Finite-temperature Perturbation Theory Research Articles

Gravitational waves and tadpole resummation: Efficient and easy convergence of finite temperature QFT

We demonstrate analytically and numerically that “optimized partial dressing” (OPD) thermal mass resummation, which uses gap equation solutions inserted into the tadpole, efficiently tames finite-temperature perturbation theory calculations of the effective thermal potential, without necessitating use of the high-temperature approximation. An analytical estimate of the scale dependence for OPD resummation, standard Parwani resummation (Daisy resummation), and dimensional reduction shows that OPD has similar scale dependence to dimensional reduction, greatly improving Parwani resummation. We also elucidate how to construct and solve the gap equation for realistic numerical calculations, and demonstrate OPD’s improved accuracy for a toy scalar model. OPD’s improved accuracy is most physically significant when the high-temperature approximation breaks down, rendering dimensional reduction unusable and Parwani resummation highly inaccurate, with the latter underestimating the maximal gravitational wave amplitude for the model by 2 orders of magnitude compared to OPD. Our work highlights the need to bring theoretical uncertainties under control even when analyzing broad features of a model. Given the simplicity of the OPD compared to two-loop dimensional reduction, as well as the ease with which this scheme handles departures from the high-temperature expansion, we argue this scheme has great potential in analyzing the parameter space of realistic beyond the Standard Model models. Published by the American Physical Society 2024

Complex analysis of divergent perturbation theory at finite temperature.

We investigate the convergence properties of finite-temperature perturbation theory by considering the mathematical structure of thermodynamic potentials using complex analysis. We discover that zeros of the partition function lead to poles in the internal energy and logarithmic singularities in the Helmholtz free energy that create divergent expansions in the canonical ensemble. Analyzing these zeros reveals that the radius of convergence increases at higher temperatures. In contrast, when the reference state is degenerate, these poles in the internal energy create a zero radius of convergence in the zero-temperature limit. Finally, by showing that the poles in the internal energy reduce to exceptional points in the zero-temperature limit, we unify the two main mathematical representations of quantum phase transitions.

Finite Temperature Coupled Cluster Theories for Extended Systems.

At zero temperature, coupled cluster theories are widely used to predict total energies, ground state expectation values, and even excited states for molecules and extended systems. However, for systems with a small band gap, such as metals, the zero-temperature approximation does not necessarily hold. Thermal effects may even give rise to interesting chemistry on metal surfaces. Most approaches to temperature dependent electronic properties employ finite temperature perturbation theory in the Matsubara frequency formulation. Computations require a large number of Matsubara frequencies to yield sufficiently accurate results, especially at low temperatures. This work, and independently the work of White and Chan J. Chem. Theory Comput. 2018 , DOI: 10.1021/acs.jctc.8b00773 , proposes a coupled cluster implementation directly in the imaginary time domain on the compact interval [0, β], closely related to the thermal cluster cumulant approach of Sanyal et al. [ Chem. Phys. Lett. 1992 , 192 , 55 - 61 ] , Sanyal et al. [ Phys. Rev. E 1993 , 48 , 3373 - 3389 ], and Mandal et al. [ Int. J. Mod. Phys. B 2003 , 17 , 5367 - 5377 ]. Here, the arising imaginary time dependent coupled cluster amplitude integral equations are solved in the linearized direct ring doubles approximation, also referred to as Tamm-Dancoff approximation with second order (linearized) screened exchange. In this framework, the transition from finite to zero temperature is uniform and comes at no extra costs, allowing to go to temperatures as low as room temperature. In this approximation, correlation grand potentials are calculated over a wide range of temperatures for solid lithium, a metallic system, and for solid silicon, a semiconductor.

Theory of excitations of dipolar Bose–Einstein condensate at finite temperature

We present a systematic study of dilute three-dimensional dipolar Bose gas employing a finite temperature perturbation theory (beyond the mean field). We analyze in particular the behavior of the anomalous density, we find that this quantity has a finite value in the limit of weak interactions at both zero and finite temperatures. We show that the presence of the dipole–dipole interaction enhances fluctuations, the second order correlation function and thermodynamic quantities such as the chemical potential, the ground state energy, the compressibility and the superfluid fraction. We identify the validity criterion of the small parameter of the theory for Bose-condensed dipolar gases.

