Matsubara Summation Research Articles

Factorizing two-loop vacuum sum-integrals

We derive analytic results for scalar massless bosonic vacuum sum-integrals at two loops. Building upon a recent factorization proof of massive two-loop vacuum integrals, we are able to solve the corresponding Matsubara sums and map the result onto one-loop structures, thereby proving factorization also in the sum-integral setting. Analytic results are provided for generic integer-valued propagator- and numerator-powers of the class of sum-integrals under consideration, allowing to eliminate them from any perturbative expansion, dramatically simplifying the evaluation of some observables encountered e.g. in hot QCD.

Finite temperature effects in quantum systems with competing scalar orders

The study of the competition or coexistence of different ground states in many-body systems is an exciting and actual topic of research, both experimentally and theoretically. Quantum fluctuations of a given phase can suppress or enhance another phase depending on the nature of the coupling between the order parameters, their dynamics and the dimensionality of the system. The zero temperature phase diagrams of systems with competing scalar order parameters with quartic and bilinear coupling terms have been previously studied for the cases of a zero temperature bicritical point and of coexisting orders. In this work, we apply the Matsubara summation technique from finite temperature quantum field theory to introduce the effects of thermal fluctuations on the effective potential of these systems. This is essential to make contact with experiments. We consider two and three-dimensional materials characterized by a Lorentz invariant quantum critical theory, i.e., with dynamic critical exponent z = 1, such that time and space scale in the same way. We obtain that in both cases, thermal fluctuations lead to weak first-order temperature phase transitions, at which coexisting phases arising from quantum corrections become unstable. We show that above this critical temperature (Tc), the system presents scaling behavior consistent with that approaching a quantum critical point. Below the transition the specific heat has a thermally activated contribution with a gap related to the size of the domains of the ordered phases. We obtain that Tc decreases as a function of the distance to the zero temperature classical bicritical point (ZTCBP) in the coexistence region, implying that in our approach, the system attains the highest Tc above the fine tuned value of this ZTCBP.

Nernst heat theorem for an atom interacting with graphene: Dirac model with nonzero energy gap and chemical potential

We derive the low-temperature behavior of the Casimir-Polder free energy for a polarizable atom interacting with graphene sheet which possesses the nonzero energy gap $\Delta$ and chemical potential $\mu$. The response of graphene to the electromagnetic field is described by means of the polarization tensor in the framework of Dirac model on the basis of first principles of thermal quantum field theory in the Matsubara formulation. It is shown that the thermal correction to the Casimir-Polder energy consists of three contributions. The first of them is determined by the Matsubara summation using the polarization tensor defined at zero temperature, whereas the second and third contributions are caused by an explicit temperature dependence of the polarization tensor and originate from the zero-frequency Matsubara term and the sum of all Matsubara terms with nonzero frequencies, respectively. The asymptotic behavior for each of the three contributions at low temperature is found analytically for any value of the energy gap and chemical potential. According to our results, the Nernst heat theorem for the Casimir-Polder free energy and entropy is satisfied for both $\Delta > 2\mu$ and $\Delta < 2\mu$. We also reveal an entropic anomaly arising in the case $\Delta = 2\mu$. The obtained results are discussed in connection with the long-standing fundamental problem in Casimir physics regarding the proper description of the dielectric response of matter to the electromagnetic field.

Minimax isometry method: A compressive sensing approach for Matsubara summation in many-body perturbation theory

We present a compressive sensing approach for the long-standing problem of Matsubara summation in many-body perturbation theory. By constructing low-dimensional, almost isometric subspaces of the Hilbert space we obtain optimum imaginary time and frequency grids that allow for extreme data compression of fermionic and bosonic functions in a broad temperature regime. The method is applied to the random phase and self-consistent $GW$ approximation of the grand potential. Integration and transformation errors are investigated for Si and ${\mathrm{SrVO}}_{3}$.

