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arXiv:2202.04096 · Physical review. B./Physical review. B

Berry phase in the rigid rotor: Emergent physics of odd antiferromagnets

Subhankar Khatua, R. Ganesh

Published May 2, 2022• 5 citations
arXivHTMLPDF

Abstract

The rigid rotor is a classic problem in quantum mechanics, describing the dynamics of a rigid body with its center of mass held fixed. It is well known to describe the rotational spectra of molecules. The configuration space of this problem is SO(3), the space of all rotations in three dimensions. This is a topological space with two types of closed loops: trivial loops that can be adiabatically shrunk to a point and nontrivial loops that cannot. In the traditional formulation of the problem, stationary states are periodic over both types of closed loops. However, periodicity conditions may change if Berry phases are introduced. We argue that time-reversal symmetry allows for only one new possibility---a Berry phase of $\ensuremath{\pi}$ attached to all nontrivial loops. We derive the corresponding stationary states by exploiting the connection between SO(3) and SU(2) spaces. The solutions are antiperiodic over any nontrivial loop; i.e., stationary states reverse sign under a $2\ensuremath{\pi}$ rotation about any axis. Remarkably, this framework is realized in the low-energy physics of certain quantum magnets. The magnets must satisfy the following conditions: (a) the classical ground states are unpolarized, carrying no net magnetization, (b) the set of classical ground states is indexed by SO(3), and (c) the product $N\ifmmode\times\else\texttimes\fi{}S$ is a half-integer, where $N$ is the number of spins and $S$ is the spin quantum number. We demonstrate this result in a family of Heisenberg antiferromagnets defined on polygons with an odd number of vertices. At each vertex, we have a spin-$S$ moment that is coupled to its nearest neighbors. In the classical limit, these magnets have coplanar ground states. Their quantum spectra, at low energies, correspond to ``spherical top'' and ``symmetric top'' rigid rotors. For integer values of $S$, we recover traditional rigid-rotor spectra. With half-integer $S$, we obtain rotor spectra with a Berry phase of $\ensuremath{\pi}$.

