Hubbard-Stratonovich Transformation Research Articles

Path integrals, diffusion onSU(2) and the fully frustrated antiferromagnetic spin cluster

We study the path-integral treatment of the quantum mechanics of a fully frustrated cluster of spins: a cluster in which every pair of spins is coupled equally by antiferromagnetic Heisenberg interactions. Such clusters are interesting partly because they are the building blocks of geometrically frustrated spin systems. Using a Hubbard–Stratonovich transformation to decouple the interactions, the Boltzmann factor for the spin cluster is written in terms of the time-evolution operator for a single spin in a stochastically varying magnetic field. The time-evolution operator follows a random walk in SU(2): by switching from a Langevin to a Fokker–Planck description of this walk and computing the probability distribution of its end-point, we arrive at an expression for the partition function as a finite sum of single integrals. The calculation provides an analytically tractable illustration of the auxiliary field approach, as used in quantum Monte Carlo calculations, and may potentially be extended to treat more complex frustrated spin systems.

Continuous-time quantum Monte Carlo method for fermions: Beyond auxiliary field framework

A numerically exact continuous-time quantum Monte Carlo algorithm for finite fermionic systems with nonlocal interactions is proposed. The scheme is particularly applicable for general multiband time-dependent correlations, since it does not invoke Hubbard-Stratonovich transformation. The present determinantal grand-canonical method is based on a stochastic series expansion for the partition function in the interaction representation. The results for the Green function and for the time-dependent susceptibility of multiorbital supersymmetric impurity model with a spin-flip interaction are presented.

Decoupling quantum dissipation interaction via stochastic fields.

Based on the Hubbard-Stratonovich transformation, the dissipative interaction between the system of interest and the heat bath is decoupled and the separated system and bath thus evolve in common classical random fields. This manipulation allows us to establish a novel theoretical methodology by which the reduced density matrix is formulated as an ensemble average of its random realizations in the auxiliary white noise fields. Within the stochastic description, the interaction between the system and the bath is reflected in the mutually induced mean fields. The relationship between the bath-induced field and the influence functional in the path integral framework is revealed. As a demonstration of this approach, we derive the exact master equations for two model systems.

A universal result on central charges in the presence of double-trace deformations

We study large N conformal field theories perturbed by relevant double-trace deformations. Using the auxiliary field trick, or Hubbard–Stratonovich transformation, we show that in the infrared the theory flows to another CFT. The generating functionals of planar correlators in the ultraviolet and infrared CFTs are shown to be related by a Legendre transform. Our main result is a universal expression for the difference of the scale anomalies between the ultraviolet and infrared fixed points, which is of order 1 in the large N expansion. Our computations are entirely field theoretic, and the results are shown to agree with predictions from AdS/CFT. We also remark that a certain two-point function can be computed for all energy scales on both sides of the duality, with full agreement between the two and no scheme dependence.

Loop Effects in AdS/CFT and Beyond

Double-trace operators which are relevant deformations of large N conformal field theories give rise to renormalization group flows that can be studied on both sides of the AdS/CFT duality. One-loop calculations in AdS can be compared successfully to results derived in field theory using the Hubbard-Stratonovich transformation. The calculations do not rely on supersymmetry, and they apply in any dimension, provided the original conformal field theory and its AdS dual are well-defined. A speculative idea is proposed for solving the cosmological constant problem based on non-gravitational strings that become light as the universe expands. This is a summary of my talk presented at the Nishinomiya-Yukawa Symposium in Nishinomiya, Japan. It is based primarily on [S. S. Gubser and I. Mitra, hep-th/0210093; S. S. Gubser and I. R. Klebanov, hep-th/0212138].

Statistical field theory for simple fluids: mean field and Gaussian approximations

We present an exact field theoretical representation of the statistical mechanics of simple classical liquids with short-range pairwise additive interactions. The action of the field theory is obtained by performing a Hubbard-Stratonovich transformation of the configurational Boltzmann factor. The mean field and Gaussian approximations of the theory are derived and applications to the liquid-vapour transition considered.

Extended dynamical mean-field theory andGWmethod

We develop the extended dynamical mean field theory (E-DMFT) with a view towards realistic applications. {\bf 1)} We introduce an intuitive derivation of the E-DMFT formalism. By identifying the Hartree contributions before the E-DMFT treatment, it allows to handle systems in symmetry breaking phases within a simple formalism. {\bf 2)} We make a new implementation of E-DMFT through real Hubbard-Stratonovich transformation to decouple the non-local two-particle interactions. We apply it to a 3D U-V model and investigate the behavior of the various Green's functions, especially the density susceptibility, as the density instability is approached. We obtain the phase diagram at a finite temperature. {\bf 3)} We present a formalism incorporating E-DMFT with Cellular DMFT. {\bf 4)} We suggest an improvement of the E-DMFT approach by combining it with a generalized GW method. The method combines the local self-energy from E-DMFT and the non-local ones from the perturbative calculation of GW. We apply the method to a 1D U-V model with two sublattices carrying different chemical potentials. By comparing with those from DMRG, we show the results are shifted in the correct direction due to the GW contributions. {\bf 5)} In order to handle the generic Coulomb repulsion within E-DMFT, we describe a method to tailor E-DMFT so that proper momentum dependence can be kept in general response functions.