Finite-temperature perturbation theory for the random directed polymer problem

We study the random directed polymer problem—the short-scale behavior of an elastic string (or polymer) in one transverse dimension subject to a disorder potential and finite temperature fluctuations. We are interested in the polymer short-scale wandering expressed through the displacement correlator 〈[δu(X)]2〉, with δu(X) being the difference in the displacements at two points separated by a distance X. While this object can be calculated at short scales using the perturbation theory in higher dimensions d > 2, this approach becomes ill-defined and the problem turns out to be nonperturbative in the lower dimensions and for an infinite-length polymer. In order to make progress, we redefine the task and analyze the wandering of a string of a finite length L. At zero temperature, we find that the displacement fluctuations 〈[δu(X)]2〉 ∝ LX2 depend on L and scale with the square of the segment length X, which differs from a straightforward Larkin-type scaling. The result is best understood in terms of a typical squared angle 〈α2〉 ∝ L, where α = ∂xu, from which the displacement scaling for the segment X follows naturally, 〈[δu(X)]2〉 ∝ 〈α2〉 X2. At high temperatures, thermal fluctuations smear the disorder potential and the lowest-order results for disorder-induced fluctuations in both the displacement field and the angle vanish in the thermodynamic limit L → ∞. The calculation up to the second order allows us to identify the regime of validity of the perturbative approach and provides a finite expression for the displacement correlator, albeit depending on the boundary conditions and the location relative to the boundaries.

Three-loop HTL QCD thermodynamics

The hard-thermal-loop perturbation theory (HTLpt) framework is used to calculate the thermodynamic functions of a quark-gluon plasma to three-loop order. This is the highest order accessible by finite temperature perturbation theory applied to a non-Abelian gauge theory before the high-temperature infrared catastrophe. All ultraviolet divergences are eliminated by renormalization of the vacuum, the HTL mass parameters, and the strong coupling constant. After choosing a prescription for the mass parameters, the three-loop results for the pressure and trace anomaly are found to be in very good agreement with recent lattice data down to T ~ 2 − 3T c , which are temperatures accessible by current and forthcoming heavy-ion collision experiments.

Thermally induced coherence in a Mott insulator of bosonic atoms

Conventional wisdom is that increasing temperature causes quantum coherence to decrease. Using finite temperature perturbation theory and exact calculations for the strongly correlated bosonic Mott insulating state we show a practical counter-example that can be explored in optical lattice experiments: the short-range coherence of the Mott insulating phase can increase substantially with increasing temperature. We demonstrate that this phenomenon originates from thermally produced defects that can tunnel with ease. Since the near zero temperature coherence properties have been measured with high precision we expect these results to be verifiable in current experiments.

Three-loop hard-thermal-loop free energy for QED

We calculate the free energy of a hot gas of electrons and photons to three loops using the hard-thermal-loop perturbation theory reorganization of finite-temperature perturbation theory. We calculate the free energy through three loops by expanding in a power series in ${m}_{D}/T$, ${m}_{f}/T$, and ${e}^{2}$, where ${m}_{D}$ and ${m}_{f}$ are thermal masses and $e$ is the coupling constant. We demonstrate that the hard-thermal-loop perturbation reorganization improves the convergence of the successive approximations to the QED free energy at large coupling, $e\ensuremath{\sim}2$. The reorganization is gauge invariant by construction, and due to cancellation among various contributions, we obtain a completely analytic result for the resummed thermodynamic potential at three loops. Finally, we compare our result with similar calculations that use the $\ensuremath{\Phi}$-derivable approach.

Equation of State of Gluon Plasma from a Fundamental Modular Region

Despite considerable practical success in dealing with the gluon plasma, finite-temperature perturbation theory suffers at the fundamental level from infrared divergences discovered by Linde. However, if gauge or Gribov copies are properly eliminated from the physical state space, infrared modes are strongly suppressed. We describe the gluon plasma in zeroth order as a gas of free quasiparticles with a temperature-independent dispersion relation of Gribov type, E(k)=k(2)sqrt[+M4 / k(2)], that results from the reduction of the physical state space. The effective mass M2 / k controls infrared divergences and allows finite calculable corrections. The equation of state of this gas is calculated and compared with numerical lattice data.

