Matsubara Formalism Research Articles

Pseudoscalar meson nonet at zero and finite temperature

Theoretical understanding of experimental results from relativistic heavy-ion collisions requires a microscopic approach to the behavior of QCD n-point functions at finite temperatures, as given by the hierarchy of Dyson-Schwinger equations, properly generalized within the Matsubara formalism. The convergence of sums over Matsubara modes is studied. The technical complexity of finite-temperature calculations mandates modeling. We present a model where the QCD interaction in the infrared, nonperturbative domain is modeled by a separable form. Results for the mass spectrum of light quark flavors (u, d, s) and for the pseudoscalar bound-state amplitudes at finite temperature are presented.

Thermal Casimir effect in ideal metal rectangular boxes

The thermal Casimir effect in ideal metal rectangular boxes is considered using the method of zeta functional regularization. A renormalization procedure is suggested which provides the finite expression for the Casimir free energy in any restricted quantization volume. This expression satisfies the classical limit at high temperature and leads to zero thermal Casimir force for systems with infinite characteristic dimensions. In the case of two parallel ideal metal planes the results, as derived previously using thermal quantum field theory in Matsubara formulation and other methods, are reproduced starting from the expression obtained. It is shown that for rectangular boxes the temperature-dependent contribution to the electromagnetic Casimir force can be both positive and negative depending on side lengths. Numerical computations of the scalar and electromagnetic Casimir free energy and force are performed for cubes.

Dynamical chiral symmetry breaking, color superconductivity, and Bose–Einstein condensation in an [formula omitted]-invariant supersymmetric Nambu–Jona-Lasinio model at finite temperature and density

We investigate the phenomena of the dynamical chiral symmetry breaking (DCSB), color superconductivity (CSC), and Bose–Einstein condensation (BEC) in a supersymmetric (SUSY) vector-like SU ( N c ) gauge model at finite temperature and density. Both the N = 1 four-dimensional and N = 2 three-dimensional cases are considered. We employ the N = 1 four-dimensional generalized SUSY Nambu–Jona-Lasinio model ( N = 1 generalized SNJL 4 ) with a chemical potential as the model Lagrangian. The N = 2 three-dimensional theory is obtained by a simple dimensional reduction scheme of the four-dimensional counterpart. In order to realize the DCSB and BCS-type CSC in this model, we introduce a SUSY soft mass term. After adopting the method of SUSY auxiliary fields with the Fierz transformation in color and flavor spaces, we discuss several possible breaking schemes of the global symmetries of the model. The integrations of the auxiliary fields of composites in the effective potential are performed by using the steepest descent approximation. Under the finite-temperature Matsubara formalism, the gap equations are derived and solved. The roles of both the boson and fermion sectors in the BEC, DCSB and CSC are examined by the quasiparticle excitation spectra and the gap equations. The physical properties of the DCSB and the CSC are studied in detail. Examination on the physical property of the BEC is beyond scope of this paper. It is found that the BEC, DCSB and CSC can coexist under a condition of model parameters. Several important observations are obtained in the method of the construction of the SUSY BCS-type theory, starting from a SUSY field theoretical framework.

Dual path integral representation for finite temperature quantum field theory

We impose the periodicity conditions corresponding to the Matsubara formalism for thermal field theory as constraints in the imaginary-time path integral. These constraints are introduced by means of time-independent auxiliary fields which, by integration of the original variables, become dynamical fields in the resulting ``dual'' representation for the theory. This alternative representation has the appealing property of involving fields that live in one dimension less than the original ones, with a quantum partition function whose integration measure is identical to the one of its classical counterpart, albeit with a different (spatially nonlocal) action.

Polarization tensor of charged gluons in color magnetic background field at finite temperature

We calculate the polarization tensor of charged gluons in a Abelian homogeneous magnetic background field at finite temperature in one loop order Lorentz background field gauge in full generality. Thereby we first determine the ten independent tensor structures. For the calculation of the corresponding form factors we use the Schwinger representation and represent form factors as double parametric integrals and a sum resulting from the Matsubara formalism used. The integrands are given explicitly in terms of hyperbolic trigonometric functions. Like in the case of neutral gluons, the polarization tensor is not transversal. Out of the tensor structures, seven are transversal and three are not. The nontransversal part follows explicitly from our calculations.

Analytic approach to the thermal Casimir force between metal and dielectric

The analytic asymptotic expressions for the Casimir free energy, pressure and entropy at low temperature in the configuration of one metal and one dielectric plate are obtained. For this purpose we develop the perturbation theory in a small parameter proportional to the product of the separation between the plates and the temperature. This is done using both the simplified model of an ideal metal and of a dielectric with constant dielectric permittivity and for the realistic case of the metal and dielectric with frequency-dependent dielectric permittivities. The analytic expressions for all related physical quantities at high temperature are also provided. The obtained analytic results are compared with numerical computations and good agreement is found. We demonstrate for the first time that the Lifshitz theory, when applied to the configuration of metal–dielectric, satisfies the requirements of thermodynamics if the static dielectric permittivity of a dielectric plate is finite. If it is infinitely large, the Lifshitz formula is shown to violate the Nernst heat theorem. The implications of these results for the thermal quantum field theory in Matsubara formulation and for the recent measurements of the Casimir force between metal and semiconductor surfaces are discussed.