DOI
10.1103/physrevb.105.184401

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The rigid rotor is a classic problem in quantum mechanics, describing the dynamics of a rigid body with its center of mass held fixed. It is well known to describe the rotational spectra of molecules. The configuration space of this problem is SO(3), the space of all rotations in three dimensions. This is a topological space with two types of closed loops: trivial loops that can be adiabatically shrunk to a point and nontrivial loops that cannot. In the traditional formulation of the problem, stationary states are periodic over both types of closed loops. However, periodicity conditions may change if Berry phases are introduced. We argue that time-reversal symmetry allows for only one new possibility---a Berry phase of \ensuremathπ\ensuremath{\pi}\ensuremathπ attached to all nontrivial loops. We derive the corresponding stationary states by exploiting the connection between SO(3) and SU(2) spaces. The solutions are antiperiodic over any nontrivial loop; i.e., stationary states reverse sign under a 2\ensuremathπ2\ensuremath{\pi}2\ensuremathπ rotation about any axis. Remarkably, this framework is realized in the low-energy physics of certain quantum magnets. The magnets must satisfy the following conditions: (a) the classical ground states are unpolarized, carrying no net magnetization, (b) the set of classical ground states is indexed by SO(3), and (c) the product N\ifmmode×\else\texttimes\fiSN\ifmmode\times\else\texttimes\fi{}SN\ifmmode×\else\texttimes\fiS is a half-integer, where NNN is the number of spins and SSS is the spin quantum number. We demonstrate this result in a family of Heisenberg antiferromagnets defined on polygons with an odd number of vertices. At each vertex, we have a spin-SSS moment that is coupled to its nearest neighbors. In the classical limit, these magnets have coplanar ground states. Their quantum spectra, at low energies, correspond to spherical top'' and symmetric top'' rigid rotors. For integer values of SSS, we recover traditional rigid-rotor spectra. With half-integer SSS, we obtain rotor spectra with a Berry phase of \ensuremathπ\ensuremath{\pi}\ensuremathπ.
Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.
We derive the general formula giving Berry's phase for an arbitrary spin, having both magnetic-dipole and electric-quadrupole couplings with external time-dependent fields. We assume that the 'effective' electric and magnetic fields remain orthogonal during the quantum cycles. This mild restriction has many advantages. It provides simple symmetries leading to useful selection rules. Furthermore, the Hamiltonian parameter space coincides with the density matrix space for a spin S = 1. This implies a mathematical identity between Berry's phase and the Aharonov–Anandan phase, which is lost for S > 1. We have found that new physical features of Berry's phases emerge for integer spins ⩾2. We provide explicit numerical results of Berry's phases for S = 2, 3, 4. For any spin, one easily finds well-defined regions of the parameter space where instantaneous eigenstates remain widely separated. We have decided to stay within them and not deal with non-Abelian Berry's phases. We present a thorough and precise analysis of the non-adiabatic corrections with separate treatment for periodic and non-periodic Hamiltonian parameters. In both cases, the accuracy for satisfying the adiabatic approximation within a definite time interval is considerably improved if one chooses for the time derivatives of the parameters a time-dependence having a Blackman pulse shape. This has the effect of taming the non-adiabatic oscillation corrections which could be generated by a linear ramping. For realistic experimental conditions, the non-adibatic corrections can be kept below the 0.1% level. For a class of quantum cycles, involving as a sole periodic parameter the precession angle of the electric field around the magnetic field, the corrections odd upon the reversal of the associated rotation velocity can be canceled exactly if the quadrupole to dipole coupling is chosen appropriately ('magic values'). The even ones are eliminated by taking the difference of the Berry phases generated by two 'mirror' cycles. We end by indicating a possible application of the theoretical tools developed in this paper. We propose a way to perform a holonomic entanglement of N non-correlated 1/2-spins by performing adiabatic cycles governed by a Hamiltonian given as a nonlinear function of the total spin operator S , defined as the sum of the N individual spin operators. The basic idea behind this proposal is the mathematical fact that any non-correlated states can be expanded into eigenstates of S 2 and S z . The same eigenvalues appear several times in the decomposition when N > 2 but all these states differ by their symmetry properties under the N -spin permutations. The case N = 4 and S z = 1 is treated explicitly and a maximum entanglement is achieved.
We derive the general formula giving the Berry phase for an arbitrary spin, having both magnetic-dipole and electric-quadrupole couplings with external time-dependent fields. We assume that the effective E and B fields remain orthogonal during the quantum cycles. This mild restriction has many advantages. It provides simple symmetries leading to selection rules and the Hamiltonian-parameter and density-matrix spaces coincide for S=1. This implies the identity of the Berry and Aharonov-Anandan phases, which is lost for S>1. We have found that new features of Berry phases emerge for integer spins>2. We provide explicit numerical results of Berry phases for S=2,3,4. We give a precise analysis of the non-adiabatic corrections. The accuracy for satisfying adiabaticity is greatly improved if one chooses for the time derivatives of the parameters a time-dependence having a Blackman pulse shape. This has the effect of taming the non-adiabatic oscillation corrections which could be generated by a linear ramping. For realistic experimental conditions, the non-adibatic corrections can be kept < 0.1%. For quantum cycles,involving as sole periodic parameter the precession angle of E around B, the corrections odd upon the reversal of the angular velocity can be cancelled exactly if the quadrupole to dipole coupling ratio takes a "magic" value. The even ones are cancelled by subtraction of the phases relative to opposite velocities. As a possible application of the results of this paper we suggest a route to holonomic entanglement of N non-correlated 1/2-spins by performing adiabatic cycles governed by a Hamiltonian which is a non-linear function of the total spin operator S defined as the sum of the N spin operators. The case N=4 and Sz=1 is treated explicitly and maximum entanglement is achieved.
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.
In this second paper in a series, we show that the the general statistical approach to nonrelativistic quantum mechanics developed in the first paper yields a representation of quantum spin and magnetic moments based on classical nonrelativistic spinning top models, using Euler angle coordinates. The models allow half-odd-integer spin and predict supraluminal speeds only for electrons and other leptons, which must be treated relativistically. The spin operators in the space-fixed frame satisfy the usual commutation rules, while those in the rotating body-fixed frame satisfy "left-handed" rules. The commutation rules are independent of the structure of the top, so all nonrelativistic rigidly rotating objects must have integer or odd-half-integer spin. Physical boundary conditions restrict all mixed spin states to involve only half-odd-integer or only integer spin eigenstates. For spin 1/2, the theory automatically yields a modified Pauli-Schrödinger equation. The Hamiltonian operator in this equation contains a rigid rotator term and a term involving the square of the magmetic field, as well as an interaction term having the usual form in spherically symmetric and some cylindrically symmetric models, valid for any magnetogyric ratio.
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.