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Quantum Particle in a Diffusive Electron Gas
Published in Andrei D. Zaikin, Dmitry S. Golubev, Dissipative Quantum Mechanics of Nanostructures, 2019
Andrei D. Zaikin, Dmitry S. Golubev
To begin with, we notice that in both limits of zero Matsubara frequency and zero imaginary time, the current–current correlator can be conveniently related to the free energy of the system F = −T ln Z. Making use of Eq. (8.27), which defines the current expectation value I=〈I^〉 and employing the identity () ∂(e−βH^)∂ϕx=−∫0βdτe−(β−τ)H^∂H^∂ϕxe−τH^
Basic Physical Concepts of Organic Conductors
Published in Jean-Pierre Farges, Organic Conductors, 2022
where τ is the usual imaginary time of statistical physics, ωm = 2πTm is a Matsubara frequency, and β = 1/kBT. In one dimension the free-electron susceptibilities are all logarithmic, behaving like χCDW,SDW∘(2kF+q,ωm)χSS,TS∘(q,ωm)≈ℓπvFℓ=1n[ζ]ζ2=E02(πT)2+(vFq)2+(ωm)2
Measuring the imaginary-time dynamics of quantum materials
Published in Philosophical Magazine, 2020
S. Lederer, D. Jost, T. Böhm, R. Hackl, E. Berg, S. A. Kivelson
For any observables and , it follows from linear-response theory and the fluctuation dissipation theorem that there is a relation between the dissipative part of the linear response function, , and the imaginary-time-ordered correlation function, : where (in units in which ) is the Fourier transform of and, for with . The relation between and the imaginary-time correlation function in the Matsubara frequency domain is where . Because is a bosonic correlator, . Thus, if we are interested in the ‘long-time’ behaviour of , we mean we are interested in the longest-possible times, i.e.. The important point to note about Equation (1) is that for , the integral is dominated by the range of frequencies , so the long-imaginary time dynamics can be computed from measurements of the response function in a very limited range of frequencies.