Matsubara Frequencies Research Articles

Efficient ab initio Migdal-Eliashberg calculation considering the retardation effect in phonon-mediated superconductors

We formulate an efficient scheme to perform Migdal-Eliashberg calculation considering the retardation effect from first principles. While the conventional approach requires a huge number of Matsubara frequencies, we show that the intermediate representation of the Green's function [H. Shinaoka et al., Phys. Rev. B 96, 035147 (2017)] dramatically reduces the numerical cost to solve the linearized gap equation. Without introducing any empirical parameter, we demonstrate that we can successfully reproduce the experimental superconducting transition temperature of elemental Nb ($\sim 10$ K) very accurately. The present result indicates that our approach has a superior performance for many superconductors for which $T_{\rm c}$ is lower than ${\mathcal O}(10)$ K

The unbalanced phonon-induced superconducting state on a square lattice beyond the static boundary

The paper presents our verification of induction of the superconducting state on a square lattice by the linear electron–phonon interaction for values of the unbalance parameter (γ=λD∕λND) less than γC=0.42. Symbols λD and λND denote the values of the coupling constant in the diagonal and the non-diagonal channel of the self-energy. Calculations were carried out using the Eliashberg equations, in which the order parameter (Δkiωn) and the wave function renormalising factor (Zkiωn) depend explicitly on the Matsubara frequency (ωn) and the wave vector (k). The value of γC in the static boundary (Δkiωn→Δkiωn=1), equal to (0.93), is significantly greater than the obtained limit value of 0.42. Values of the thermodynamic functions of the superconducting state determined for our assumptions are significantly different from the values calculated in accordance with the BCS theory. The results were obtained for the electron–phonon interaction function explicitly dependent on the momentum transfer between electron states.

Possible odd-frequency Amperean magnon-mediated superconductivity in topological insulator–ferromagnetic insulator bilayer

We study the magnon-mediated pairing between fermions on the surface of a topological insulator (TI) coupled to a ferromagnetic insulator with a tilted mean field magnetization. Tilting the magnetization towards the interfacial plane reduces the magnetic band gap and leads to a shift in the effective TI dispersions. We derive and solve the self-consistency equation for the superconducting gap in two different situations, where we neglect or include the frequency dependence of the magnon propagator. Neglecting the frequency dependence results in p-wave Amperean solutions. We also find that tilting the magnetization into the interface plane favors Cooper pairs with center of mass momenta parallel to the magnetization vector, increasing $T_c$ compared to the out-of-plane case. Including the frequency dependence of the magnon propagator, and solving for a low number of Matsubara frequencies, we find that the eigenvectors of the Amperean solutions at the critical temperature are dominantly odd in frequency and even in momentum, thus opening the possibility for odd-frequency Amperean pairing.

Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. II. The γ model at a finite T for 0<γ<1

In this paper we continue the analysis of the interplay between non-Fermi liquid and superconductivity for quantum-critical systems, the low-energy physics of which is described by an effective model with dynamical electron-electron interaction $V(\Omega_m) \propto 1/|\Omega_m|^\gamma$ (the $\gamma$ model). In paper I [A. Abanov and A. V. Chubukov, Phys Rev B. 102, 024524 (2020)] two of us analyzed the $\gamma$ model at $T=0$ for $0<\gamma <1$ and argued that there exist a discrete, infinite set of topologically distinct solutions for the superconducting gap, all with the same spatial symmetry. The gap function $\Delta_n (\omega_m)$ for the $n$th solution changes sign $n$ times as the function of Matsubara frequency. In this paper we analyze the linearized gap equation at a finite $T$. We show that there exist an infinite set of pairing instability temperatures, $T_{p,n}$, and the eigenfunction $\Delta_n (\omega_{m})$ changes sign $n$ times as a function of a Matsubara number $m$. We argue that $\Delta_n (\omega_{m})$ retains its functional form below $T_{p,n}$ and at $T=0$ coincides with the $n$th solution of the nonlinear gap equation. Like in paper I, we extend the model to the case when the interaction in the pairing channel has an additional factor $1/N$ compared to that in the particle-hole channel. We show that $T_{p,0}$ remains finite at large $N$ due to special properties of fermions with Matsubara frequencies $\pm \pi T$, but all other $T_{p,n}$ terminate at $N_{cr} = O(1)$. The gap function vanishes at $T \to 0$ for $N > N_{cr}$ and remains finite for $N < N_{cr}$. This is consistent with the $T =0$ analysis.

Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. I. The γ model and its phase diagram at T=0 : The case 0&lt;γ&lt;1

We analyze a class of quantum-critical models, in which momentum integration and the selection of a particular pairing symmetry can be done explicitly, and the competition between non-Fermi liquid and pairing can be analyzed within an effective model with dynamical electron-electron interaction $V(\Omega_m)\sim 1/|\Omega_m|^\gamma$ (the $\gamma$-model). In this paper, the first in the series, we consider the case $T=0$ and $0<\gamma <1$. We argue that tendency to pairing is stronger, and the ground state is a superconductor. We argue, however, that superconducting state is highly non-trivial as there exists a discrete set of topologically distinct solutions for the pairing gap $\Delta_n (\omega_m)$ ($n = 0, 1, 2..., \infty$). All solutions have the same spatial pairing symmetry, but differ in the time domain: $\Delta_n (\omega_m)$ changes sign $n$ times as a function of Matsubara frequency $\omega_m$. The $n =0$ solution $\Delta_0 (\omega_m)$ is sign-preserving and tends to a finite value at $\omega_m =0$, like in BCS theory. The $n = \infty$ solution corresponds to an infinitesimally small $\Delta (\omega_m)$. As a proof, we obtain the exact solution of the linearized gap equation at $T=0$ on the entire frequency axis for all $0<\gamma <1$, and an approximate solution of the non-linear gap equation.We argue that the presence of an infinite set of solutions opens up a new channel of gap fluctuations. We extend the analysis to the case where the pairing component of the interaction has additional factor $1/N$ and show that there exists a critical $N_{cr} >1$, above which superconductivity disappears, and the ground state becomes a non-Fermi liquid.We show that all solutions develop simultaneously once $N$ gets smaller than $N_{cr}$.

Quantum field theoretical description of the Casimir effect between two real graphene sheets and thermodynamics

The analytic asymptotic expressions for the Casimir free energy and entropy for two parallel graphene sheets possessing nonzero energy gap $\Delta$ and chemical potential $\mu$ are derived at arbitrarily low temperature. Graphene is described in the framework of thermal quantum field theory in the Matsubara formulation by means of the polarization tensor in (2+1)-dimensional space-time. Different asymptotic expressions are found under the conditions $\Delta>2\mu$, $\Delta=2\mu$, and $\Delta<2\mu$ taking into account both the implicit temperature dependence due to a summation over the Matsubara frequencies and the explicit one caused by a dependence of the polarization tensor on temperature as a parameter. It is shown that for both $\Delta>2\mu$ and $\Delta<2\mu$ the Casimir entropy satisfies the third law of thermodynamics (the Nernst heat theorem), whereas for $\Delta=2\mu$ this fundamental requirement is violated. The physical meaning of the discovered anomaly is considered in the context of thermodynamic properties of the Casimir effect between metallic and dielectric bodies.

Algorithmic approach to diagrammatic expansions for real-frequency evaluation of susceptibility functions

We systematically generate the perturbative expansion for the two-particle spin susceptibility in the Feynman diagrammatic formalism and apply this expansion to a model system---the single-band Hubbard model on a square lattice. We make use of algorithmic Matsubara integration (AMI) [A. Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 99, 035120 (2019)] to analytically evaluate Matsubara frequency summations, allowing us to symbolically impose analytic continuation to the real-frequency axis. We minimize our computational expense by applying graph invariant transformations [A. Taheridehkordi, S. H. Curnoe, and J. P. F. LeBlanc, Phys. Rev. B 101, 125109 (2020)]. We highlight extensions of the random-phase approximation and T-matrix methods that, due to AMI, become tractable. We present results for weak interaction strength where the direct perturbative expansion is convergent, and verify our results on the Matsubara axis by comparison to other numerical methods. By examining the spin susceptibility as a function of real frequency via an order-by-order expansion, we can identify precisely what role higher-order corrections play on spin susceptibility and demonstrate the utility and limitations of our approach.

