variable. The implications of this interpretation for Berry’s adiabatic phase and its semi-classical limit are also discussed. Probably one of the best recognized applications of the quantum adiabatic approximation [1, 2] is in Berry’s derivation of the adiabatic geometrical phases [3]. Following Berry’s, by now, classical article on the adiabatic geometric phase [3], Hannay proposed a classical analogue of Berry’s phase [4] and Berry [5] and Anandan [6] explored the semiclassical limit of Berry’s phase and the classical analogue of the general, non-adiabatic geometric phase [7], respectively. The purpose of this note is to provide an alternative interpretation of the quantum adiabatic approximation which yields a natural approach to study the semi-classical limit of this approximation and consequently Berry’s phase.

Jan 1, 19964 citations

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.

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\otimes \mathbb{Z}_2

). It has an additional subtlety for half-integer

S

values, wherein both rings must be threaded by

\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 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

\pi

-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.