Assigning temperatures to eigenstates

In the study of thermalization in finite isolated quantum systems, an inescapable issue is the definition of temperature. We examine and compare different possible ways of assigning temperatures to energies or equivalently to eigenstates in such systems. A commonly used assignment of temperature in the context of thermalization is based on the canonical energy-temperature relationship, which depends only on energy eigenvalues and not on the structure of eigenstates. For eigenstates, we consider defining temperature by minimizing the distance between (full or reduced) eigenstate density matrices and canonical density matrices. We show that for full eigenstates, the minimizing temperature depends on the distance measure chosen and matches the canonical temperature for the trace distance; however, the two matrices are not close. With reduced density matrices, the minimizing temperature has fluctuations that scale with subsystem and system size but appears to be independent of distance measure. In particular limits, the two matrices become equivalent while the temperature tends to the canonical temperature.

Figure 1

Eigenstate temperature results for staggered field XXZ-chain: $h_x=h_z=0.5$, $\mathrm{\Delta }=0.95$, $L=10$. (A) ${\beta }_E$ against energy, for 20 eigenstates which are equally spaced in energy across the spectrum, with curves showing ${\beta }_C/p$. (Highest and lowest state not visible.) (B) $d_1(\rho ,{\rho }_C)$ vs $\beta$ curve for ground state ($E_1$), midspectrum state ($E_3$), and $E_2$ in between the two. (C) The minimum of $d_p(\rho ,{\rho }_C)$ plotted against eigenenergy, for the same eigenstates used in panel (A).

Figure 2

Finite window eigenstate temperature results for spin chains with $L=10$, namely: (A)–(C) Chaotic Ising model with $h_z=0.5$, $h_x=0.75$ and $\mathrm{\Delta }E\sim 0.059$. (D)–(F) Staggered field XXZ-chain with $\mathrm{\Delta }=0.95$, $J=1$, $h_x=h_z=0.5$, and $\mathrm{\Delta }E\sim 0.0757$. (A), (D) ${\beta }_{\text{MC}}$ against energy, for 20 energy windows which are equally spaced in energy across the spectrum, with curves showing ${\beta }_C/p$. (Highest and lowest state not visible.) (B), (E) $d_1({\rho }_{\text{MC}},{\rho }_C)$ vs $\beta$ curve for energy windows: $E_1$ near the ground state, $E_2$ in the middle of spectrum, and $E_2$ in between the two. (C), (F) The minimum of $d_p({\rho }_{\text{MC}},{\rho }_C)$ plotted against energy, for the same energy windows used in panels (A), (D).

Figure 3

Subsystem temperature results for the chaotic Ising model with $h_z=0.5$ and $h_x=0.75$. (A) $\beta$ that minimizes $d_1({\rho }^A,{\rho }_C^A)$ (${\beta }_S$) vs energy, plotted alongside the canonical ${\beta }_C$ curve, for the given $L$ and $L_A$. (B)–(D) $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$ plotted vs energy, each row illustrating a different scaling of system and subsystem size.

Figure 4

Subsystem temperature results for square lattice model with the addition of staggered $S^x$ fields with $h_x=0.5$. The geometry of each system is illustrated above each plot, in which, red and black sites correspond to the subsystems $A$ and $B$, respectively. The results in each plot are of a particular realization: (A) $\beta$ that minimizes $d_1({\rho }^A,{\rho }_C^A)$ (${\beta }_S$) vs energy, plotted alongside the canonical ${\beta }_C$ curve, for the given $L$ and $L_A$. (B), (C) $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$ plotted vs energy, each row illustrating a different scaling of system and subsystem size.

Figure 5

Subsystem temperature scaling for (A)–(C) disordered field XXZ-chain with $W=0.25$, and $L_A=2$, over many disorder realizations. (D)–(F) chaotic Ising model with $h_x=0.75$, $h_z=0.5$, and $L_A=2$. For both models, statistics are taken from the central 20% of the spectrum. (A), (D) Mean of $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$ and its standard deviation vs $D_B$ for $p=1$. (B), (E) Width of ${\beta }_S$ vs $D_B$ for $p=1,2$. (C), (F) RMS-distance from the linear fit of ${\beta }_S$, to ${\beta }_C$ curve vs $D_B$, for $p=1,2$.

Figure 6

Finite window eigenstate temperature ${\beta }_{\mathrm{\Delta }E}$ calculated using Bures distance $d_B({\rho }_{\text{MC}},{\rho }_C)$, for two models: (A) Random, real, and symmetric matrix; (B) chaotic Ising model with $h_x=0.5$, $h_z=0.75$. In both cases, $L=9$ ($D=2^9$) and 20 energy windows are uniformly chosen from the spectrum of the given Hamiltonian.

