Auxiliary Field Approach Research Articles

Real-Time Thermal Field Theory Analyses of 2D Gross-Neveu Model**The project partially supported by National Natural Science Foundation of China and by Grant No. LWTZ-1298 of the Chinese Academy of Sciences

The discrete symmetry breaking and possible restoration at finite temperature are analyzed in 2D Gross-Neveu model by the real-time thermal field theory in the fermion bubble approximation. The dynamical fermion mass m is proven to, be scale-independent and this fact indicates the equivalence between the fermion bubble diagram approximation and the meanfield approximation used in the auxiliary scalar field approach. Reproducing of the nonzero critical temperature is the dynamical fermion mass at , shows the equivalence between the real-time and the imaginary-time thermal. field theories in this problem. However, in the real-time formalism, more results including absence of scalar bound state, the equation of criticality curve of chemical potential-temperature and the behavior of at can be easily obtained. The last one indicates the second-order phase transition feature of the symmetry restoration.

Time discretization of functional integrals

Numerical evaluation of functional integrals usually involves a finite (L-slice) discretization of the imaginary-time axis. In the auxiliary-field method, the L-slice approximant to the density matrix can be evaluated as a function of inverse temperature at any finite L as $\rho_L(\beta)=[\rho_1(\beta/L)]^L$, if the density matrix $\rho_1(\beta)$ in the static approximation is known. We investigate the convergence of the partition function $Z_L(\beta)=Tr\rho_L(\beta)$, the internal energy and the density of states $g_L(E)$ (the inverse Laplace transform of $Z_L$), as $L\to\infty$. For the simple harmonic oscillator, $g_L(E)$ is a normalized truncated Fourier series for the exact density of states. When the auxiliary-field approach is applied to spin systems, approximants to the density of states and heat capacity can be negative. Approximants to the density matrix for a spin-1/2 dimer are found in closed form for all L by appending a self-interaction to the divergent Gaussian integral and analytically continuing to zero self-interaction. Because of this continuation, the coefficient of the singlet projector in the approximate density matrix can be negative. For a spin dimer, $Z_L$ is an even function of the coupling constant for L<3: ferromagnetic and antiferromagnetic coupling can be distinguished only for $L\ge 3$, where a Berry phase appears in the functional integral. At any non-zero temperature, the exact partition function is recovered as $L\to\infty$.

Hund's rule coupling as the microscopic origin of the spin-triplet pairing in a correlated and degenerate band system

We propose that the spin-triplet pairing can originate from the intraatomic Hund's rule exchange in a degenerate d-band system. The role of this interaction in the pairing is decisive when accounted for in conjunction with rather strong correlations induced by the direct Coulomb interactions. The superconducting-gap value is obtained in the saddle-point approximation for the Coulomb correlations, treated in the auxiliary Bose field approach, which is combined with the mean-field approximation of the BCS type for the pairing part.

Discrete symmetry breaking and restoration at finite temperature in 3D Gross-Neveu model

Dynamical spontaneous breaking of some discrete symmetries including special parities and time reversal and their restoration at finite temperature T are researched in 3D Gross-Neveu model by means of Schwinger-Dyson equation in the real-time thermal field theory in the fermion bubble diagram approximation. When the momentum cut-off Λ is large enough, the equation of critical chemical potential μ c and critical temperature T c will be Λ-independent and identical to the one obtained by auxiliary scalar field approach. The dynamical fermion mass m≡ m( T, μ), as the order parameter of symmetry breaking, has the same ( T c − T) 1/2 behavior at T≲ T c as one in 4D NJL-model and this shows the second-order phase transition feature of the symmetry restoration at T> T c . It is also proven that no scalar bound state could exist in this model.