Quantum field theory of van der Waals friction

van der Waals friction between two semi-infinite solids, and between a small neutral particle and semi-infinite solid is studied using thermal quantum field theory in the Matsubara formulation. We show that the friction to linear order in the sliding velocity can be obtained from the equilibrium Green functions and that our treatment can be extended for bodies with complex geometry. The calculated friction agrees with the friction obtained using a dynamical modification of the Lifshitz theory, which is based on the fluctuation-dissipation theorem. We show that it should be possible to measure the van der Waals friction in noncontact friction experiment using state-of-the-art equipment.

Dynamical chiral symmetry breaking and superconductivity in the supersymmetric Nambu–Jona-Lasinio model at finite temperature and density

We investigate the dynamical chiral symmetry breaking (DCSB) and superconductivity in a supersymmetric model at finite temperature and density. We employ the N=1 four-dimensional generalized supersymmetric Nambu–Jona-Lasinio model (N=1 generalized SNJL4) with a chemical potential as the model Lagrangian, and select the gauge freedom as U(1). In order to realize the DCSB and BCS-type superconductivity in this model, we introduce a SUSY soft mass term. Under the finite-temperature Matsubara formalism, the effective potential and the gap equations are derived in the flamework of the large-N expansion. The finite-density effect in the DCSB is shown by the critical coupling. The roles of both the boson and fermion sectors in the superconductivity are examined by the quasiparticle excitation spectra and the gap equations.

Thermal quantum field theory and the Casimir interaction between dielectrics

The Casimir and van der Waals interaction between two dissimilar thick dielectric plates is reconsidered on the basis of thermal quantum field theory in Matsubara formulation. We briefly review two main derivations of the Lifshitz formula in the framework of thermal quantum field theory without use of the fluctuation-dissipation theorem. A set of special conditions is formulated under which these derivations remain valid in the presence of dissipation. The low-temperature behavior of the Casimir and van der Waals interactions between dissimilar dielectrics is found analytically from the Lifshitz theory for both an idealized model of dilute dielectrics and for real dielectrics with finite static dielectric permittivities. The free energy, pressure and entropy of the Casimir and van der Waals interactions at low temperatures demonstrate the same universal dependence on the temperature as was previously discovered for ideal metals. The entropy vanishes when temperature goes to zero proving the validity of the Nernst heat theorem. This solves the long-standing problem on the consistency of the Lifshitz theory with thermodynamics in the case of dielectric plates. The obtained asymptotic expressions are compared with numerical computations for both dissimilar and similar real dielectrics and found to be in excellent agreement. The role of the zero-frequency term in Matsubara sum is investigated in the case of dielectric plates. It is shown that the inclusion of conductivity in the model of dielectric response leads to the violation of the Nernst heat theorem. The applications of this result to the topical problems of noncontact atomic friction and the Casimir interaction between real metals are discussed.

Nonlocal electron-phonon correlations in a dispersive Holstein model

Due to the dispersion of optical phonons, long range electron-phonon $(e\text{\ensuremath{-}}\mathit{ph})$ correlations renormalize downwards the coupling strength in the Holstein model. We evaluate the size of this effect both in a linear chain and in a square lattice for a time averaged $e\text{\ensuremath{-}}\mathit{ph}$ potential, where the time variable is introduced according to the Matsubara formalism. Mapping the Holstein Hamiltonian onto the time scale we derive the perturbing source current which appears to be nontime retarded. This property permits to disentangle phonon and electron coordinates in the general path integral for an electron coupled to dispersive phonons. While the phonon paths can be integrated out analytically, the electron path integrations have to be done numerically. The equilibrium thermodynamic properties of the model are thus obtained as a function of the electron hopping value and of the phonon spectrum parameters. We derive the $e\text{\ensuremath{-}}\mathit{ph}$ corrections to the phonon free energy and show that its temperature derivatives do not depend on the $e\text{\ensuremath{-}}\mathit{ph}$ effective coupling, hence, the Holstein phonon heat capacity is strictly harmonic. A significant upturn in the low temperature total heat capacity over $T$ ratio is attributed to the electron hopping which largely contributes to the action.

