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Register a new userAccuracy of the microcanonical Lanczos method to compute real-frequency dynamical spectral functions of quantum models at finite temperatures
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We examine the accuracy of the microcanonical Lanczos method (MCLM) developed by Long, {it et al.} [Phys. Rev. B {bf 68}, 235106 (2003)] to compute dynamical spectral functions of interacting quantum models at finite temperatures. The MCLM is based on the microcanonical ensemble, which becomes exact in the thermodynamic limit. To apply the microcanonical ensemble at a fixed temperature, one has to find energy eigenstates with the energy eigenvalue corresponding to the internal energy in the canonical ensemble. Here, we propose to use thermal pure quantum state methods by Sugiura and Shimizu [Phys. Rev. Lett. {bf 111}, 010401 (2013)] to obtain the internal energy. After obtaining the energy eigenstates using the Lanczos diagonalization method, dynamical quantities are computed via a continued fraction expansion, a standard procedure for Lanczos-based numerical methods. Using one-dimensional antiferromagnetic Heisenberg chains with $S=1/2$, we demonstrate that the proposed procedure is reasonably accurate even for relatively small systems.
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We study trace estimators for equilibrium thermodynamic observables that rely on the idea of typicality and derivatives thereof such as the finite-temperature Lanczos method (FTLM). As numerical examples quantum spin systems are studied. Our initial aim was to identify pathological examples or circumstances, such as strong frustration or unusual densities of states, where these methods could fail. Instead we failed with the attempt. All investigated systems allow such approximations, only at temperatures of the order of the lowest energy gap the convergence is somewhat slower in the number of random vectors over which observables are averaged.
Building on previous developments, we show that the Diagrammatic Monte Carlo technique allows to compute finite temperature response functions directly on the real-frequency axis within any field-theoretical formulation of the interacting fermion problem. There are no limitations on the type and nature of the systems action or whether partial summation and self-consistent treatment of certain diagram classes are used. In particular, by eliminating the need for numerical analytic continuation from a Matsubara representation, our scheme allows to study spectral densities of arbitrary complexity with controlled accuracy in models with frequency-dependent effective interactions. For illustrative purposes we consider the problem of the plasmon line-width in a homogeneous electron gas (jellium).
We present a combination method based on orignal version of Davidson algorithm for extracting few of the lowest eigenvalues and eigenvectors of a sparse symmetric Hamiltonian matrix and the simplest version of Lanczos technique for obtaining a tridiagonal representation of the Hamiltonian to compute the continued fraction expansion of the Greens function at a very low temperature. We compare the Davidson$+$Lanczos method with the full diagonalization on a one-band Hubbard model on a Bethe lattice of infinite-coordination using dynamical mean field theory.
Using the numerical renormalization group (NRG), we analyze the temperature dependence of the spectral function of a magnetic impurity described by the single-impurity Anderson model coupled to superconducting contacts. With increasing temperature the spectral weight is gradually transferred from the $delta$-peak (Shiba/Yu-Shiba-Rusinov/Andreev bound state) to the continuous sub-gap background, but both spectral features coexist at any finite temperature, i.e., the $delta$-peak itself persists to temperatures of order $Delta$. The continuous background is due to inelastic exchange scattering of Bogoliubov quasiparticles off the impurity and it is thermally activated since it requires a finite thermal population of quasiparticles above the gap. In the singlet regime for strong hybridization (charge-fluctuation regime) we detect the presence of an additional sub-gap structure just below the gap edges with thermally activated behavior, but with an activation energy equal to the Shiba state excitation energy. These peaks can be tentatively interpreted as Shiba bound states arising from the scattering of quasiparticles off the thermally excited sub-gap doublet Shiba states, i.e., as high-order Shiba states.
It is virtually impossible to evaluate the magnetic properties of large anisotropic magnetic molecules numerically exactly due to the huge Hilbert space dimensions as well as due to the absence of symmetries. Here we propose to advance the Finite-Temperature Lanczos Method (FTLM) to the case of single-ion anisotropy. The main obstacle, namely the loss of the spin rotational symmetry about the field axis, can be overcome by choosing symmetry related random vectors for the approximate evaluation of the partition function. We demonstrate that now thermodynamic functions for anisotropic magnetic molecules of unprecedented size can be evaluated.
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