Fermionic pole-skipping in holography

We examine thermal Green’s functions of fermionic operators in quantum field theories with gravity duals. The calculations are performed on the gravity side using ingoing Eddington-Finkelstein coordinates. We find that at negative imaginary Matsubara frequencies and special values of the wavenumber, there are multiple solutions to the bulk equations of motion that are ingoing at the horizon and thus the boundary Green’s function is not uniquely defined. At these points in Fourier space a line of poles and a line of zeros of the correlator intersect. We analyze these ‘pole-skipping’ points in three-dimensional asymptotically anti-de Sitter spacetimes where exact Green’s functions are known. We then generalize the procedure to higher-dimensional spacetimes and derive the generic form the boundary correlator takes near the pole-skipping points in momentum space. We also discuss the special case of a fermion with half-integer mass in the BTZ background. We discuss the implications and possible generalizations of the results.

Sparse modeling approach to obtaining the shear viscosity from smeared correlation functions

We propose the sparse modeling method to estimate the spectral function from the smeared correlation functions. We give a description of how to obtain the shear viscosity from the correlation function of the renormalized energy-momentum tensor (EMT) measured by the gradient flow method (C (t, τ )) for the quenched QCD at finite temperature. The measurement of the renormalized EMT in the gradient flow method reduces a statistical uncertainty thanks to its property of the smearing. However, the smearing breaks the sum rule of the spectral function and the over-smeared data in the correlation function may have to be eliminated from the analyzing process of physical observables. In this work, we demonstrate the sparse modeling analysis in the intermediate-representation basis (IR basis), which connects between the Matsubara frequency data and real frequency data. It works well even using very limited data of C (t, τ ) only in the fiducial window of the gradient flow. We utilize the ADMM algorithm which is useful to solve the LASSO problem under some constraints. We show that the obtained spectral function reproduces the input smeared correlation function at finite flow-time. Several systematic and statistical errors and the flow-time dependence are also discussed.

Quantum chaos, pole-skipping and hydrodynamics in a holographic system with chiral anomaly

It is well-known that chiral anomaly can be macroscopically detected through the energy and charge transport, due to the chiral magnetic effect. On the other hand, in a holographic many body system, the chaotic modes might be only associated with the energy conservation. This suggests that, perhaps, one can detect microscopic anomalies through the diagnosis of quantum chaos in such systems. To investigate this idea, we consider a magnetized brane in AdS space time with a Chern-Simons coupling in the bulk. By studying the shock wave geometry in this background, we first compute the corresponding butterfly velocities, in the presence of an external magnetic field B, in μ « T and B « T2 limit. We find that the butterfly propagation in the direction of B has a different velocity than in the opposite direction; the difference is ∆vB = (log(4)−1)∆vsound with ∆vsound being the difference between the velocity of two sound modes propagating in the system. The splitting of butterfly velocities confirms the idea that chiral anomaly can be macroscopically manifested via quantum chaos. We then show that the pole-skipping points of energy density Green’s function of the boundary theory coincide precisely with the chaos points. This might be regarded as the hydrodynamic origin of quantum chaos in an anomalous system. Additionally, by studying the near horizon dynamics of a scalar field on the above background, we find the spectrum of pole-skipping points associated with the two-point function of dual boundary operator. We find that the sum of wavenumbers corresponding to pole-skipping points at a specific Matsubara frequency is a universal quantity, which is independent of the scaling dimension of the dual boundary operator. We then show that this quantity follows from a closed formula and can be regarded as another macroscopic manifestation of the chiral anomaly.

