We consider a model Hamiltonian describing a rotor as fixed and weakly interacting with a bath of oscillators. From the basic principles of statistical mechanics, we derive the corresponding master equation for the rotor density matrix operator. Two relevant limit regimes, imposed by the weak-coupling assumptions, are then examined in detail. The first regime, corresponding to the classical Brownian limit, leads to the same electrical susceptibility formulae as deduced from the well known Fokker-Planck-Kramers equation for the rotational Brownian motion. The second regime appears as the Van Hove limit for the master equation in the interaction picture. Based on the application of a mathematical theorem by E.B. Davies (1976), this limit provides an elegant Van Vleck-Weisskopf lineform for the electrical susceptibility, explicitly expressed for the model considered here.We consider a model Hamiltonian describing a rotor as fixed and weakly interacting with a bath of oscillators. From the basic principles of statistical mechanics, we derive the corresponding master equation for the rotor density matrix operator. Two relevant limit regimes, imposed by the weak-coupling assumptions, are then examined in detail. The first regime, corresponding to the classical Brownian limit, leads to the same electrical susceptibility formulae as deduced from the well known Fokker-Planck-Kramers equation for the rotational Brownian motion. The second regime appears as the Van Hove limit for the master equation in the interaction picture. Based on the application of a mathematical theorem by E.B. Davies (1976), this limit provides an elegant Van Vleck-Weisskopf lineform for the electrical susceptibility, explicitly expressed for the model considered here.

Journal of Physics A Mathematical and GeneralNov 21, 19956 citations

On the weak coupling limit problem

Luigi Accardi, A. Frigerio, Yun Gang Lu

Lecture notes in mathematicsJan 1, 198915 citations

Superconductivity near two-dimensional Van Hove singularities: A determinant quantum Monte Carlo study

Gustav Romare, Daniel J. Shaffer, Alex Levchenko +2 more

The superconducting transition temperature $T_c$ of the two-dimensional attractive Hubbard model is computed in the vicinity of both ordinary (logarithmic) and higher-order (power-law) Van Hove singularities using determinant quantum Monte Carlo simulations. For interaction strengths $|U| \lesssim W/3$ , where $W$ is the electronic bandwidth, $T_c$ is enhanced in the neighborhood of the Van Hove point, albeit more weakly than expected from weak-coupling BCS theory. Enhancing the Van Hove singularity from logarithmic to power-law yields only a minor additional enhancement of $T_c$ . For $|U| \gtrsim W/3$ , the maximum $T_c$ shifts away from the Van Hove point and instead occurs at a density unrelated to any features in the non-interacting density of states, consistent with a strong-coupling interpretation. We find that the maximal $T_c$ in the model is achieved at intermediate $U$ and at a density away from the Van Hove point.The superconducting transition temperature $T_c$ of the two-dimensional attractive Hubbard model is computed in the vicinity of both ordinary (logarithmic) and higher-order (power-law) Van Hove singularities using determinant quantum Monte Carlo simulations. For interaction strengths $|U| \lesssim W/3$, where $W$ is the electronic bandwidth, $T_c$ is enhanced in the neighborhood of the Van Hove point, albeit more weakly than expected from weak-coupling BCS theory. Enhancing the Van Hove singularity from logarithmic to power-law yields only a minor additional enhancement of $T_c$. For $|U| \gtrsim W/3$, the maximum $T_c$ shifts away from the Van Hove point and instead occurs at a density unrelated to any features in the non-interacting density of states, consistent with a strong-coupling interpretation. We find that the maximal $T_c$ in the model is achieved at intermediate $U$ and at a density away from the Van Hove point.

arXiv:2604.13161Physical review. B./Physical review. BOct 1, 20260 citations

The superconducting transition temperature

T_c

of the two-dimensional attractive Hubbard model is computed in the vicinity of both ordinary (logarithmic) and higher-order (power-law) Van Hove singularities using determinant quantum Monte Carlo simulations. For interaction strengths

|U| \lesssim W/3

, where

W

is the electronic bandwidth,

T_c

is enhanced in the neighborhood of the Van Hove point, albeit more weakly than expected from weak-coupling BCS theory. Enhancing the Van Hove singularity from logarithmic to power-law yields only a minor additional enhancement of

T_c

. For

|U| \gtrsim W/3

, the maximum

T_c

shifts away from the Van Hove point and instead occurs at a density unrelated to any features in the non-interacting density of states, consistent with a strong-coupling interpretation. We find that the maximal

T_c

in the model is achieved at intermediate

U

and at a density away from the Van Hove point.