A field-theoretical approach to simulation in the classical canonical and grand canonical ensemble

In this paper we present a new approach to simulation methods for classical statistical mechanics relying on a field-theoretical formalism. It is based on applying the complex Hubbard–Stratonovich transformation to the canonical and grand-canonical partition function, which allows one to reexpress their particle representation in terms of a functional integral over a fluctuating auxiliary field. The thermodynamic averages from the resulting field representations can then be calculated with a conventional Monte Carlo algorithm. We explored the applicability of the auxiliary field methodology for both the canonical and grand-canonical ensemble using a system of particles interacting through a purely repulsive Gaussian pair potential in a broad range of external parameters. In the grand-canonical case this technique represents an alternative to standard grand-canonical Monte Carlo methods. Generally providing a framework for simulating classical particle systems within a continuum formalism can be useful for multiscale modeling where the field or continuum description naturally appears within quantum mechanics on smaller length scales and within classical mechanics on larger ones.

Coherent state path integral and Langevin equation of interacting fermions

Interacting fermions, electrons and holes in a semiconductor, are coupled to a thermal reservoir of bosons which yield the fluctuating noise. We use a coherent state path integral formulation on the time contour for non-equilibrium systems in terms of anticommuting variables which replace the fermionic creation- and annihilation operators in the time development operator. An auxiliary commuting field σ x ( t p ), defined on the time contour, is introduced by a Hubbard–Stratonovich transformation. In terms of this new field, a Langevin equation is derived which is similar to a saddle-point equation with a random force f x ( t). In comparison to the bosonic case of excitons in a semiconductor previously described, one obtains a different expression for the Langevin equation and, especially, a different relation for the probability distribution of the noise term f x ( t). The calculation for density matrix elements with the Langevin approach can also be interpreted as an average over modified non-equilibrium Green functions with the appropriately derived probability distribution of the noise.

Clustering and signalling of cell receptors

As a response to ligand binding, transmembrane cell receptors often enhance their clustering, or oligomerization, during the signalling process. Here we present a statistical mechanical model which combines the aspects of both clustering and signalling. In this model, receptors float on the surface, while for two neighboring receptors, there is an interaction energy dependent on their conformational states. On the other hand, ligand binding of a receptor shifts the energy difference between the two conformational states. Due to thermal fluctuation, the effects of clustering and signalling are statistical average quantities. This model reduces to a floating Ising model with a random field. We calculate the signalling in a grand canonical ensemble mean-field approach, using Hubbard–Stratonovich transformation and replica method. Monte Carlo simulations are also performed. Essential biological features are obtained in our model.

Characteristic polynomials of random Hermitian matrices and Duistermaat–Heckman localisation on non-compact Kähler manifolds

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N× N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard “supersymmetry” approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard–Stratonovich transformation in the “bosonic” sector. The method, suggested recently by J.V. Fyodorov [Nucl. Phys. B 621 [PM] (2002) 643], is shown to be capable of calculation when reinforced with a generalisation of the Itzykson–Zuber integral to a non-compact integration manifold. We arrive to such a generalisation by discussing the Duistermaat–Heckman localisation principle for integrals over non-compact homogeneous Kähler manifolds. In the limit of large- N the asymptotic expression for the correlation function reproduces the result outlined earlier by A.V. Andreev and B.D. Simons [Phys. Rev. Lett. 75 (1995) 2304].

Functional integral approach: a third formulation of quantum statistical mechanics.

Quantum statistical mechanics has developed primarily through two approaches, pioneered by Gibbs and Feynman, respectively. In Gibbs' method one calculates partition functions from phase-space integrations or sums over stationary states. Alternatively, in Feynman's approach, the focus is on the path-integral formulation. The Hubbard-Stratonovich transformation leads to a functional-integral formulation for calculating partition functions. We outline here the functional integral approach to quantum statistical mechanics, including generalizations and improvements to Hubbard's formulation. We show how the dimensionality of the integrals is reduced exactly, how the problem of assuming an unknown canonical transformation is avoided, how the reality of the partition function in the complex representation is guaranteed, and how the extremum conditions are simplified. This formulation can be applied to general systems, including superconductors.

Self-consistent quantal treatment of decay rates within the perturbed static path approximation

The framework of the perturbed static path approximation is used to calculate the partition function of a finite Fermi system from a Hamiltonian with a separable two body interaction. Therein, the collective degree of freedom is introduced in self-consistent fashion through a Hubbard-Stratonovich transformation. In this way, all transport coefficients that dominate the decay of a metastable system are defined and calculated microscopically. Otherwise the same formalism is applied as in the Caldeira-Leggett model to deduce the decay rate from the free energy above the so called crossover temperature T(0).

