Finite-temperature density matrix embedding theory

We describe a formulation of the density matrix embedding theory at finite temperature. We present a generalization of the ground-state bath orbital construction that embeds a mean-field finite-temperature density matrix up to a given order in the Hamiltonian, or the Hamiltonian up to a given order in the density matrix. We assess the performance of the finite-temperature density matrix embedding on the one-dimensional Hubbard model both at half-filling and away from it, and the two-dimensional Hubbard model at half-filling, comparing to exact data where available, as well as results from finite-temperature density matrix renormalization group, dynamical mean-field theory, and dynamical cluster approximations. The accuracy of finite-temperature density matrix embedding appears comparable to that of the ground-state theory, with, at most, a modest increase in bath size, and competitive with that of cluster dynamical mean-field theory.

Figure 1

Error in energy per site (units of $t)$ of FT-DMET for the 1D Hubbard model at $U=2$ and $U=4$ (two impurity sites and half-filling) with bath orbitals generated via the density matrix $\gamma$ [Eq. (15)] (blue lines) or lattice Hamiltonian $h$ [Eq. (13)] (orange lines) as a function of inverse temperature $\beta$. The numbers in parentheses denote the number of bath orbitals. The grey area denotes the ground-state error with two impurity orbitals.

Figure 2

Percentage error of the FT-DMET (with two impurity sites and two bath orbitals) energy per site vs ED (four sites) on a nonembedded cluster with PBC and APBC boundary conditions for the 1D Hubbard model at various $U$ and $\beta$.

Figure 3

Absolute error of the FT-DMET energy per site of the 1D Hubbard model at half-filling as a function of impurity and bath size. $InBm$ denotes $n$ impurity sites and $m$ bath orbitals. Increasing impurity (blue lines); increasing bath (orange lines). The grey band depicts the ground-state error with two impurity sites and two bath orbitals.

Figure 4

Absolute error of the FT-DMET entropy per site of the 1D Hubbard model at half-filling as a function of the number of bath sites. The right panels show the absolute entropy.

Figure 5

Energy per site (units of $t)$ of a 16-site Hubbard chain with periodic boundary conditions at $U=4$ as a function of the chemical potential $\mu$ at various $\beta$ values. The difference between the DMRG and DMET $\left(I2B6\right)$ energies per site is $0.01–1.4\%$. Solid lines: DMRG energies; dashed lines: DMET energies; pentagons: Bethe Ansatz.

Figure 6

Energy per site versus $U$ (units of $t)$ of the 2D Hubbard model at half-filling with $\mathrm{FT}\text{−}\mathrm{DMET}(2\times 2$ cluster with four and eight bath orbitals), DCA (34, 72, and $2\times 2$ site clusters).

Figure 7

$\text{N}\acute{\mathrm{e}}\text{el}$ transition for the 2D Hubbard model within quantum impurity simulations. (a) Antiferromagnetic moment $m$ as a function of $T$ with various $U$ values (units of $t)$; (b) $\text{N}\acute{\mathrm{e}}\text{el}$ temperature $T_N$ calculated with FT-DMET, single-site DMFT, and DCA. DMFT data are taken from Ref. [32], DCA/NCA data for $U=4$ are taken from Ref. [31], DCA/QMC data for $U=6$ are taken from Ref. [33], and DCA/QMC data for $U=8$ are taken from Ref. [34].

Figure 8

2D Hubbard antiferromagnetic moment (color scale) as a function of $T$ and $U$ (units of $t)$ in FT-DMET $(2\times 2$ impurity, four bath orbitals.)