Matsubara frequency

The original idea of approximating the quantum mechanical partition function by a classical one belongs to Feynman [Feynman and Vernon 1963 Feynman and Kleinert 1986]. Expanding an arbitrary /S-periodic orbit, entering into the partition-function path integral, in a Fourier series in Matsubara frequencies v . 

Fig. 1. Momentum shell in the space of (discrete) Matsubara frequencies (w ) and momenta (fc). Only modes of the phase tp in the stripe 1/6 < fc < 1 with b = 1 + 0+ are integrate in one RG step.

It is important to emphasize (see Ref. [2]) that the TC in this case differs from the TC realized in the superfluid He3 and, for example, in materials like Sr2RuC>4 [4], The triplet-type superconducting condensate we predict here is symmetric in momentum and therefore is insensitive to non-magnetic impurities. It is odd in frequency and is called sometimes odd superconductivity. This type of the pairing has been proposed by Berezinskii in 1975 [5] as a possible candidate for the mechanism of superfluidity in He3. However, it turned out that another type of pairing was realized in He3 triplet, odd in momentum p (sensitive to ordinary impurities) and even in the Matsubara frequencies w. It is also important to note that while the symmetry of the order parameter A in Refs. [4, 5] differs from that of the BCS order parameter, in our case A is nonzero only in the S layers and is of the BCS type. It is determined by the amplitude of the singlet component. Since the triplet and singlet components are connected which each other, the TC affects A in an indirect way. 

But when does the scaling process stop It must stop when (1) the quantum coherence length = %0el, where 0 Vp/E0, becomes equal to the thermal coherence length th = Vpl nT, at which point the thermal fluctuations take over, (2) E0(/) = E0e l becomes equal to the energy vFq of the quasiparticles involved in the q dependent response functions [the q of Eq. (9a)] or (3) equal to the external Matsubara frequency [Pg.39]

It is obviously ideally suited to measuring the effect of the electron quantum fluctuations on the phonon frequency. What one immediately learns from Eq. (26) is that the propagator is quasistatic that is, the >m = 0 component dominates for T > co /2tt. This comes from the definition of the Matsubara frequencies for bosons [under Eq. (8)]. As far as the electrons are concerned, the atoms move very slowly (the adiabatic limit). If 2g2 gi> - g3 (see Fig. 5), the electrons are able to screen the slow lattice motion and thus soften the interactions. We are obviously interested in the 2kF phonons, which will be screened most effectively by the dominant 2kF charge response of the one-dimensional electron gas. 

Let us start with the E e JT polaron in the weak-coupling region (or for the case of small gEigie), in which the perturbation approach in momentum representation is useful. The thermal one-electron Green s function Gkya(ico ) with co the fermion Matsubara frequency is defined at temperature T by... 

Feynman-Kleinert formalism Golden ratio Matsubara frequencies Matter s stability path integrals for chemistry Riemann s series Wiegner expansion... 

Xii(o>) = magnetic susceptibility x(q, a>) = electronic susceptibility XzziQ ") = longitudinal susceptibility 1/ (2) = trigamma function ip z) = digamma function o) = Matsubara frequencies op(fl) = dispersion of optical phonons oj(q) = dispersion of magnetic excitons phonon mode frequencies = antisymmetric part of deformation tensor... 

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