Real-frequency diagrammatic Monte Carlo at finite temperature

Diagrammatic expansions are a central tool for treating correlated electron systems. At thermal equilibrium, they are most naturally defined within the Matsubara formalism. However, extracting any dynamic response function from a Matsubara calculation ultimately requires the ill-defined analytical continuation from the imaginary- to the real-frequency domain. It was recently proposed [Phys. Rev. B 99, 035120 (2019)] that the internal Matsubara summations of any interaction-expansion diagram can be performed analytically by using symbolic algebra algorithms. The result of the summations is then an analytical function of the complex frequency rather than Matsubara frequency. Here we apply this principle and develop a diagrammatic Monte Carlo technique which yields results directly on the real-frequency axis. We present results for the self-energy $\Sigma(\omega)$ of the doped 32x32 cyclic square-lattice Hubbard model in a non-trivial parameter regime, where signatures of the pseudogap appear close to the antinode. We discuss the behavior of the perturbation series on the real-frequency axis and in particular show that one must be very careful when using the maximum entropy method on truncated perturbation series. Our approach holds great promise for future application in cases when analytical continuation is difficult and moderate-order perturbation theory may be sufficient to converge the result.

Dual parquet scheme for the two-dimensional Hubbard model: Modeling low-energy physics of high- Tc cuprates with high momentum resolution

We present a method to treat the two-dimensional (2D) Hubbard model for parameter regimes which are relevant for the physics of the high-${T}_{c}$ superconducting cuprates. Unlike previous attempts to attack this problem, our approach takes into account all fluctuations in different channels on equal footing and is able to treat reasonable large lattice sizes up to $32\ifmmode\times\else\texttimes\fi{}32$. This is achieved by the following three-step procedure: (i) We transform the original problem to a representation (dual fermions) in which all purely local correlation effects from the dynamical mean field theory are already considered in the bare propagator and bare interaction of the new problem. (ii) The strong $1/{(i\ensuremath{\nu})}^{2}$ decay of the bare propagator allows us to integrate out all higher Matsubara frequencies aside from the lowest using low-order diagrams. The new effective action depends only on the two lowest Matsubara frequencies which allows us to (iii) apply the two-particle self-consistent parquet formalism, which takes into account the competition between different low-energy bosonic modes in an unbiased way, on much finer momentum grids than usual. In this way, we were able to map out the phase diagram of the 2D Hubbard model as a function of temperature and doping. Consistently with the experimental evidence for hole-doped cuprates and previous dynamical cluster approximation calculations, we find an antiferromagnetic region at low doping and a superconducting dome at higher doping. Our results also support the role of the van Hove singularity as an important ingredient for the high value of ${T}_{c}$ at optimal doping. At small doping, the destruction of antiferromagnetism is accompanied by an increase of charge fluctuations supporting the scenario of a phase-separated state driven by quantum critical fluctuations.

Sparse sampling and tensor network representation of two-particle Green's functions

Many-body calculations at the two-particle level require a compact representation of two-particle Green’s functions. In this paper, we introduce a sparse sampling scheme in the Matsubara frequency domain as well as a tensor network representation for two-particle Green’s functions. The sparse sampling is based on the intermediate representation basis and allows an accurate extraction of the generalized susceptibility from a reduced set of Matsubara frequencies. The tensor network representation provides a system independent way to compress the information carried by two-particle Green’s functions. We demonstrate efficiency of the present scheme for calculations of static and dynamic susceptibilities in single- and two-band Hubbard models in the framework of dynamical mean-field theory.

Sparse sampling approach to efficient ab initio calculations at finite temperature

Efficient ab initio calculations of correlated materials at finite temperature require compact representations of the Green's functions both in imaginary time and Matsubara frequency. In this paper, we introduce a general procedure which generates sparse sampling points in time and frequency from compact orthogonal basis representations, such as Chebyshev polynomials and intermediate representation (IR) basis functions. These sampling points accurately resolve the information contained in the Green's function, and efficient transforms between different representations are formulated with minimal loss of information. As a demonstration, we apply the sparse sampling scheme to diagrammatic $GW$ and GF2 calculations of a hydrogen chain, of noble gas atoms and of a silicon crystal.

