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

arXiv:cond-mat/9701103 · Journal of Physics A Mathematical and General

On the path integral representation for spin systems

D. C. Cabra, A. Dobry, Andrés Greco, G. L. Rossini

Published Apr 21, 1997• 11 citations
arXivHTMLPDF

Abstract

We propose a classical constrained Hamiltonian theory for the spin. After the Dirac treatment we show that due to the existence of second-class constraints the Dirac brackets of the proposed theory represent the commutation relations for the spin. We show that the corresponding partition function, obtained via the Fadeev - Senjanovic procedure, coincides with that obtained using coherent states. We also evaluate this partition function for the case of a single spin in a magnetic field.

DOI
10.1088/0305-4470/30/8/016

Related papers

A Bosonic Coherent State Path Integral Representation for Spin Systems based on a Projection Operator Implementation of the Schwinger boson number constraint

Seong Jae Kim

The path integral single spin partition function in the basis of boson coherent states, for a general normal ordered two-mode Schwinger boson Hamiltonian, has been computed. Within the initial phase of the calculational process, by a specific approach, the Schwinger boson constraint by means of the projection operator has been implemented.\nAfter, the case of the Zeeman Hamiltonian has been taken on. Using the expression for the partition function obtained for the general case, the appropriate single spin partition function has been produced. Additionally, partition functions have been computed for a few specific spin quantum numbers and then compared with those calculated by the straightforward means, which brought about great confidence in the projection operator implementation; both results match perfectly. \nFinally, the Heisenberg Model has been tackled, and the appropriate expressions for the partition functions have been computed. Their validity has not been verified for any specific cases, but the Zeeman Hamiltonian case dealt with earlier appears to be a strong indicator that these are indeed the correct expressions. \nMany open questions remain, however, for future research to address. All in all, it has been discovered that using the projection operator as a means to enforce the Schwinger boson constraint, veritably works for a Zeeman Hamiltonian problem for specific spin quantum numbers.

•Jan 1, 2018•0 citations

Coherent State Path Integral for the Harmonic Oscillator and a Spin Particle in a Constant Magnetic field

Mario Bergeron

Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms (in 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP1 model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.

Fortschritte der Physik/Progress of Physics•Jan 1, 1992•11 citations

Grassmann coherent states representation of the path integral: Evaluation of the generating function for spin systems

Pablo G. O. Anicich, Horacio Grinberg

Abstract An interacting spin system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a discrete time formalism are used in combination with a basis set involving Grassmann variables coherent states to get a many‐body analytic propagator. The generating function thus obtained leads, after an adequate tracing over Grassmann variables in the imaginary time domain, to the partition function. A spin 1/2 Hamiltonian involving the whole set of interactions is considered. Fermion operators satisfying the standard anticommutation relations are constructed from the raising and lowering spin operators via the Jordan–Wigner transformation. The partition function obtained is more general than the partition function of the traditional Ising model involving only first‐neighbor interactions. Computations were performed assuming that the coupling as a function of the distance can be reasonably well represented by an Airy function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

International Journal of Quantum Chemistry•Jan 1, 2002•8 citations

Path integral representation for spin systens

Naoum Karchev

The present paper is a short review of different path integral representations of the partition function of quantum spin systems. To begin with, I consider coherent states for SU(2) algebra. Different parameterizations of the coherent states lead to different path integral representations. They all are unified within an U(1) gauge theory of quantum spin systems.

arXiv:1211.4509•Nov 19, 2012•2 citations

Path integral representations for a system constrained on a manifold

Yoshio Ohnuki

A systematic study is made of path integral representations for a particle constrained to move on a manifold diffeomorphic to S D by applying the irreducible representations of the Dirac algebra on it. Especially, we derive two types of path integral representation for this system, one of which is of a new form with a simple and compact expression, and the other is a rigorous version of the Faddeev–Senjanovic formula. It is also shown that the parameter α [0, 1) specifying the irreducible representation for D = 1 has a close connection with the Aharonov–Bohm gauge potential produced by the magnetic flux .