Ab initio electron-two-phonon scattering in GaAs from next-to-leading order perturbation theory

Electron-phonon (e–ph) interactions are usually treated in the lowest order of perturbation theory. Here we derive next-to-leading order e–ph interactions, and compute from first principles the associated electron-two-phonon (2ph) scattering rates. The derivations involve Matsubara sums of two-loop Feynman diagrams, and the numerical calculations are challenging as they involve Brillouin zone integrals over two crystal momenta and depend critically on the intermediate state lifetimes. Using Monte Carlo integration together with a self-consistent update of the intermediate state lifetimes, we compute and converge the 2ph scattering rates, and analyze their energy and temperature dependence. We apply our method to GaAs, a weakly polar semiconductor with dominant optical-mode long-range e–ph interactions. We find that the 2ph scattering rates are as large as nearly half the value of the one-phonon rates, and that including the 2ph processes is necessary to accurately predict the electron mobility in GaAs from first principles.

Algorithmic Matsubara integration for Hubbard-like models

We present an algorithm to evaluate Matsubara sums for Feynman diagrams comprised of bare Green's functions with single-band dispersions with local U Hubbard interaction vertices. The algorithm provides an exact construction of the analytic result for the frequency integrals of a diagram that can then be evaluated for all parameters $U$, temperature $T$, chemical potential $\mu$, external frequencies and internal/external momenta. This method allows for symbolic analytic continuation of results to the real frequency axis, avoiding any ill-posed numerical procedure. When combined with diagrammatic Monte-Carlo, this method can be used to simultaneously evaluate diagrams throughout the entire $T-U-\mu$ phase space of Hubbard-like models at minimal computational expense.

Reduced-Shifted Conjugate-Gradient Method for a Green’s Function: Efficient Numerical Approach in a Nano-Structured Superconductor

We propose the reduced-shifted Conjugate-Gradient (RSCG) method, which is numerically efficient to calculate a matrix element of a Green's function defined as a resolvent of a Hamiltonian operator, by solving linear equations with a desired accuracy. This method does not calculate solution vectors of linear equations but does directly calculate a matrix element of the resolvent. The matrix elements with different frequencies are simultaneously obtained. Thus, it is easy to calculate the exception value expressed as a Matsubara summation of these elements. To illustrate a power of our method, we choose a nano-structured superconducting system with a mean-field Bogoliubov-de Gennes (BdG) approach. This method allows us to treat with the system with the fabrication potential, where one can not effectively use the kernel-polynomial-based method. We consider the d-wave nano-island superconductor by simultaneously solving the linear equations with a large number (~ 50000) of Matsubara frequencies.

Bose-Einstein condensation and Silver Blaze property from the two-loopΦ-derivable approximation

We extend our previous investigation of the two-loop $\mathrm{\ensuremath{\Phi}}$-derivable approximation to the case of a finite chemical potential $\ensuremath{\mu}$ and discuss Bose-Einstein condensation in the case of a charged scalar field with $O(2)$ symmetry. We show that the approximation is renormalizable by means of counterterms which are independent of both the temperature and the chemical potential. We point out the presence of an additional skew contribution to the propagator as compared to the $\ensuremath{\mu}=0$ case, which comes with its own gap equation (except at Hartree level). We solve this equation together with the field equation, and the usual longitudinal and transversal gap equations to find that the transition is second order, in agreement with recent lattice results to which we compare. We also discuss a general criterion an approximation should obey for the so-called Silver Blaze property to hold, and we show that any $\mathrm{\ensuremath{\Phi}}$-derivable approximation at finite temperature and density obeys this criterion if one chooses an UV regularization that does not cut off the Matsubara sums.

Dependence of the strength of van der Waals interactions on the details of the dielectric response variation

ABSTRACTWe will present a simplified approximate model showing how even small changes in the dielectric response result in substantial variations in the Hamaker coefficient of the van der Waals interactions. Since all the terms in the Matsubara summation depends on the variation of the dielectric response spectra at one particular frequency, the total change in the Hamaker coefficient depends on the spectral changes not only at that frequency but also at the rest of the spectrum properly weighted. The Matsubara terms most affected by the addition of a single peak are not those close to the position of the added peak, but are distributed over the entire range of frequencies. We comment on the possibility of eliminating van der Waals interactions and/or drastically reducing them by spectral variation in a narrow regime of frequencies.

Dielectric response variation and the strength of van der Waals interactions

Small changes in the dielectric response of a material result in substantial variations in the Hamaker coefficient of the van der Waals interactions, as demonstrated in a simplified approximate model as well as a realistic example of amorphous silica with and without an exciton peak. Variation of the dielectric response spectra at one particular frequency influences all terms in the Matsubara summation, making the total change in the Hamaker coefficient depend on the spectral changes not only at that frequency but also at the rest of the spectrum, properly weighted. The Matsubara terms most affected by the addition of a single peak are not those close to the position of the added peak, but are distributed doubly non-locally over the entire range of frequencies. A possibility of eliminating van der Waals interactions or at least drastically reducing them by spectral variation in a narrow regime of frequencies thus seems very remote.