Spontaneous symmetry breaking in compactified λϕ4 theory

We consider the massive vector N-component ( λϕ 4) D theory in Euclidean space and, using an extended Matsubara formalism we perform a compactification on a d-dimensional subspace, d⩽ D. This allows us to treat jointly the effect of temperature and spatial confinement in the effective potential of the model, setting forth grounds for an analysis of phase transitions driven by temperature and spatial boundary. For d=2, which corresponds to the heated system confined between two parallel planes (separation L), we obtain, in the large N limit at one-loop order, formulas for temperature- and boundary-dependent mass and coupling constant. The equation for the critical curve in the β× L plane is also derived.

Boundary dependence of the coupling constant and the mass in the vectorN-component (λφ4)Dtheory

Using the Matsubara formalism, we consider the massive (λ4)D vector N-component model in the large-N limit, the system being confined between two infinite parallel planes. We investigate the behaviour of the coupling constant as a function of the separation L between the planes. For the Wick-ordered model in D = 3 we are able to give an exact formula to the L dependence of the coupling constant. For the non-Wick-ordered model we indicate how expressions for the coupling constant and the mass can be obtained for arbitrary dimension D in the small-L regime. Closed exact formulae for the L-dependent renormalized coupling constant and mass are obtained in D = 3 and their behaviours as functions of L are displayed. We are also able to obtain, in generic dimension D, an equation for the critical value of L corresponding to a second-order phase transition in terms of the Riemann zeta-function. In D = 3 an implicit formula for the critical L is given.

Semiclassical Casimir energies at finite temperature

We study the dependence on the temperature T of Casimir effects for a range of systems, and in particular for a pair of ideal parallel conducting plates, separated by a vacuum. We study the Helmholtz free energy, combining Matsubara's formalism, in which the temperature appears as a periodic Euclidean fourth dimension of circumference 1/T, with the semiclassical periodic orbital approximation of Gutzwiller. By inspecting the known results for the Casimir energy at T=0 for a rectangular parallelepiped, one is led to guess at the expression for the free energy of two ideal parallel conductors without performing any calculation. The result is a new form for the free energy in terms of the lengths of periodic classical paths on a two-dimensional cylinder section. This expression for the free energy is equivalent to others that have been obtained in the literature. Slightly extending the domain of applicability of Gutzwiller's semiclassical periodic orbit approach, we evaluate the free energy at T>0 in terms of periodic classical paths in a four-dimensional cavity that is the tensor product of the original cavity and a circle. The validity of this approach is at present restricted to particular systems. We also discuss the origin of the classical form of the free energy at high temperatures.

CASIMIR FORCE UNDER THE INFLUENCE OF REAL CONDITIONS

The Casimir force is calculated analytically for configurations of two parallel plates and a spherical lens (sphere) above a plate with an account of nonzero temperature, finite conductivity of the boundary metal and surface roughness. The permittivity of the metal is described by the plasma model. It is proved that in case of the plasma model the scattering formalism of quantum field theory in Matsubara formulation underlying Lifshitz formula is well defined and no modifications are needed concerning the zero-frequency contribution. The temperature correction to the Casimir force is found completely with respect to temperature and perturbatively (up to the second order in the relative penetration depth of electromagnetic zero-point oscillations into the metal) with respect to finite conductivity. The asymptotics of low and high temperatures are presented and contributions of longitudinal and perpendicular modes are determined separately. Serving as an example, aluminium test bodies are considered, showing good agreement between the obtained analytical results and previously performed numerical computations. The roughness correction is formally included and formulas are given permitting us to calculate the Casimir force under the influence of all relevant factors.

Chiral quark model with infrared cut-off for the description of meson properties in hot matter

A simple chiral quark model of the Nambu--Jona-Lasinio (NJL) type with a quark confinement mechanism is constructed for the description of the light meson sector of QCD at finite temperature. Unphysical quark production thresholds in the NJL model are excluded by an infrared cut-off in the momentum integration within quark loop diagrams. This chiral quark model satisfies the low energy theorems. Using the vacuum masses and decay widths of \pi- and \rho-mesons for fixing the model parameters, the properties of the \sigma- meson are derived. Within the Matsubara formalism, the model is systematically extended to finite temperatures where chiral symmetry restoration due to a dropping constituent quark mass entails a vanishing of the infrared cut-off (deconfinement) at the pion Mott temperature T_c=186 MeV. Besides of the pion mass and weak decay constant, the masses, coupling constants and decay widths of \sigma- and \rho-mesons in hot matter are investigated. The quark-antiquark decay channel of the light mesons is opened for T>T_c only and becomes particularly strong for the \rho- meson. The two-pion decay channel below T_c has almost constant width for the \rho- meson up to T_c, but for the \sigma-meson it closes below T_c such that a scalar meson state with vanishing width is obtained as a precursor of the chiral/deconfinement transition.

