Entropy and temperature in finite isolated quantum systems

We investigate how the temperature calculated from the microcanonical entropy compares with the canonical temperature for finite isolated quantum systems. We concentrate on systems with sizes that make them accessible to numerical exact diagonalization. We thus characterize the deviations from ensemble equivalence at finite sizes. We describe multiple ways to compute the microcanonical entropy and present numerical results for the entropy and temperature computed in these various ways. We show that using an energy window whose width has a particular energy dependence results in a temperature with minimal deviations from the canonical temperature.

Figure 1

Illustration of two ways of choosing the energy window $\mathrm{\Delta }E$ used to define the microcanonical entropy. The top figure shows the obvious choice: the width $\mathrm{\Delta }E$ is the same at all energies. The bottom schematic illustrates an energy-dependent width: $\mathrm{\Delta }E\left(E\right)\propto \sqrt{T_c^2{C}_c}$, where the temperature $T_c$ is the canonical temperature corresponding to energy $E$ and $C_c$ is the heat capacity at $T=T_c$. The window is shown at three energy values $E_a, E_b$, and $E_c$, which are in the regions of low temperature, infinite temperature, and low negative temperature. Each vertical tick marks an eigenvalue.

Figure 2

(a) Microcanonical entropy $S\left(E\right)$ calculated with constant (energy-independent) $\mathrm{\Delta }E$. Evaluated only at the eigenenergies $E=E_n$. (b) Resultant inverse temperature $\beta \left(E\right)$ obtained as derivative of sixth-order polynomial fitted to $S\left(E\right)$ data. 25 eigenstates from both ends of the spectrum [shown in gray in (a)] were excluded from the fit. (a), (b) $5\times 4$ square XXZ lattice, Eq. (13), with $L=5\times 4, N=4$. (c) Root-mean-square (RMS) distance between $\beta \left(E\right)$ and the canonical temperature, versus Hilbert space dimension $D=2^L$, with $\mathrm{\Delta }E=0.5$. Spin chain with $L$ spins, Eq. (12), $J=1, \mathrm{\Delta }=0.95, h_z=h_x=0.5$. Units used in this and all subsequent figures are explained in Sec. 2c.

Figure 3

(a) Specific heat $C=\partial E/\partial T$. (b) $\mathrm{\Delta }E\left(E\right)={\alpha }^{-1}\sqrt{2\pi k_B{T}_c^2{C}_c}$, for $\alpha =5$, numerically computed. Dashed curve: 12th order polynomial fit, used to avoid the spurious divergence. Data for $5\times 4$ square XXZ lattice, Eq. (13), with $N=4$.

Figure 4

Using an energy-dependent window, Eq. (15). Left panels (a), (b) and right panels (c), (d) show data for two different systems. (a), (c) Microcanonical entropy calculated with $\mathrm{\Delta }E\left(E\right)={\alpha }^{-1}\sqrt{2\pi k_B{T}_c^2{C}_c}$ along with sixth-order polynomial fit. Fit excludes gray points. (b), (d) Resultant temperature $\tilde{\beta }\left(E\right)$, with two different values of $\alpha$ in each case, compared with canonical temperature ${\beta }_C\left(E\right)$. (a), (b) Data for $5\times 4$ square XXZ lattice, Eq. (13), with $N=4$. (c), (d) Fully connected lattice, Eq. (14), with $L=16, N=5$. (e) RMS distance between $\tilde{\beta }\left(E\right)$ and ${\beta }_C\left(E\right)$ plotted against $\alpha$ for two square lattices, Eq. (13), a $5\times 4$ lattice with $N=4$, and a $7\times 4$ lattice with $N=5$. Inset: logarithmic scale.

Figure 5

Top: Cumulative density of states $\mathrm{\Omega }\left(E\right)$ (normalized by Hilbert space dimension), fitted by a polynomial $f\left(E\right)$ of degree 10. Inset: zoom. Bottom: ${\mathrm{\Omega }}^{\prime }\left(E\right)/D$ from derivative of fitted function, compared with histogram (distribution) of eigenvalues, $P\left(E_n\right)$. Data for $7\times 4$ square XXZ lattice, Eq. (13), with $N=5$.

Figure 6

Entropy and temperature using a continuous approximation to the cumulative density of states. (a), (c) $S_{\mathrm{\Omega }}=ln{\mathrm{\Omega }}^{\prime }\left(E\right)$. (b), (d) Resultant inverse temperature ${\beta }_{\mathrm{\Omega }}={\partial }_E{S}_{\mathrm{\Omega }}$ compared with the canonical temperature ${\beta }_C$. (a), (b) Staggered field XXZ chain, Eq. (12), with $L=12, J=1, \mathrm{\Delta }=0.95, h_z=h_x=0.5$. (c). (d) $5\times 4$ square XXZ lattice, Eq. (13), with $N=4$. (e) RMS distance between ${\beta }_{\mathrm{\Omega }}\left(E\right)$ and ${\beta }_C\left(E\right)$, versus $D\left(2^L\right)$, for the spin chain, Eq. (12), with the same parameters as in (a), (b) and variable $L$.

Figure 7

Entropy and temperature using a continuous approximation to the cumulative density of states and an energy-dependent $\mathrm{\Delta }E\left(E\right)=\sqrt{T_c^2{C}_c}$. (a), (c) $S_{\mathrm{\Omega }}\left(E\right)=ln({\mathrm{\Omega }}^{\prime }\left(E\right)\mathrm{\Delta }E\left(E\right))$. (b), (d) Resultant inverse temperature ${\beta }_{\mathrm{\Omega }}\left(E\right)={\partial }_E{S}_{\mathrm{\Omega }}\left(E\right)$ compared with the canonical temperature ${\beta }_C$. (a), (b) Staggered field XXZ chain, Eq. (12), with $L=12, J=1, \mathrm{\Delta }=0.95, h_z=h_x=0.5$. (c), (d) $5\times 4$ square XXZ lattice, Eq. (13), with $N=4$.