Finite-temperature Renormalization Group Research Articles

Frequency-temperature crossover in the conductivity of disordered Luttinger liquids

The temperature (T) and frequency ({omega}) dependent conductivity of weakly disordered Luttinger liquids is calculated in a systematic way both by perturbation theory and from a finite temperature renormalization group (RG) treatment to leading order in the disorder strength. Whereas perturbation theory results in {omega}/T scaling of the conductivity, such scaling is violated in the RG treatment. We also determine the nonlinear field dependence of the conductivity, whose power law scaling is different from that of temperature and frequency dependence.

Quantum nonlinear sigma model with a damping term; finite temperature renormalization group analysis

Quantum nonlinear sigma model with a damping term; finite temperature renormalization group analysis

Quantum Nonlinear Sigma Model with a Damping Term; Finite Temperature Renormalization Group Analysis

We study the behavior of the finite temperature quantum nonlinear sigma model in two dimensions in the presence of the damping of the form $$f (|\omega_n|) = \gamma |\omega_n|^\alpha$$ , where α satisfies $$\alpha \ge 1$$ . The analytical calculations will be performed using $$\alpha = 2$$ and the results will be compared with the standard results obtained for the standard quantum nonlinear sigma model. The behavior of such a system is connected with the pseudogap which appears in the normal state of the cuprate superconductors.

Dimensional reduction, hard thermal loops, and the renormalization group

We study the realization of dimensional reduction and the validity of the hard thermal loop expansion for lambda phi^4 theory at finite temperature, using an environmentally friendly finite-temperature renormalization group with a fiducial temperature as flow parameter. The one-loop renormalization group allows for a consistent description of the system at low and high temperatures, and in particular of the phase transition. The main results are that dimensional reduction applies, apart from a range of temperatures around the phase transition, at high temperatures (compared to the zero temperature mass) only for sufficiently small coupling constants, while the HTL expansion is valid below (and rather far from) the phase transition, and, again, at high temperatures only in the case of sufficiently small coupling constants. We emphasize that close to the critical temperature, physics is completely dominated by thermal fluctuations that are not resummed in the hard thermal loop approach and where universal quantities are independent of the parameters of the fundamental four-dimensional theory.

Finite-temperature renormalization group analysis of interaction effects in 2D lattices of Bose–Einstein condensates

By using a renormalization group analysis, we study the effect of interparticle interactions on the critical temperature TBKT at which the Berezinskii–Kosterlitz–Thouless (BKT) transition occurs for Bose–Einstein condensates loaded at finite temperature in a 2D optical lattice. We find that TBKT decreases as the interaction energy decreases; when U/J = 36/π one has TBKT = 0, signalling the possibility of a quantum phase transition of BKT type.

Finite temperature renormalization group effective potentials for the linear sigma model

We derive an approximate renormalization group (RG) solution for the linear sigma model of Gell-Mann and Levy at finite temperature. For our purposes, the Fermionic degrees of freedom of the model are interpreted as quarks which interact via the $\ensuremath{\sigma}$ and $\ensuremath{\pi}$ mesons to provide a phenomenological description of hot nuclear matter at temperatures from 1 GeV to approximately 100 MeV. Our solution consists of two coupled, nonlinear flow equations for the Yukawa coupling, g, of the model and the Bosonic effective potential, U. These equations are solved numerically to study the behavior of the model as it evolves from high to low energy scales at finite temperature T. This allows us to determine the critical temperature, ${T}_{c},$ at which chiral symmetry breaking occurs, and to assess the relative sensitivity of this quantity with respect to variations in the input parameters of the model. Our results are consistent with values for ${T}_{c}$ obtained from other theoretical approaches such as lattice gauge theory or other RG techniques.