Journal of Physics A Mathematical and General•Jan 9, 2004•1 citation

SPINOR DESCRIPTION OF A GENERAL SPIN-J SYSTEM

Vitor Rocha Vieira, Pedro D Sacramento

We consider a spin coherent states description of a general quantum spin system. It is shown that it is possible to use the spin-1/2 representation to study the general spin-J case. We identify the 1/2 spinor components as the homogeneous coordinates of the projective space associated to the complex variable that labels the coherent states and establish a relation between the two-component spinor and the bosonic Schwinger representation of a spin operator. We rewrite the equations of motion, obtained from the path integral for the evolution operator or partition function, in terms of the 1/2 spinor and define the effective Hamiltonian of its evolution.

arXiv:cond-mat/0012226•WORLD SCIENTIFIC eBooks•Nov 1, 2000•0 citations
We propose a classical constrained Hamiltonian theory for the spin. After the Dirac treatment we show that due to the existence of second-class constraints the Dirac brackets of the proposed theory represent the commutation relations for the spin. We show that the corresponding partition function, obtained via the Fadeev - Senjanovic procedure, coincides with that obtained using coherent states. We also evaluate this partition function for the case of a single spin in a magnetic field.
The path integral single spin partition function in the basis of boson coherent states, for a general normal ordered two-mode Schwinger boson Hamiltonian, has been computed. Within the initial phase of the calculational process, by a specific approach, the Schwinger boson constraint by means of the projection operator has been implemented.\nAfter, the case of the Zeeman Hamiltonian has been taken on. Using the expression for the partition function obtained for the general case, the appropriate single spin partition function has been produced. Additionally, partition functions have been computed for a few specific spin quantum numbers and then compared with those calculated by the straightforward means, which brought about great confidence in the projection operator implementation; both results match perfectly. \nFinally, the Heisenberg Model has been tackled, and the appropriate expressions for the partition functions have been computed. Their validity has not been verified for any specific cases, but the Zeeman Hamiltonian case dealt with earlier appears to be a strong indicator that these are indeed the correct expressions. \nMany open questions remain, however, for future research to address. All in all, it has been discovered that using the projection operator as a means to enforce the Schwinger boson constraint, veritably works for a Zeeman Hamiltonian problem for specific spin quantum numbers.
Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms (in 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP1 model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.
Abstract An interacting spin system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a discrete time formalism are used in combination with a basis set involving Grassmann variables coherent states to get a many‐body analytic propagator. The generating function thus obtained leads, after an adequate tracing over Grassmann variables in the imaginary time domain, to the partition function. A spin 1/2 Hamiltonian involving the whole set of interactions is considered. Fermion operators satisfying the standard anticommutation relations are constructed from the raising and lowering spin operators via the Jordan–Wigner transformation. The partition function obtained is more general than the partition function of the traditional Ising model involving only first‐neighbor interactions. Computations were performed assuming that the coupling as a function of the distance can be reasonably well represented by an Airy function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002
A systematic study is made of path integral representations for a particle constrained to move on a manifold diffeomorphic to S D by applying the irreducible representations of the Dirac algebra on it. Especially, we derive two types of path integral representation for this system, one of which is of a new form with a simple and compact expression, and the other is a rigorous version of the Faddeev–Senjanovic formula. It is also shown that the parameter α [0, 1) specifying the irreducible representation for D = 1 has a close connection with the Aharonov–Bohm gauge potential produced by the magnetic flux .
We consider a spin coherent states description of a general quantum spin system. It is shown that it is possible to use the spin-1/2 representation to study the general spin-J case. We identify the 1/2 spinor components as the homogeneous coordinates of the projective space associated to the complex variable that labels the coherent states and establish a relation between the two-component spinor and the bosonic Schwinger representation of a spin operator. We rewrite the equations of motion, obtained from the path integral for the evolution operator or partition function, in terms of the 1/2 spinor and define the effective Hamiltonian of its evolution.