Nonequilibrium perturbation theory for complex scalar fields

Real-time perturbation theory is formulated for complex scalar fields away from thermal equilibrium in such a way that dissipative effects arising from the absorptive parts of loop diagrams are approximately resummed into the unperturbed propagators. Low order calculations of physical quantities then involve quasiparticle occupation numbers which evolve with the changing state of the field system, in contrast to standard perturbation theory, where these occupation numbers are frozen at their initial values. The evolution equation of the occupation numbers can be cast approximately in the form of a Boltzmann equation. Particular attention is given to the effects of a non-zero chemical potential, and it is found that the thermal masses and decay widths of quasiparticle modes are different for particles and antiparticles.Comment: 15 pages using RevTeX; 2 figures in 1 Postscript file; Submitted to Phys. Rev.

Perturbative nonequilibrium dynamics of phase transitions in an expanding universe

A complete set of Feynman rules is derived, which permits a perturbative description of the nonequilibrium dynamics of a symmetry-breaking phase transition in $\lambda\phi^4$ theory in an expanding universe. In contrast to a naive expansion in powers of the coupling constant, this approximation scheme provides for (a) a description of the nonequilibrium state in terms of its own finite-width quasiparticle excitations, thus correctly incorporating dissipative effects in low-order calculations, and (b) the emergence from a symmetric initial state of a final state exhibiting the properties of spontaneous symmetry breaking, while maintaining the constraint $\equiv 0$. Earlier work on dissipative perturbation theory and spontaneous symmetry breaking in Minkowski spacetime is reviewed. The central problem addressed is the construction of a perturbative approximation scheme which treats the initial symmetric state in terms of the field $\phi$, while the state that emerges at later times is treated in terms of a field $\zeta$, linearly related to $\phi^2$. The connection between early and late times involves an infinite sequence of composite propagators. Explicit one-loop calculations are given of the gap equations that determine quasiparticle masses and of the equation of motion for $$ and the renormalization of these equations is described. The perturbation series needed to describe the symmetric and broken-symmetry states are not equivalent, and this leads to ambiguities intrinsic to any perturbative approach. These ambiguities are discussed in detail and a systematic procedure for matching the two approximations is described.Comment: 22 pages, using RevTeX. 6 figures. Submitted to Physical Review

Crossover from the pair contact process with diffusion to directed percolation

Crossover behaviors from the pair contact process with diffusion (PCPD) and the driven PCPD (DPCPD) to the directed percolation (DP) are studied in one dimension by introducing a single particle annihilation/branching dynamics. The crossover exponents $\phi$ are estimated numerically as $1/\phi \simeq 0.58\pm0.03$ for the PCPD and $1/\phi \simeq 0.49 \pm 0.02$ for the DPCPD. Nontriviality of the PCPD crossover exponent strongly supports non-DP nature of the PCPD critical scaling, which is further evidenced by the anomalous critical amplitude scaling near the PCPD point. In addition, we find that the DPCPD crossover is consistent with the mean field prediction of the tricritical DP class as expected

Crossover from the parity-conserving pair contact process with diffusion to other universality classes

The pair contact process with diffusion (PCPD) with modulo 2 conservation (\pcpdt) [$2A\to 4A$, $2A\to 0$] is studied in one dimension, focused on the crossover to other well established universality classes: the directed Ising (DI) and the directed percolation (DP). First, we show that the \pcpdt shares the critical behaviors with the PCPD, both with and without directional bias. Second, the crossover from the \pcpdt to the DI is studied by including a parity-conserving single-particle process ($A \to 3A$). We find the crossover exponent $1/\phi_1 = 0.57(3)$, which is argued to be identical to that of the PCPD-to-DP crossover by adding $A \to 2A$. This suggests that the PCPD universality class has a well defined fixed point distinct from the DP. Third, we study the crossover from a hybrid-type reaction-diffusion process belonging to the DP [$3A\to 5A$, $2A\to 0$] to the DI by adding $A \to 3A$. We find $1/\phi_2 = 0.73(4)$ for the DP-to-DI crossover. The inequality of $\phi_1$ and $\phi_2$ further supports the non-DP nature of the PCPD scaling. Finally, we introduce a symmetry-breaking field in the dual spin language to study the crossover from the \pcpdt to the DP. We find $1/\phi_3 = 1.23(10)$, which is associated with a new independent route from the PCPD to the DP.Comment: 8 pages, 8 figure

Nontrivial critical crossover between directed percolation models: Effect of infinitely many absorbing states