Finite-temperature perturbation theory for quasi-one-dimensional spin-12Heisenberg antiferromagnets

We develop a finite-temperature perturbation theory for quasi-one-dimensional quantum spin systems, in the manner suggested by H.J. Schulz (1996) and use this formalism to study their dynamical response. The corrections to the random-phase approximation formula for the dynamical magnetic susceptibility obtained with this method involve multi-point correlation functions of the one-dimensional theory on which the random-phase approximation expansion is built. This ``anisotropic'' perturbation theory takes the form of a systematic high-temperature expansion. This formalism is first applied to the estimation of the N\'eel temperature of S=1/2 cubic lattice Heisenberg antiferromagnets. It is then applied to the compound Cs$_2$CuCl$_4$, a frustrated S=1/2 antiferromagnet with a Dzyaloshinskii-Moriya anisotropy. Using the next leading order to the random-phase approximation, we determine the improved values for the critical temperature and incommensurability. Despite the non-universal character of these quantities, the calculated values are different by less than a few percent from the experimental values for both compounds.

Microscopic theory of dipole-exchange spin waves in ferromagnetic films: Linear and nonlinear processes

The linear and nonlinear processes in ferromagnetic films at low temperatures $(T\ensuremath{\ll}{T}_{c})$ are studied in a microscopic theory. Both the long-range magnetic dipole-dipole and the Heisenberg exchange interactions to nearest and next-nearest neighbors are included. The results obtained for the linearized spin-wave spectrum are compared with previous macroscopic theories. For ultrathin films (or for large wave vectors) the microscopic theory provides important corrections. The nonlinear dynamics of the spin waves are studied through a finite-temperature perturbation theory based on Feynman diagrams. We obtain explicit results for the energy shift and damping (or reciprocal lifetime) of the dipole-exchange spin waves due to all possible three-magnon and four-magnon processes involving combinations of the surface and quantized bulk spin waves at low temperatures. To investigate different dipole interaction strengths (relative to the exchange) numerical results are presented using parameters for Fe, EuO, and ${\mathrm{GdCl}}_{3}.$

Quantum, many-body, finite-temperature perturbation theory for an electron–ion system

The expansion in powers of the electron charge, e, for a neutral system of electrons (fermions) and ions (Maxwell–Boltzmann particles) is extended to order e4 for arbitrary values of temperature and density. The methods of calculation of the series terms will be illustrated, and some of the consequences of these results will be discussed. The ionization profile so derived, at least at high temperatures, will be contrasted with Saha theory. Some special features of hydrogen related to the possible plasma phase transition will be noted.

Dynamic equations for individual spins in materials with magnetic order: The ideal ferromagnet

The time and temperature dependence of individual spins is studied in an ideal ferromagnet. Starting from the Heisenberg Hamiltonian in a magnetic field and, building on linear-response theory, we derive the linearized microscopic analogy to the macroscopic Landau-Lifshitz equations of motion for magnetization. The dynamical equations take into account the influence of spin correlations in addition to the molecular field. In order to illustrate the validity of our approach, we calculate the temperature-dependent spin-wave spectrum and Landau damping by employing a finite-temperature perturbation theory for spin operators. The latter is adapted to suit higher-order spin-correlation functions which can be described by simple quasiparticle interactions with the spin system. Our results agree with those established on the thermodynamics of the Heisenberg model. In addition, we provide a method of how to deduce macroscopic equations of the Landau-Lifshitz type from first principles, including temperature dependence and damping.

Field-dependent thermodynamic properties of spin-ladder and bilayer systems

Field-dependent thermodynamic properties of two gapped spin systems, two-leg spin-ladder and square-lattice bilayer, are studied using finite-temperature perturbation theory in the strong-coupling limit. The treatment gives reasonable description of the gapped disordered phase of the systems. The free energy $F(H,T)$, susceptibility $\ensuremath{\chi}(H,T)$ and specific heat $C(H,T)$ are calculated. The susceptibility displays a maximum depending strongly on the external field, while it is insensitive to H at higher temperature. With the external field close to the lower critical value, the specific heat exhibits two-round-peak structure. These results agree quite well with the recent experimental measurements on the strongly coupled ladder material ${\mathrm{Cu}}_{2}({\mathrm{C}}_{5}{\mathrm{H}}_{12}{\mathrm{N}}_{2}{)}_{2}{\mathrm{Cl}}_{4}$ by Hammar et al.

