Algorithm for Linear Response Functions at Finite Temperatures: Application to ESR Spectrum of $s=\frac{1}{2}$ Antiferromagnet Cu Benzoate

We introduce an efficient and numerically stable method for calculating linear response functions $\chi (\mathbf{q},\omega )$ of quantum systems at finite temperatures. The method is a combination of numerical solution of the time-dependent Schrödinger equation, random vector representation of trace, and Chebyshev polynomial expansion of Boltzmann operator. This method should be very useful for a wide range of strongly correlated quantum systems at finite temperatures. We present an application to the ESR spectrum of $s=\frac{1}{2}$ antiferromagnet Cu benzoate.

Figure 1

ESR spectra of normal polarization, $I^z(\omega )$, calculated with $\beta =2–16$, $H_x=1.0$, $N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=16$, $\eta =0.01$, and $N_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}=16$. $S$ and $B_1$ stand for spinon excitation and first breather excitation, respectively.

Figure 2

(a) Peak width (FWHM) of spinon ($S$) and first breather ($B_1$) excitations calculated with $\beta =0.5–9.5$, $H_x=1.0$, $N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=16$, $\eta =0.01$, and $N_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}=16$. The horizontal solid line represents the width due to the finite resolution $\eta =0.01$. (b) Peak frequency of spinon ($S$) and first breather ($B_1$) excitations calculated with $\beta =0.5–9.5$, $H_x=1.0$, $N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=16$, $\eta =0.01$, and $N_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}=16$.

Figure 3

The solid curve with filled squares shows $E_g(H_x)$ for the $H_x$ parallel to the $c$ axis calculated from our results with $\beta =16$, $N_{\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}}=16$, $\eta =0.01$, and $N_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}=16$. The solid curve with filled circles shows $E_g(H_x)$ for the $H_x\parallel c\mathrm{\text{axis}}$ calculated from the experimental $B_1$ peak read from Fig. 4 of Ref. 19. The thin line shows a fit $\omega =c{H}_x^{2/3}$ to the experimental result [19].