At non-equilibrium phase transitions into absorbing (trapped) states, it is well known that the directed percolation (DP) critical scaling is shared by two classes of models with a single (S) absorbing state and with infinitely many (IM) absorbing states. We study the crossover behavior in one dimension, arising from a considerable reduction of the number of absorbing states (typically from the IM-type to the S-type DP models), by following two different (excitatory or inhibitory) routes which make the auxiliary field density abruptly jump at the crossover. Along the excitatory route, the system becomes overly activated even for an infinitesimal perturbation and its crossover becomes discontinuous. Along the inhibitory route, we find continuous crossover with the universal crossover exponent $\phi\simeq 1.78(6)$, which is argued to be equal to $\nu_\|$, the relaxation time exponent of the DP universality class on a general footing. This conjecture is also confirmed in the case of the directed Ising (parity-conserving) class. Finally, we discuss the effect of diffusion to the IM-type models and suggest an argument why diffusive models with some hybrid-type reactions should belong to the DP class.Comment: 8 pages, 9 figure

Nonequilibrium perturbation theory for spin-1/2 fields

A partial resummation of perturbation theory is described for field theories containing spin-1/2 particles in states that may be far from thermal equilibrium. This allows the nonequilibrium state to be characterized in terms of quasiparticles that approximate its true elementary excitations. In particular, the quasiparticles have dispersion relations that differ from those of free particles, finite thermal widths and occupation numbers which, in contrast to those of standard perturbation theory evolve with the changing nonequilibrium environment. A description of this kind is essential for estimating the evolution of the system over extended periods of time. In contrast to the corresponding description of scalar particles, the structure of nonequilibrium fermion propagators exhibits features which have no counterpart in the equilibrium theory.Comment: 16 pages; no figures; submitted to Phys. Rev.

Dissipation in equations of motion of scalar fields

The methods of non-equilibrium quantum field theory are used to investigate the possibility of representing dissipation in the equation of motion for the expectation value of a scalar field by a friction term, such as is commonly included in phenomenological inflaton equations of motion. A sequence of approximations is exhibited which reduces the non-equilibrium theory to a set of local evolution equations. However, the adiabatic solution to these evolution equations which is needed to obtain a local equation of motion for the expectation value is not well defined; nor, therefore, is the friction coefficient. Thus, a non-equilibrium treatment is essential, even for a system that remains close to thermal equilibrium, and the formalism developed here provides one means of achieving this numerically.Comment: 17 pages, 5 figure

Numerical investigation of friction in inflaton equations of motion

The equation of motion for the expectation value of a scalar quantum field does not have the local form that is commonly assumed in studies of inflationary cosmology. We have recently argued that the true, temporally non-local equation of motion does not possess a time-derivative expansion and that the conversion of inflaton energy into particles is not, in principle, described by the friction term estimated from linear response theory. Here, we use numerical methods to investigate whether this obstacle to deriving a local equation of motion is purely formal, or of some quantitative importance. Using a simple scalar-field model, we find that, although the non-equilibrium evolution can exhibit significant damping, this damping is not well described by the local equation of motion obtained from linear response theory. It is possible that linear response theory does not apply to the situation we study only because thermalization turns out to be slow, but we argue that that the large discrepancies we observe indicate a failure of the local approximation at a more fundamental level.Comment: 13 pages, 7 figure

Scaling in high-temperature superconductors

A Hartree approximation is used to study the interplay of two kinds of scaling which arise in high-temperature superconductors, namely critical-point scaling and that due to the confinement of electron pairs to their lowest Landau level in the presence of an applied magnetic field. In the neighbourhood of the zero-field critical point, thermodynamic functions scale with the scaling variable $(T-T_{c2}(B))/B^{1/2\nu}$, which differs from the variable $(T - T_c(0))/B^{1/2\nu}$ suggested by the gaussian approximation. Lowest-Landau-level (LLL) scaling occurs in a region of high field surrounding the upper critical field line but not in the vicinity of the zero-field transition. For YBaCuO in particular, a field of at least 10 T is needed to observe LLL scaling. These results are consistent with a range of recent experimental measurements of the magnetization, transport properties and, especially, the specific heat of high-$T_c$ materials.Comment: 22 pages + 1 figure appended as postscript fil

An Analytic Equation of State for Ising-like Models

Using an Environmentally Friendly Renormalization we derive, from an underlying field theory representation, a formal expression for the equation of state, $y=f(x)$, that exhibits all desired asymptotic and analyticity properties in the three limits $x\to 0$, $x\to \infty$ and $x\to -1$. The only necessary inputs are the Wilson functions $\gamma_\lambda$, $\gamma_\phi$ and $\gamma_{\phi^2}$, associated with a renormalization of the transverse vertex functions. These Wilson functions exhibit a crossover between the Wilson-Fisher fixed point and the fixed point that controls the coexistence curve. Restricting to the case N=1, we derive a one-loop equation of state for $2< d<4$ naturally parameterized by a ratio of non-linear scaling fields. For $d=3$ we show that a non-parameterized analytic form can be deduced. Various asymptotic amplitudes are calculated directly from the equation of state in all three asymptotic limits of interest and comparison made with known results. By positing a scaling form for the equation of state inspired by the one-loop result, but adjusted to fit the known values of the critical exponents, we obtain better agreement with known asymptotic amplitudes.Comment: 10 pages, 2 figure