Negative moments of characteristic polynomials of random matrices: Ingham–Siegel integral as an alternative to Hubbard–Stratonovich transformation

We reconsider the problem of calculating arbitrary negative integer moments of the (regularised) characteristic polynomial for N× N random matrices taken from the Gaussian Unitary Ensemble (GUE). A very compact and convenient integral representation is found via the use of a matrix integral close to that considered by Ingham and Siegel. We find the asymptotic expression for the discussed moments in the limit of large N. The latter is of interest because of a conjectured relation to properties of the Riemann ζ-function zeroes. Our method reveals a striking similarity between the structure of the negative and positive integer moments which is usually obscured by the use of the Hubbard–Stratonovich transformation. This sheds a new light on “bosonic” versus “fermionic” replica trick and has some implications for the supersymmetry method. We briefly discuss the case of the chiral GUE model from that perspective.

Extended cluster algorithm in quantum simulations

We propose an extension of the quantum cluster algorithm by a combined use of the Fortuin-Kasteleyn mapping and the Hubbard-Stratonovich transformation. To describe the idea, we consider the $S=\frac{1}{2}$ $\mathrm{XXZ}$ chain model and express the partition function as the sum in an extended configuration space of spins, graphs, and fields. Then it is clarified that this algorithm possesses a computationally tractable continuous-time limit and maintains virtues of the quantum cluster algorithm. Numerical simulations are performed and its applicability is demonstrated. Further, we discuss potential gains in our algorithm.

Some Findings on the Minus-Sign Problem for the Hubbard Model in the Projector Quantum Monte Carlo Method

The quantum Monte Carlo method for the Hubbard model using the Hubbard-Stratonovich transformation is reexamined. It is observed that the value of the ground state energy for a 3×2 and 4×4 system is estimated correctly by ignoring the minus-sign of the negative sampling weight appearing in the simulation using the projector method. It is also observed that the average of sign is positive and the correlation between the energy and the sign is weak for these systems. The reasons why the ground state energy is estimated correctly is given on a phenomenological base.

Quantum Monte Carlo study of the alternating extended Peierls–Hubbard model applied to the trans-polyacetylene

The one-dimensional alternating Peierls–Hubbard model is especially interesting as nontrivial model for conjugated polymer chains, such as polyacetylene. We study this model for chains of 64 sites using the determinantal method based on Hubbard–Stratonovich transformation. We obtain the first electronic energies and their mean fluctuations at half-filling as a function of the on-site electron–electron interaction (both short and long range U, V coupling are considered). We also study the effect of the electron–electron interaction on the dimerization by investigating some of the important correlation functions, such as spin–spin correlation, on-site charge and the specific heat. The effect of nonlinear excitations is also examined, particularly soliton excitation.

Coherent state path integral and Langevin equations of interacting bosons

Interacting excitons in a semiconductor coupled to a thermal reservoir are treated as bosons. We use a coherent state path integral formulation on the time contour and transform the quantum mechanical system of bosonic excitons to a classical Langevin equation with an inhomogenous random force which represents the influence of the thermal reservoir. However, this classical Langevin equation cannot take into account the different order of annihilation and creation operators or their correlations. This is achieved by an application of a Hubbard–Stratonovich transformation which introduces an auxiliary variable σ x ( t p ) related to the self-energy. In terms of this new field, an additional Langevin equation is derived which is similar to a saddle-point equation with a random force f x ( t). This equation contains the mean value σ ̄ x(t) of the auxiliary fields σ x(t +), σ x(t −) as a fluctuating potential or self-energy in matrix-form on the time contour.

Numerical study of the one-dimensional Hubbard model in determination of trans-polyacetylene properties

Quantum Monte Carlo simulation on one-dimensional Hubbard model is developed to study the effect of electron–electron interaction on trans-polyacetylene (t-PA) properties. The numerical calculation is based on the evaluation of the trace of the partition function using a special version of the so-called determinantal method. We use the Blakenbecler, Scalapino, Sugar and Hirsch (BSSH) algorithm based on the Hubbard–Stratonovich transformation to overcome the original problem involving anticommuting fermion operators. This numerical study allowed us to calculate mean values of different observables of physical interest such as the energies and their fluctuations. The behavior of quantum fluctuations is confirmed and interpreted by determining the specific heat. The system exhibits different ordering depending on different ranges of repulsion energy of Coulomb (U) and the on-site energy (E).

The effective action and the spectrum of collective modes in the antiferromagnetic phase of the three-band repulsive hubbard model

The paper is concerned with the application of path integration (successive integration over “fast” and “slow” variables and the Hubbard-Stratonovich transformation) to the antiferromagnetic state in the three-band, two-dimensional repulsive Hubbard model. It is shown that, in the lowest approximation, the model under consideration is equivalent to the half-filled single-band Hubbard model. The effective action for the collective Bose-fields, the critical temperature of the phase transition, and the spectra of excitations are obtained. Bibliography: 24 titles.