Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Berry phase and supersymmetric topological index

Kirill Ilinski, Gleb Kalinin, V. V. Melezhik

Published Dec 1, 1995• 2 citations
Publisher

Abstract

The sequences of supersymmetry for a cyclic adiabatic evolution governed by the supersymmetric quantum mechanical Hamiltonian are revised. The condition (so-called supersymmetric adiabatic evolution condition) under which the supersymmetric reductions of Berry (nondegenerated case) or Wilczek–Zee (degenerated case) phases of superpartners are taking place is pointed out. The analog of the Witten index (supersymmetric Berry index) is determined. The final expression for the new index has compact form of indBH=sDet U≡Det Uτ, where U is the cyclic evolution operator generated by supersymmetric Hamiltonian H and τ is a supersymmetric involution. As the examples of the suggested concept of the supersymmetric adiabatic evolution the holomorphic quantum mechanics on a complex plane and meromorphic quantum mechanics on Riemann surfaces are considered. The supersymmetric Berry indices for the models are calculated.

DOI
10.1063/1.531176

Related papers

Berry phase in Heisenberg representation

V. A. Andreev, A. B. Klimov, P. B. Lerner

We define the Berry phase for the Heisenberg operators. This definition is motivated by the calculation of the phase shifts by different techniques. These techniques are: the solution of the Heisenberg equations of motion, the solution of the Schrodinger equation in coherent-state representation, and the direct computation of the evolution operator. Our definition of the Berry phase in the Heisenberg representation is consistent with the underlying supersymmetry of the model in the following sense. The structural blocks of the Hamiltonians of supersymmetrical quantum mechanics ('superpairs') are connected by transformations which conserve the similarity in structure of the energy levels of superpairs. These transformations include transformation of phase of the creation-annihilation operators, which are generated by adiabatic cyclic evolution of the parameters of the system.

•May 1, 1994•0 citations

Certain properties of Berry's phases in supersymmetric quantum mechanics. II

K M Cheng, P. C. W. Fung

For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.

Journal of Physics A Mathematical and General•Aug 21, 1992•1 citation

Certain properties of Berry's phases in supersymmetric quantum mechanics

K M Cheng, P. C. W. Fung

The authors discuss certain properties of Berry's phases in two quantum systems with supersymmetrically related Hamiltonians and derive an explicit expression for the difference in their Berry's phases. Moreover, they obtain an expression for a topological quantity which can be interpreted in term of holonomy, and which contains the Berry's phase and is invariant in the two supersymmetrically related quantum systems. They take two examples to illustrate their findings. The first one is the example of the Aharonov-Bohm effect in which the Berry's phases of the two supersymmetrically related quantum systems are equal. The second example is a system with a spin-1/2 particle in a time varying magnetic field. They obtain the stated topological quantity and, in their analysis, they discover that this quantity is corresponding to the essential gauge transformation in such systems.

Journal of Physics A Mathematical and General•Mar 21, 1992•3 citations

Phase change during a cyclic quantum evolution

Yakir Aharonov, J. Anandan

A new geometric phase factor is defined for any cyclic evolution of a quantum system. This is independent of the phase factor relating the initial and final state vectors and the Hamiltonian, for a given projection of the evolution on the projective space of rays of the Hilbert space. Some applications, including the Aharonov-Bohm effect, are considered. For the special case of adiabatic evolution, this phase factor is a gauge-invariant generalization of the one found by Berry.

Physical Review Letters•Apr 20, 1987•2,185 citations

1 A New Class of Adiabatic Cyclic States and Geometric Phases for Non-Hermitian Hamiltonians

For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:

•Jan 1, 1999•0 citations

Recent Developments in Supersymmetric Quantum Mechanics

Georg Junker

Some recent results in supersymmetric quantum mechanics are presented. New semi-classical approximation formulas for Witten's realization of supersymmetric quantum mechanics are discussed. Implications of the supersymmetric structure of Pauli's Hamiltonian are also considered. In particular, the paramagnetisation of a non-interacting electron gas is related to a modified version of Witten's index. We also show that the supersymmetry in this system provides a new counter example for the paramagnetic conjecture.