Chiral quark model with infrared cut-off for the description of meson properties in hot matter

A simple chiral quark model of the Nambu-Jona-Lasinio (NJL) type with a quark confinement mechanism is constructed for the description of the light-meson sector of QCD at finite temperature. Unphysical quark production thresholds in the NJL model are excluded by an infrared cut-off in the momentum integration within quark loop diagrams. This chiral quark model satisfies the low-energy theorems. Using the vacuum masses and decay widths of π- and ρ-mesons for fixing the model parameters, the properties of the σ-meson are derived. Within the Matsubara formalism, the model is systematically extended to finite temperatures where chiral symmetry restoration due to a dropping constituent quark mass entails a vanishing of the infrared cut-off (deconfinement) at the pion Mott temperature T c = 186 MeV. Besides the pion mass and weak decay constant, the masses, coupling constants and decay widths of σ- and ρ-mesons in hot matter are investigated. The quark-antiquark decay channel of the light mesons is opened for T > T c only and becomes particularly strong for the ρ-meson. The two-pion decay channel below T c has almost constant width for the ρ-meson up to T c, but for the σ-meson it closes below T c such that a scalar meson state with vanishing width is obtained as a precursor of the chiral/deconfinement transition.

FINITE T MESON CORRELATIONS AND QUARK DECONFINEMENT

The finite temperature spatial [Formula: see text] correlation modes in the π and ρ channels are studied with the rainbow-ladder truncated quark Dyson–Schwinger equation and Bethe–Salpeter equation in the Matsubara formalism. To retain the finite range of the effective interaction while facilitating summation over fermion Matsubara modes necessary to ensure continuity at T=0, a separable kernel is used. The model is fixed by T=0 properties and it implements dynamical chiral symmetry breaking and quark confinement. Transition temperatures for deconfinement (Td) and chiral restoration (Tc) are identified. Above and below these transitions we study Mπ(T), fπ(T) and the three-space transverse and longitudinal masses [Formula: see text] and [Formula: see text]. We also study the intrinsic T-dependence of the electromagnetic coupling constant gρ(T) and the strong coupling constant gρππ(T). For both this model and a related semi-analyticinfrared dominant model, we analyze the high T behavior of the obtained masses in comparison to the M(T)→2πT behavior found in lattice QCD simulations.

Reconstructing the Thermal Green Functions¶at Real Times from Those at Imaginary Times

By exploiting the analyticity and boundary value properties of the thermal Green functions that result from the KMS condition in both time and energy complex variables, we treat the general (non-perturbative) problem of recovering the thermal functions at real times from the corresponding functions at imaginary times, introduced as primary objects in the Matsubara formalism. The key property on which we rely is the fact that the Fourier transforms of the retarded and advanced functions in the energy variable have to be the `unique Carlsonian analytic interpolations' of the Fourier coefficients of the imaginary-time correlator, the latter being taken at the discrete Matsubara imaginary energies, respectively in the upper and lower half-planes. Starting from the Fourier coefficients regarded as `data set', we then develop a method based on the Pollaczek polynomials for constructing explicitly their analytic interpolations.

INFRARED DIVERGENCE IN QED3 AT FINITE TEMPERATURE

We consider various ways of treating the infrared divergence which appears in the dynamically generated fermion mass, when the transverse part of the photon propagator in N flavour QED 3 at finite temperature is included in the Matsubara formalism. This divergence is likely to be an artifact of taking into account only the leading order term in the [Formula: see text] expansion when we calculate the photon propagator and is handled here phenomenologically by means of an infrared cutoff. Inserting both the longitudinal and the transverse part of the photon propagator in the Schwinger–Dyson equation, we find the dependence of the dynamically generated fermion mass on the temperature and the cutoff parameters. It turns out that consistency with certain statistical physics arguments imposes conditions on the cutoff parameters. For parameters in the allowed range of values we find that the ratio r=2* Mass (T=0)/critical temperature is approximately 6, consistent with previous calculations which neglected the transverse photon contribution.

SUSCEPTIBILITY AND VERTEX CORRECTIONS FOR A SQUARE FERMI SURFACE

We investigate the response of an electron system which exhibits ideal nesting features. Using the standard Matsubara formalism we derive analytic expressions for the imaginary and real parts of the bare particle–hole susceptibility. The imaginary part has sharp peaks whose maxima at the nesting momenta approximately scale with (ω/T). The peak lineshapes resemble neutron scattering data on chromium and some copper oxide superconductors. The real part of the bare susceptibility at the nesting vectors diverges logarithmically at low temperatures. Analytic formulas for the first vertex correction to the susceptibility are derived for a Hubbard interaction and its momentum and temperature variations are calculated numerically. This term detracts substantially from the ordinary RPA terms for intermediate values of the Coulomb repulsion. Exact cancellation of a certain class of diagrams at half filling is shown to result from particle–hole symmetry. We discuss the consequences of these results for spin fluctuation theories of high temperature superconductors and spin density wave instabilities.