A UNIFIED DESCRIPTION OF STATIC AND DYNAMIC PROPERTIES OF FERMI LIQUIDS

In Landau's phenomenological Fermi-liquid theory (FLT), most physical quantities are derived from the knowledge of the energy variation δE[δn] corresponding to a change δn of the quasi-particle (QP) distribution function n≡{nkσ}. We show that the internal energy E[n] (or, more precisely, the thermodynamic potential Φ[n]), expressed as a function of the QP distribution n, can be interpreted as an effective potential (in the sense of field theory), which is obtained from the free energy by a Legendre transformation. This allows to obtain explicitly δΦ (or δE) starting from a microscopic Hamiltonian and to relate the Landau f function to the forward-scattering two-particle vertex without considering the collective modes as in the standard diagrammatic derivation of FLT. Out-of-equilibrium properties are obtained by extending the definition of the effective potential to space- and time-dependent configurations. Φ[n] is then a functional of the Wigner distribution function n≡{nkσ (r,t)}. It contains information about both the static and dynamic properties of the Fermi liquid. In particular, it yields the quantum Boltzmann equation satisfied by nkσ (r, t). Finally, we show how δΦ[δn] can be derived (in the static case) using a finite-temperature renormalization-group approach. In agreement with previous results based on this technique, we find that the Landau f function is defined by the fixed-point value of the Ω-limit of the forward-scattering two-particle vertex.

Consistency of blocking transformations in the finite-temperature renormalization group

The finite-temperature renormalization group is formulated via the Wilson-Kadanoff blocking transformation. Momentum modes and the Matsubara frequencies are coupled by constraints from a smearing function which plays the role of an infrared cutoff regulator. Using the scalar λφ 4 theory as an example, we consider four general types of smearing functions and show that, to zeroth order in the derivative expansion, they yield qualitatively the same temperature dependence of the running constants and the same critical exponents within numerical accuracy.

Evolution of Coupling Constant in Hot QCD

The evolution of QCD coupling constant at finite temperature is considered by making use of the finite temperature renormalization group equation up to the one-loop order in the background field method with the Feynman gauge and the imaginary time formalism. The results are compared with the ones obtained in the literature. We point out, in particular, the origin of the discrepancies between different calculations, such as the choice of gauge, the breakdown of Lorentz invariance, imaginary versus real time formalism and the applicability of the Ward identities at finite temperature.

FINITE-TEMPERATURE RENORMALIZATION OF THE (ϕ4)4–MODEL

We summarize results of the finite-temperature renormalization group approach, formalized by Matsumoto, Nakano and Umezawa in 1984, for the λ(ϕ4)4-model. The flow parameter is the reference temperature at which the mass parameter and the coupling constant of the theory are defined through renormalization conditions. We derive flow equations to one-loop order, and integrate numerically from zero temperature to above the critical temperature. The mass and the coupling constant both vanish at the critical temperature, and are positive below and above the critical region. In the critical region dimensional reduction to an effective 3D theory takes place. The leading behavior of the mass at high temperature is linear with a small logarithmic sub-leading contribution.

CRITICAL TEMPERATURE AND AMPLITUDE RATIOS FROM A FINITE TEMPERATURE RENORMALIZATION GROUP

We study λφ4 theory using an environmentally friendly finite temperature renormalization group. We derive flow equations, using a fiducial temperature as flow parameter, develop them perturbatively in an expansion free from ultraviolet and infrared divergences, then integrate them numerically from zero to temperatures above the critical temperature. The critical temperature, at which the mass vanishes, is obtained by integrating the flow equations, and is determined as a function of the zero temperature mass and coupling. We calculate the field expectation value and the minimum of the effective potential as functions of temperature and derive some universal amplitude ratios which connect the broken and symmetric phases of the theory. The latter are found to be in good agreement with those of the three-dimensional Ising model obtained from high and low temperature series expansions.

What can we learn from the finite temperature renormalization group?

The renormalization group for finite temperature quantum field theories is studied, in particular for λϕ4. It is shown that the “high” temperature limit can only be discussed perturbatively ifT dependent renormalization schemes are implemented. Zero temperature renormalization schemes or renormalization at some fixed reference temperatureTo are both inadequate as they imply perturbative expansions about fixed points of the renormalization group which are associated with a zero temperature system and a system at temperatureTo respectively.T dependent schemes give rise to an expansion about the true fixed point of the system, the resulting renormalization group allows the entire crossover between high and low temperature behaviour to be investigated.

