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arXiv:2503.16897

A formally exact real-space representation of the Berry phase on infinite lattices: Applications to dipole and quadrupole moments

Y. Onaya, F. Hamano, Takahiro Fukui

Published Mar 21, 2025• 0 citations
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Abstract

Inspired by Kitaev's real-space representation of Chern numbers, we develop a real-space formulation of the Berry phase for infinite lattices. While the well-known Resta formula for the Berry phase is defined under periodic boundary conditions for finite lattices, our approach constructs the Berry phase directly on an infinite lattice without requiring momentum-space discretization. We apply this method to several disordered models to examine its validity. Furthermore, we attempt to generalize the real-space representation to the quadrupole moment, drawing an analogy to the generalization of the Resta formula for the quadrupole moment.

DOI
10.48550/arxiv.2503.16897

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Inspired by Kitaev's real-space representation of Chern numbers, we develop a real-space formulation of the Berry phase for infinite lattices. While the well-known Resta formula for the Berry phase is defined under periodic boundary conditions for finite lattices, our approach constructs the Berry phase directly on an infinite lattice without requiring momentum-space discretization. We apply this method to several disordered models to examine its validity. Furthermore, we attempt to generalize the real-space representation to the quadrupole moment, drawing an analogy to the generalization of the Resta formula for the quadrupole moment.
We extend Kitaev's real-space formulation of the first Chern number to the second Chern number and establish a computational framework for its evaluation. To test its validity, we apply the derived formula to the disordered Wilson-Dirac model and analyze its ability to capture topological properties in the presence of disorder. Our results demonstrate that the real-space approach provides a viable method for characterizing higher-dimensional topological phases beyond momentum-space formulations.
We rewrite a lattice model of the four-dimensional Chern insulator as a family of translationally-invariant infinite chains over the three-dimensional Brillouin zone and compute its higher three-form Berry curvature using infinite matrix product states (iMPS). We calculate the topological phase diagram of the associated Dixmier--Douady--Kapustin--Spodyneiko (DDKS) number as a function of the model's mass term, and show that it is exactly congruent to the phase diagram in terms of the second Chern number, the analytic expression of which is known for this particular model. This agreement demonstrates that higher Berry curvature can be used to compute second Chern numbers in a manifestly quantized manner. Motivated by the connection between the second Chern form and the Chern--Simons axion coupling, we study magnetoelectric coupling in three dimensions and its relation to higher Berry phases.
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.
A formalism is developed for calculating Berry phases for nonadiabatic time-periodic quantum systems when a dynamical invariant is known. It is found that, when the invariant is periodic and has a nondegenerate spectrum, this method allows a convenient way to obtain generalized Berry phases and the proper cyclic initial states. The method is applied to the generalized harmonic oscillator and the two-level system, where the invariant operators are explicitly constructed. Formulae for the conventional Berry's phases are readily obtained by taking the adiabatic limit of the exact results.
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.
The quantum mechanical position operators, and their products, are not well-defined in systems obeying periodic boundary conditions. Here we extend the work of Resta [Phys. Rev. Lett. 80, 1800 (1998)], who developed a formalism to calculate the electronic polarization as an expectation value of a many-body operator, to include higher multipole moments, e.g., quadrupole and octupole. We define nthn\mathrm{th}nth-order multipole operators whose expectation values can be used to calculate the nthn\mathrm{th}nth multipole moment when all of the lower moments are vanishing (modulo a quantum). We show that changes in our operators are tied to flows of $n\ensuremath{-}1\mathrm{st}$ multipole currents, and encode the adiabatic evolution of the system in the presence of an $n\ensuremath{-}1\mathrm{st}$ gradient of the electric field. Finally, we test our operators on a set of tight-binding models to show that they correctly determine the phase diagrams of topological quadrupole and octupole models, capture an adiabatic quadrupole pump, and distinguish a bulk quadrupole moment from other mechanisms that generate corner charges.