Interface pinning and slow ordering kinetics on infinitely ramified fractal structures

We investigate the time-dependent Ginzburg-Landau (TDGL) equation for a nonconserved order parameter on an infinitely ramified (deterministic) fractal lattice employing two alternative methods: the auxiliary field approach and a numerical method of integration of the equations of evolution. In the first case the domain size evolves with time as $L(t)\ensuremath{\sim}{t}^{{1/d}_{w}}$, where ${d}_{w}$ is the anomalous random-walk exponent associated with the fractal and differs from the normal value $2$, which characterizes all Euclidean lattices. Such a power-law growth is identical to the one observed in the study of the spherical model on the same lattice, but fails to describe the asymptotic behavior of the numerical solutions of the TDGL equation for a scalar order parameter. In fact, the simulations performed on a two dimensional Sierpinski carpet indicate that, after an initial stage dominated by a curvature reduction mechanism in the manner of Allen and Cahn [Acta. Metall. 27, 1085 (1979)], the system enters in a regime where the domain walls between competing phases are pinned by lattice defects. The lack of translational invariance determines a rough free-energy landscape, the existence of many metastable minima, and the suppression of the marginally stable modes, which in translationally invariant systems lead to power-law growth and self-similar patterns. On fractal structures, as the temperature vanishes the evolution is frozen since only thermally activated processes can sustain the growth of pinned domains.

MESON-DIQUARK LOOP EXPANSION

We consider a general (nonlocal) four-fermion quantum field theory and show how the Cornwall-Jackiw-Tomboulis effective action can be systematically expanded in the number, η, of composite, bose loops. This is achieved by the introduction of auxiliary, bilocal fields which describe fermion-fermion and fermion-antifermion correlations. The η expansion can be understood as a generalization of the [Formula: see text] expansion and is of particular interest in quark models, for example, where the bilocal fields can be identified with meson and diquark degrees of freedom. Comparison with the usual loop (ħ) expansion reveals some unusual characteristics of the η expansion and throws light on recent studies of diquark degrees of freedom in which the auxiliary field approach is used.

Diffusive behavior of states in the Hubbard-Stratonovitch transformation.

The Hubbard-Stratonovitch auxiliary-field approach to projection of the ground state of an interacting fermion system from a trial state is examined. It is shown that the method is equivalent to solving a differential equation with diffusion, drift, and branching terms on the manifold of normalized Slater determinants. Explicit general expressions are given for the coefficients of the diffusion equation in terms of the Hubbard-Stratonovitch fields. The form of the equation is somewhat similar to that obtained in the continuum Green's-function Monte Carlo method, although its interpretation and relation to the physical many-body problem is quite different. The character of the representation of the many-body ground state arising in the auxiliary-field projection approach is discussed within this framework. The diffusion process is normally found to concentrate the states representing the ground state near the classical mean-field solutions. The consequences of this picture of the auxiliary-field approach for the ``fermion minus-sign'' problem in this context are discussed, and some conclusions are reached concerning the range of validity of a recently suggested approximation where minus signs are ignored.

Gauge dependence of the effective potential for bilocal composite fields in Abelian gauge theories.

The Nielsen-Fukuda-Kugo identity expressing the gauge dependence of effective potentials is generalized to the case of the effective potential for bilocal composite fields in quantum electrodynamics. The resulting identity serves to show the gauge independence of the ground-state energy in the presence of dynamical chiral-symmetry breaking. As a by-product we derive a formula giving the gauge dependence of fermion propagators. It is shown explicitly that the effective potential in the quenched planar approximation does not respect the identity. The Nielsen-Fukuda-Kugo identity is also derived for the effective potential in the auxiliary-field approach.

Minimal coupling in spin-2 field

It is shown that the spin-2 Lagrangian of Watanabe and Bhargava can be generalised by the introduction of a real parameter b associated with derivative ordering. After minimal coupling the spin-2 dynamical equations of Nath and Velo and Zwanziger can be derived as special cases. The constraint situation in these equations is then summarised and related to the first-order Lagrangian approach of Federbush, Chang, et al. Only one value of the parameter b gives a correct manifold of states after coupling. Finally it is shown that the auxiliary field approach to spin-2 proposed by Chang is dynamically inconsistent under minimal coupling.