Finite temperature single-particle Green's function in the Lieb-Liniger model

Abstract

We develop a Monte Carlo sampling algorithm to numerically evaluate the Lehmann representation for the finite temperature single-particle Green's function in the repulsive Lieb-Liniger model. This allows us to determine the spectral function in the full range of temperatures and interactions, as well as in generalized Gibbs ensembles. We test our results against known results for dynamics at infinite interaction strength and static correlators, and find excellent agreement.

Figures (19)

• Figure 1: $M_{{\boldsymbol{\lambda}},{\boldsymbol{\mu}}}/L$ sampled in a single MCMC run, at different couplings $\gamma$, temperatures $\tau$ and sizes $L$. We plot one value every 3000 MCMC steps. Dashed lines: entropy densities $s[\rho]$ associated with the thermal macrostates (Appendix \ref{['appendix:GME']}). The insets in (b) and (f) show that $M_{{\boldsymbol{\lambda}},{\boldsymbol{\mu}}}$ grows linearly with $L=N$ for a simple scalable family of $\ket {\boldsymbol{\mu}}$ states comment_on_figure.
• Figure 2: Average $m[\rho]=\braket{M_{{\boldsymbol{\lambda}},{\boldsymbol{\mu}}}}/L$ as a function of $\tau$ over single MCMC runs at $\gamma = \infty$ and $L=400$, cf. \ref{['fig:stationary']}(e). Blue symbols: entropy density $s[\rho]$ of the thermal macrostate at dimensionless temperature $\tau$, $\gamma = \infty$ and $L \to \infty$. The orange error bars indicate standard deviations $\sigma_m$. Inset: log-log plot for the standard deviation $\sigma_m$ as a function of $L$, for $\tau=20, \gamma=\infty$ (cf.\ref{['fig:stationary']}(e)). The 2-parameter linear fit (dashed line) yields $\alpha \approx 0.75$, $\beta\approx0.5$.
• Figure 3: $C(x,t)$ and $\tilde{C}(k,\omega)$ for $L=N = 200$, $\tau=5$ and $\gamma=\infty$, obtained by running 100 Markov chains in parallel with $\ell_{\rm max}=10^7$ MCMC steps each. The scale $k_F =\pi\, n$ is the Fermi momentum at $\gamma=\infty$. (a) $\text{Re}[C(x,t)]$ as a function of position at times $t = (0, 0.2, 0.4, 0.6)/n^2$ (blue, red, yellow and purple curves). The dots are exact thermodynamic-limit results korepin1993quantum_inversekorepin1990time_dependent, see Appendix \ref{['appendix:fredholm']}. Inset: same data on log-scale. (b) $\tilde{C}(k,\omega)$ as a function of frequency at momenta $k/k_F = (0, 0.16, 0.32, 0.48, 0.8)$ (respectively from lightest to darkest curves) compared to exact results (dots).
• Figure 4: Same $\tau=5, L =200$ MCMC plots as in \ref{['fig:benchmarks']}, but for finite values of $\gamma = 0.5, 4, 10$. (a), (b): $\text{Re}[C(x,t)]$ as a function of position at times $t = (0, 0.2, 0.4, 0.6)/n^2$ (blue, red, yellow and purple curves). (c), (d): $\tilde{C}(k,\omega)$ as a function of frequency at momenta $k/k_F = (0, 0.16, 0.32, 0.48, 0.8)$ (lightest to darkest curves). $\ell_{\rm max}=10^6$ in each of the 100 Markov chain runs. Dashed lines in (b) are exact $L\to\infty$ results in absence of interactions ($\gamma=0$).
• Figure 5: Density plots for $\widetilde{C}(k,\omega)/n$ at several values of $\tau$ and $\gamma$. Data obtained for $L=200$ by running 100 parallel Markov chains with $\ell_{\rm max}=10^6$ steps each. The white ‘x’ markers in (a), (e), (i) denote the continuous free dispersion $\omega=-k^2+h_{\gamma=0}$, where $h_{\gamma=0}$ is the chemical potential that sets $n=1$ at $\gamma=0$. Note the different scales of colours for each subplot.