Figure 7

Subsystem temperature results with ${\rho }^A=exp(-\beta H_A)$, for staggered field model with $h_x=h_z=0.5$, $J=1$ and $\mathrm{\Delta }=0.95$. (A) $\beta$ minimizing $d_1({\rho }^A,{\rho }_C^A)$ (${\beta }_S$) vs energy, plotted alongside the canonical ${\beta }_C$ curve, for the given system and subsystem size. (B)–(D) $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$ plotted vs energy, each row illustrating a different scaling of system and subsystem size.

Figure 8

Subsystem temperature results with ${\rho }^A=exp(-\beta H_A)$, for staggered field XXZ-chain with $J=1$, $\mathrm{\Delta }=0.95$, $h_x=h_z=0.5$ and $L_A=2$. Statistics from the central 20% of the spectrum. (A) Mean of $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$ and its standard deviation vs $D_B$ for $p=1$. (B) Width of ${\beta }_S$ vs $D_B$ for $p=1,2$. (C) RMS-distance from the linear fit of ${\beta }_S$, to ${\beta }_C$ curve vs $D_B$, for $p=1,2$.

Figure 9

Subsystem temperature results for staggered field model with $L=12$, $J=1$, $\mathrm{\Delta }=0.95$, and $h_z=h_x=h$. Top: The RMS-distance between ${\beta }_S$ and ${\beta }_C$ (Deviation) plotted vs shared field strength $h$. The RMS-distance is calculated for eigenstates in the central 20% of the spectrum. Bottom: Average restricted gap ratio value plotted vs shared field strength $h$.

Figure 10

Eigenstate temperature results for random symmetric matrix with $D=2^{10}$. Left: ${\beta }_E$ against energy, for 20 eigenstates which are equally spaced in energy across the spectrum, with curves showing ${\beta }_C/p$. (Highest and lowest state not visible.) Mid: $d_1(\rho ,{\rho }_C)$ vs $\beta$ curve for ground state ($E_1$), midspectrum state ($E_3$), and $E_2$ in between the two. Right: The minimum of $d_p(\rho ,{\rho }_C)$ plotted against eigenenergy, for the same eigenstates used in panel (A).

Figure 11

Subsystem temperature results for a random symmetric matrix with $D=2^{13}$. Left: $\beta$ minimizing $d_1({\rho }^A,{\rho }_C^A)$ (${\beta }_S$) vs energy, plotted alongside the canonical ${\beta }_C$ curve. Right: $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$ plotted vs energy.

Figure 12

Subsystem temperature results for staggered field model with $h_x=h_z=0.5$, $J=1$ and $\mathrm{\Delta }=0.95$. $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$ plotted vs energy for the captioned $L$ and $L_A$.

Figure 13

Subsystem temperature scaling with subsystem size results. Staggered field model with $h_x=h_z=0.5$, $J=1$, $\mathrm{\Delta }=0.95$, and $L=14$. (A) Mean value of $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$, (B) standard deviation of $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$, (C) width of ${\beta }_S$ data, (D) RMS-distance between ${\beta }_C$ and linear fit to ${\beta }_S$ vs $L_A$. All quantities are calculated in the central 20% of the spectrum.

Figure 14

Subsystem temperature results for chaotic Ising model with $L=10$ and $L_A=7$, $h_x=0.75$, and $h_z=0.5$. Left: ${\beta }_S$ vs $E$ with canonical ${\beta }_C$ curve shown. Right: $min\left[d_1({\rho }^A,{\rho }_C^A)\right]$ vs energy.

Figure 15

Subsystem temperature results using $p=2$ distance, for chaotic Ising model with $h_x=0.75$, $h_z=0.5$ and $L_A=2$. Statistics from the central 20% of the spectrum. (A) Mean of $min\left[d_2({\rho }^A,{\rho }_C^A)\right]$ and its standard deviation vs $D_B$. (B) Width of ${\beta }_S$ vs $D_B$. (C) RMS-distance from the linear fit of ${\beta }_S$, to ${\beta }_C$ curve vs $D_B$.

Figure 16

Distance $d_1$ at canonical temperature ${\beta }_C$, averaged over the central 20% of the spectrum, vs inverse system size $L$. We also show the mean standard deviation of the minima. With $L_A=2$ for (A) staggered field XXZ-chain, with $h_z=h_x=0.5$, and (B) chaotic Ising model with $h_z=0.5$ and $h_x=0.75$.