High temperature limit in static backgrounds

We prove that the hard thermal loop contribution to static thermal amplitudes can be obtained by setting all the external four-momenta to zero before performing the Matsubara sums and loop integrals. At the one-loop order we do an iterative procedure for all the one-particle irreducible one-loop diagrams, and at the two-loop order we consider the self-energy. Our approach is sufficiently general to the extent that it includes theories with any kind of interaction vertices, such as gravity in the weak field approximation, for $d$ space-time dimensions. This result is valid whenever the external fields are all bosonic.

Broken phase effective potential in the two-loopΦ-derivable approximation and nature of the phase transition in a scalar theory

We study the phase transition of a real scalar ${\ensuremath{\varphi}}^{4}$ theory in the two-loop $\ensuremath{\Phi}$-derivable approximation using the imaginary time formalism, extending our previous (analytical) discussion of the Hartree approximation. We combine fast Fourier transform algorithms and accelerated Matsubara sums in order to achieve a high accuracy. Our results confirm and complete earlier ones obtained in the real time formalism [A. Arrizabalaga and U. Reinosa, Nucl. Phys. A785, 234 (2007)] but which were less accurate due to the integration in Minkowski space and the discretization of the spectral density function. We also provide a complete and explicit discussion of the renormalization of the two-loop $\ensuremath{\Phi}$-derivable approximation at finite temperature, both in the symmetric and in the broken phase, which was already used in the real time approach, but never published. Our main result is that the two-loop $\ensuremath{\Phi}$-derivable approximation suffices to cure the problem of the Hartree approximation regarding the order of the transition: the transition is of the second order type, as expected on general grounds. The corresponding critical exponents are, however, of the mean-field type. Using a ``renormalization group-improved'' version of the approximation, motivated by our renormalization procedure, we find that the exponents are modified. In particular, the exponent $\ensuremath{\delta}$, which relates the field expectation value $\overline{\ensuremath{\phi}}$ to an external field $h$, changes from 3 to 5, getting then closer to its expected value 4.789, obtained from accurate numerical estimates [A. Pelissetto and E. Vicari, Phys. Rept. 368, 549 (2002)].

Model analysis on thermal UV-cutoff effects on the critical boundary in hot QCD

We study the critical boundary on the quark-mass plane associated with the chiral phase transition in QCD at finite temperature. We point out that the critical boundaries obtained from the lattice QCD simulation and the chiral effective model are significantly different; in the (Polyakov-loop coupled) Nambu--Jona-Lasinio (NJL) model we find that the critical mass is about 1 order of magnitude smaller than the value reported in the lattice QCD case. The lattice QCD study yields smaller critical mass in the continuum limit along the temporal direction. To investigate the temporal UV-cutoff effects quantitatively, we consider the (P)NJL model with only finite Matsubara summation ${N}_{\ensuremath{\tau}}$. We confirm that the critical mass in such a UV-tamed NJL model becomes larger with decreasing ${N}_{\ensuremath{\tau}}$, which demonstrates a correct tendency to explain the difference between the lattice QCD and chiral model results.

Systematics of high temperature perturbation theory: The two-loop electron self-energy in QED

In order to investigate the systematics of the loop expansion in high temperature gauge theories beyond the leading order hard thermal loop (HTL) approximation, we calculate the two-loop electron proper self-energy in high temperature QED. The two-loop bubble diagram contains a linear infrared divergence. Even if regulated with a non-zero photon mass M of order of the Debye mass, this infrared sensitivity implies that the two-loop self-energy contributes terms to the fermion dispersion relation that are comparable to or even larger than the next-to-leading-order (NLO) contributions at one-loop. Additional evidence for the necessity of a systematic restructuring of the loop expansion comes from the explicit gauge parameter dependence of the fermion damping rate at both one and two-loops. The leading terms in the high temperature expansion of the two-loop self-energy for all topologies arise from an explicit hard-soft factorization pattern, in which one of the loop integrals is hard, nested inside a second loop integral which is soft. There are no hard-hard contributions to the two-loop Sigma at leading order at high T. Provided the same factorization pattern holds for arbitrary ell loops, the NLO high temperature contributions to the electron self-energy come from ell-1 hard loops factorized with one soft loop integral. This hard-soft pattern is both a necessary condition for the resummation over ell to coincide with the one-loop self-energy calculated with HTL dressed propagators and vertices, and to yield the complete NLO correction to the self-energy at scales ~eT, which is both infrared finite and gauge invariant. We employ spectral representations and the Gaudin method for evaluating finite temperature Matsubara sums, which facilitates the analysis of multi-loop diagrams at high T.