Finite temperature renormalization of the (? 3)6- and (? 4)4-models at zero momentum

A self-consistent renormalization scheme at finite temperature and zero momentum is used together with the finite temperature renormalization group to study the temperature dependence of the mass and the coupling to one-loop order in the (φ 3)6- and (φ 4)4-models. It is found that the critical temperature is shifted relative to the naive one-loop result and the coupling constants at the critical temperature get large corrections. In the high temperature limit of the (φ 4)4-model the coupling decreases.

RENORMALIZATION GROUP EQUATIONS AND DECONFINEMENT IN FINITE TEMPERATURE QCD

Finite temperature renormalization group equations are employed to investigate the behavior of the effective coupling constant and anomalous dimension in SU c(3) gauge field at finite temperature. On the basis of this, the deconfinement of high temperature QCD is discussed.

Limitations to dimensional reduction at high temperature

The claim that renormalizable four-dimensional field theories in the infinite-temperature limit undergo a reduction to effective three-dimensional ones is analyzed in perturbation theory. An essential ingredient is the finite-temperature renormalization group in the imaginary-time formalism. A precise criterion for the occurence of complete dimensional reduction is given. This is satisfied only in exceptional cases, and is violated e.g. by ϕ 4 and QCD. These theories dimensionally reduce only up to a given order in perturbation theory. Illustrative one-loop calculations are given on the basis of a novel summation technique. The perturbative structure of QED and QCD at high temperature is examined in detail. Ward identities as well as explicit computations are used to explain why QED dimensionally reduces to all orders, but QCD does not. In addition, a potential instability deriving from an anomalous diagram is identified and cured.

Dimensional reduction at high temperature revisited

We give a perturbative analysis of dimensional reduction in the infinite-temperature limit of thermal field theory in the imaginary-time formalism. In analogy with the ordinary (Appelquist-Carazzone) decoupling theorem, we seek a class of renormalization schemes in which the heavy (non-static) modes decouple. This class turns out to consist of temperature-dependent schemes, so that the appropriate context to study dimensional reduction at high T is seen to be the finite-temperature renormalization group. As a corollary, it follows that dimensional reduction to all orders occurs only in exceptional cases; in scalar field theories and QCD such a reduction is valid only up to a given (low) order of normalization-group improved perturbation theory.

Renormalization group analysis of the temperature dependent coupling constant in massless theory

A general analysis of finite temperature renormalization group equations for massless theories is presented. It is found that in a direction where momenta and temperature are scaled up with their ratio fixed the coupling constant behaves in the same manner as in zero temperature and that the asymptotic freedom at short distances is also maintained at finite temperature.

Finite-temperature renormalization group equation in QCD. II

We obtain the finite-temperature renormalization group improved coupling constant in QCD at the one-loop level. The behaviour of the coupling constant with respect to temperature and momenta is found to differ depending on which vertex is chosen to define the coupling constant. Further it appears to depend on the choice of subtraction point for momenta.

Finite temperature renormalization group equation in QCD

A scheme of renormalization group approach at finite temeperature is applied to QCD to investigate the behaviour of the effective coupling constant with respect to both temperature and momentum scale. It is found that the coupling constant decreases with temperature for fixed momenta. We also find that in a particular direction where both momenta and temperature are changed at the same rate the coupling constant behaves in the same manner as at zero temperature.

Higher order contributions in finite temperature renormalization group

We apply to the O(N) λφ4 model the finite temperature renormalization group method, in which the temperature-dependent effective coupling constant and mass are introduced. It is pointed out that the improvement by use of the two-loop renormalization-group coefficients is important at high temperature and drastically changes the lowest order result. We also show that the two-loop renormalization-group coefficients are exact in the large-N limit.