Generalized quadrature for finite temperature Green’s function methods

In electronic structure and quantum transport calculations, many physical quantities are integrations over the electron energy, weighted by the Fermi–Dirac distribution function. Green’s function based approaches commonly circumvent the numerically difficult real energy integration by extending the integrand analytically into the complex energy plane, and using a Gaussian quadrature integration over a complex energy contour for zero temperature. For finite temperatures, a much slower convergent sum over the Matsubara frequencies is necessary. We present a generalized quadrature method that uses orthogonal polynomials on the manifold of Matsubara frequencies to enable rapid convergence. Both Gaussian quadrature integration and Matsubara frequency summation methods are shown to be limiting cases of the generalized method. Tests on an all-electron ab initio code and total energy calculation of interacting Anderson impurity model show convergence with a small number of energy mesh points.

Prerequisites for Relevant Spectral Density and Convergence of Reduced Density Matrices at Low Temperatures

Hierarchical equations of motion approach with the Drude-Lorentz spectral density has been widely employed in investigating quantum dissipative phenomena. However, it is often computationally costly for low-temperature systems because a number of Matsubara frequencies are involved. In this note, we examine a prerequisite required for spectral density, and demonstrate that relevant spectral density may significantly reduce the number of Matsubara terms to obtain convergent results for low temperatures.

π-π scattering lengths: thermo-magnetic corrections in the linear sigma model

We present a discussion of the behavior of π-π scattering lengths under the presence of an external magnetic field, including also finite temperature effects. In this way, we are able to analize the interplay between temperature and strength of the magnetic field. Our analysis is done in the frame of the linear sigma-model, and our calculations are exact within this context. No approximations are introduced for handling the summation over Landau levels and Matsubara frequencies. Although the effects are comparatively small, it is interesting to remark that magnetic field and temperature display opposite effects over the scattering lengths.

Horizon constraints on holographic Green\u2019s functions

We explore a new class of general properties of thermal holographic Green’s functions that can be deduced from the near-horizon behaviour of classical perturbations in asymptotically anti-de Sitter spacetimes. We show that at negative imaginary Matsubara frequencies and appropriate complex values of the wavenumber the retarded Green’s functions of generic operators are not uniquely defined, due to the lack of a unique ingoing solution for the bulk perturbations. From a boundary perspective these ‘pole-skipping’ points correspond to locations in the complex frequency and momentum planes at which a line of poles of the retarded Green’s function intersects with a line of zeroes. As a consequence the dispersion relations of collective modes in the boundary theory at energy scales ω ∼ T are directly constrained by the bulk dynamics near the black-brane horizon. For the case of conserved U (1) current and energy-momentum tensor operators we give examples where the dispersion relations of hydrodynamic modes pass through a succession of pole- skipping points as real wavenumber is increased. We discuss implications of our results for transport, hydrodynamics and quantum chaos in holographic systems.

Finite temperature Landau gauge lattice quark propagator

The quark propagator at finite temperature is investigated using quenched gauge configurations. The propagator form factors are investigated for temperatures above and below the gluon deconfinement temperature T_c and for the various Matsubara frequencies. Significant differences between the functional behaviour below and above T_c are observed both for the quark wave function and the running quark mass. The results for the running quark mass indicate a link between gluon dynamics, the mechanism for chiral symmetry breaking and the deconfinement mechanism. For temperatures above T_c and for low momenta, our results support also a description of quarks as free quasiparticles.

Modelling the ultra-strongly coupled spin-boson model with unphysical modes

A quantum system weakly coupled to a zero-temperature environment will relax, via spontaneous emission, to its ground-state. However, when the coupling to the environment is ultra-strong the ground-state is expected to become dressed with virtual excitations. This regime is difficult to capture with some traditional methods because of the explosion in the number of Matsubara frequencies, i.e., exponential terms in the free-bath correlation function. To access this regime we generalize both the hierarchical equations of motion and pseudomode methods, taking into account this explosion using only a biexponential fitting function. We compare these methods to the reaction coordinate mapping, which helps show how these sometimes neglected Matsubara terms are important to regulate detailed balance and prevent the unphysical emission of virtual excitations. For the pseudomode method, we present a general proof of validity for the use of superficially unphysical Matsubara-modes, which mirror the mathematical essence of the Matsubara frequencies.