Thermal pure quantum states of many-particle systems

We generalize the thermal pure quantum (TPQ) formulation of statistical mechanics, in such a way that it is applicable to systems whose Hilbert space is infinite dimensional. Assuming particle systems, we construct the grand-canonical TPQ (gTPQ) state, which is the counterpart of the grand-canonical Gibbs state of the ensemble formulation. A single realization of the gTPQ state gives all quantities of statistical-mechanical interest, with exponentially small probability of error. This formulation not only sheds new light on quantum statistical mechanics but also is useful for practical computations. As an illustration, we apply it to the Hubbard model, on a one-dimensional (1D) chain and on a two-dimensional (2D) triangular lattice. For the 1D chain, our results agree well with the exact solutions over wide ranges of temperature, chemical potential, and the on-site interaction. For the 2D triangular lattice, for which exact results are unknown, we obtain reliable results over a wide range of temperature. We also find that finite-size effects are much smaller in the gTPQ state than in the canonical TPQ state. This also shows that in the ensemble formulation the grand-canonical Gibbs state of a finite-size system simulates an infinite system much better than the canonical Gibbs state.

Figure 1

$n$ versus $T$, obtained by gTPQ states with $L=15$, for $(U,\mu ,M)=(8,2,18)$, $(8,3,20)$, $(1,0.5,18)$, and $(1,2,22)$. Error bars show estimated errors, which can be made smaller by increasing $M$. Exact results for $L=\infty$ are also plotted.

Figure 2

$c$ versus $T$, obtained by gTPQ states with $L=15$, for $(U,\mu ,M)=(8,0,14)$, $(8,3,20)$, $(1,0,12)$, and $(1,2,22)$. Error bars show estimated errors, which can be made smaller by increasing $M$. Exact results for $L=\infty$ are also plotted.

Figure 3

${\varphi }_{\pm }$ versus $i$ for $U=8,\mu =0$, obtained by gTPQ states with $L=14$ and $M=21$, at $T=0.1,0.2,1.0$.

Figure 4

$c$ versus $T$, for the 2D triangular lattice with $U=3,\mu =1$, obtained by gTPQ states with $V=8$ ($M=1024$) and with $V=15$ ($M=10$), and by ND for $V=8$. Error bars show estimated errors, which can be made smaller by increasing $M$.

Figure 5

$c$ versus $T$, obtained by the gTPQ states with $L=8$ ($M=1024$) and $L=14$ ($M=26$), and by the cTPQ states with $L=8$ ($M=1024$) and $L=14$ ($M=22$). Error bars show estimated errors, which can be made smaller by increasing $M$. Exact results for $L=\infty$ are also plotted.