arXiv:cond-mat/9403088•Mar 29, 1994•0 citations
The sequences of supersymmetry for a cyclic adiabatic evolution governed by the supersymmetric quantum mechanical Hamiltonian are revised. The condition (so-called supersymmetric adiabatic evolution condition) under which the supersymmetric reductions of Berry (nondegenerated case) or Wilczek–Zee (degenerated case) phases of superpartners are taking place is pointed out. The analog of the Witten index (supersymmetric Berry index) is determined. The final expression for the new index has compact form of indBH=sDet U≡Det Uτ, where U is the cyclic evolution operator generated by supersymmetric Hamiltonian H and τ is a supersymmetric involution. As the examples of the suggested concept of the supersymmetric adiabatic evolution the holomorphic quantum mechanics on a complex plane and meromorphic quantum mechanics on Riemann surfaces are considered. The supersymmetric Berry indices for the models are calculated.
We define the Berry phase for the Heisenberg operators. This definition is motivated by the calculation of the phase shifts by different techniques. These techniques are: the solution of the Heisenberg equations of motion, the solution of the Schrodinger equation in coherent-state representation, and the direct computation of the evolution operator. Our definition of the Berry phase in the Heisenberg representation is consistent with the underlying supersymmetry of the model in the following sense. The structural blocks of the Hamiltonians of supersymmetrical quantum mechanics ('superpairs') are connected by transformations which conserve the similarity in structure of the energy levels of superpairs. These transformations include transformation of phase of the creation-annihilation operators, which are generated by adiabatic cyclic evolution of the parameters of the system.
For pt.I see ibid., vol.25, p.1745 (1992). In the first paper the authors studied certain crucial mathematical properties of Abelian Berry's phases in two quantum systems with supersymmetrically related Hamiltonians. They follow up such investigation by extending their analysis to nonAbelian Berry's phases in this present study. They present the derivation of an explicit expression for the difference in the relevant nonAbelian Berry's connections. Also they have derived an expression for a connection one form, which contain the non-Abelian Berry's connection, and which is invariant in the two supersymmetrically related quantum systems. To illustrate their findings mentioned above, they take the example of a Hamiltonian expressible in spin quadrupole. In this example, with a particular choice of supersymmetric partner, they show that the two Berry's connection one forms of the two supersymmetrically related systems may be related by gauge transformation.
The authors discuss certain properties of Berry's phases in two quantum systems with supersymmetrically related Hamiltonians and derive an explicit expression for the difference in their Berry's phases. Moreover, they obtain an expression for a topological quantity which can be interpreted in term of holonomy, and which contains the Berry's phase and is invariant in the two supersymmetrically related quantum systems. They take two examples to illustrate their findings. The first one is the example of the Aharonov-Bohm effect in which the Berry's phases of the two supersymmetrically related quantum systems are equal. The second example is a system with a spin-1/2 particle in a time varying magnetic field. They obtain the stated topological quantity and, in their analysis, they discover that this quantity is corresponding to the essential gauge transformation in such systems.
For a T-periodic non-Hermitian Hamiltonian H(t), we construct a class of adiabatic cyclic states of period T which are not eigenstates of the initial Hamiltonian H(0). We show that the corresponding adiabatic geometric phase angles are real and discuss their relationship with the conventional complex adiabatic geometric phase angles. We present a detailed calculation of the new adiabatic cyclic states and their geometric phases for a non-Hermitian analog of the spin 1/2 particle in a precessing magnetic field. Since the publication of Berry’s paper [1] on the adiabatic geometrical phase, the subject has undergone a rapid development. Berry’s adiabatic geometric phase for periodic Hermitian Hamiltonians with a discrete nondegenerate spectrum has been generalized to arbitrary changes of a quantum state. In particular, the conditions on the adiabaticity [2] and cyclicity [3, 4, 5] of the evolution, Hermiticity of the Hamiltonian [6], and degeneracy [7] and discreteness [8] of its spectrum have been lifted. Moreover, the classical [9] and relativistic [10] analogues of the geometric phase have been considered. The purpose of this note is to offer an alternative generalization of Berry’s treatment of the adiabatic geometric phase for a non-Hermitian parametric Hamiltonian H[R] with a nondegenerate discrete spectrum. The parameters R = (R1, · · ·, Rn) are assumed to be real ∗E-mail address:
Some recent results in supersymmetric quantum mechanics are presented. New semi-classical approximation formulas for Witten's realization of supersymmetric quantum mechanics are discussed. Implications of the supersymmetric structure of Pauli's Hamiltonian are also considered. In particular, the paramagnetisation of a non-interacting electron gas is related to a modified version of Witten's index. We also show that the supersymmetry in this system provides a new counter example for the paramagnetic conjecture.