Bond-operator approach to the bilayer Heisenberg antiferromagnet

The bilayer spin-$1/2$ antiferromagnetic Heisenberg model is studied by the bond-operator mean-field theory and perturbation theory. From both methods, the zero-temperature quantum phase transition occurs at a critical ratio of ${(J}_{2}{/J}_{1}{)}_{c}\ensuremath{\approx}2.3,$ where ${J}_{1}$ and ${J}_{2}$ are the intralayer and interlayer coupling constants. Under the mean-field method, the obtained spin-wave mass, uniform susceptibility, and inverse correlation length at the critical point show good linear behavior versus temperature at low temperatures, in agreement with results of the nonlinear $\ensuremath{\sigma}$ model and quantum Monte Carlo simulations. On the other hand, the finite temperature perturbation theory gives a good description of the disordered phase in quite a large temperature range. The specific heat and uniform susceptibility exhibit broad maxima versus temperature, consistent with series expansion results.

Perturbative study of thermodynamic properties for the two-leg spin ladder

We apply finite-temperature perturbation theory to study thermodynamic properties of the two-leg antiferromagnetic spin ladder in the strong interchain coupling limit. The internal energy, specific heat and uniform susceptibility are calculated analytically by third-order perturbation expansions. At zero temperature, the present method results in the same ground state energy as that obtained by the strong coupling expansion without temperature. At finite-temperature, we obtain a peak in the specific heat and a broad maximum in the uniform susceptibility. The results agree quite well with experimental data for the material Cu 2(C 5H 12N 2) 2Cl 4 and the numerical data of 8-order series expansion theory.

Renormalisation of the non-perturbative thermal pressure

We show how the fully resummed thermal pressure is rendered ultraviolet finite by standard zero-temperature renormalisation. The analysis is developed in a six-dimensional scalar model that mimics QED and has N flavours. The N → ∞ limit of the model can be calculated completely. At a critical temperature, one of the degrees of freedom has vanishing screening mass like the transverse gauge bosons in four-dimensional finite-temperature perturbation theory. The renormalised non-perturbative interaction pressure of this model is evaluated numerically.

Finite-temperature perturbation theory for a spatially inhomogeneous Bose-condensed gas

We develop a finite-temperature perturbation theory (beyond the mean field) for a Bose-condensed gas and calculate temperature-dependent damping rates and energy shifts for Bogolyubov excitations of any energy. The theory is generalized for the case of excitations in a spatially inhomogeneous (trapped) Bose-condensed gas, where we emphasize the principal importance of inhomogeneity of the condensate density profile and develop the method of calculating the self-energy functions. The use of the theory is demonstrated by calculating the damping rates and energy shifts of low-energy quasiclassical excitations, i.e., the quasiclassical excitations with energies much smaller than the mean-field interaction between particles. In this case the boundary region of the condensate plays a crucial role and the result for the damping rates and energy shifts is completely different from that in spatially homogeneous gases. We also analyze the frequency shifts and damping of sound waves in cylindrical Bose condensates and discuss the role of damping in the recent MIT experiment on the sound propagation.

Birefringence induced by a static electric field gradient

A quantum theory of the birefringence induced in a molecular gas by a static electric field gradient is presented; it is an application of the finite temperature perturbation theory to molecular light scattering described in the preceding paper. The kinematics of the experiment are dealt with explicitly through the use of a space-fixed frame based on a set of canonically conjugate variables for the centre-of-mass and internal motions of the molecule. This leads to a definition of the molecular multipoles that is different from the conventional one and there are no ‘origin-dependence problems’ at any stage of the calculation. The susceptibility that governs the birefringence is analysed in the spirit of Van Vleck's quantum mechanical account of polarity in terms of the magnitude, relative to k B T, of the energy level separations associated with the dominant matrix elements of the external field V and the current operator j(k). As in previous treatments, the main contribution is inversely proportional to the temperature, and there is also a temperature independent term. The birefringence depends on the matrix elements of the molecular quadrupole moment operator; in the framework used here, this has a definition independent of the polarity of the molecule.

Finite temperature perturbation theory and large gauge invariance

We examine finite temperature perturbation theory for Chern-Simons theories, in the context of an analogue 0+1-dimensional model. In particular, we show how nonextensive terms arise in the perturbative finite temperature effective action, using both the real-time and imaginary-time formalisms. We illustrate how large gauge invariance is restored at all orders, despite being broken at any given order in perturbation theory. We discuss which aspects generalize to a perturbative analysis of finite temperature Chern-Simons terms in higher dimensions.