On the evaluation of Matsubara sums

Given a connected (multi)graph G, consisting of V vertices and I lines, we consider a class of multidimensional sums of the general form S G := ∑ ∞ n1=-∞ ∑ ∞ n2=-∞ ... ∑ ∞ ni=-∞ δ G (n 1 ,n 2 ,...,n I ;{N v }) / (n 2 1 + q 2 1 )(n 2 2+ q 2 2 ) ... (n 2 1 + q 2 1 ) , where the variables q i (i = 1, ... ,I) are real and positive and the variables N v (v = 1,..., V) are integer-valued. δ G (n 1 , n 2 , ..... n I ; {N v }) is a function valued in {0,1} which imposes a series of linear constraints among the summation variables n i , determined by the topology of the graph G. We prove that these sums, which we call Matsubara sums, can be explicitly evaluated by applying a G-dependent linear operator O' G to the evaluation of the integral obtained from S G by replacing the discrete variables n i by continuous real variables x i and replacing the sums by integrals.

Continued fraction representation of the Fermi-Dirac function for large-scale electronic structure calculations

An efficient and accurate contour integration method is presented for large-scale electronic structure calculations based on the Green function. By introducing a continued fraction representation of the Fermi-Dirac function derived from a hypergeometric function, the Matsubara summation is generalized with respect to distribution of poles so that the integration of the Green function can converge rapidly. Numerical illustrations, evaluation of the density matrix for a simple model Green function and a total energy calculation for aluminum bulk within density functional theory, clearly show that the method provides remarkable convergence with a small number of poles, indicating that the method can be applied to not only the electronic structure calculations, but also a wide variety of problems.

Andreev spectrum and supercurrent in a multilayer ferromagnetic clean Josephson junction

In this work we consider a ballistic superconductor/multilayer–ferromagnet/superconductor junction, with interface scattering potential, which can be different in strength. We develop, for an arbitrary number of layers, compact analytic formulae for the Andreev bound state spectrum and the supercurrent. The phase dependence of the supercurrent is obtained from the Andreev amplitudes and results in an extremely simple form for any number of ferromagnetic layers. In the Matsubara summation it enters the denominator as a cos plus a -independent term. The reduced formulae are amenable to efficient numerical computation. For a few cases we also give simple equations for the numerical determination of the bound states, obtained with algebraic matrix multiplication and numerical results are discussed.

Infrared behavior of the dispersion relations in high-temperature scalar QED

We investigate the infrared properties of the next-to-leading-order dispersion relations in scalar quantum electrodynamics at high temperature in the context of hard-thermal-loop perturbation theory. Specifically, we determine the damping rate and the energy for scalars with ultrasoft momenta. We show by explicit calculations that an early external-momentum expansion, before the Matsubara sum is performed, gives exactly the same result as a late one. The damping rate is obtained up to fourth order included in the ultrasoft momentum and the energy up to second order. The damping rate is found sensitive in the infrared whereas the energy not.

Alpha matter on a lattice

We obtain the equation of state of interacting alpha matter and the critical temperature of Bose–Einstein condensation of alpha particles within an effective scalar field theory. We start from a non-relativistic model of uniform alpha matter interacting with attractive two-body and repulsive three-body potentials and reformulate this model as a O ( 2 ) symmetric scalar ϕ 6 field theory with negative quartic and positive sextic interactions. Upon restricting the Matsubara sums, near the temperature of Bose–Einstein condensation, to the zeroth order modes we further obtain an effective classical theory in three spatial dimensions. The phase diagram of the alpha matter is obtained from simulations of this effective field theory on a lattice using local Monte Carlo algorithms.

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.