Benchmarkings for a semiclassical impurity solver for dynamical-mean-field theory: Self-energies and magnetic transitions of the single-orbital Hubbard model

An investigation is presented of the utility of semiclassical approximations for solving the quantum-impurity problems arising in the dynamical-mean-field approach to correlated-electron models. The method is based on performing an exact numerical integral over the zero-Matsubara-frequency component of the spin part of a continuous Hubbard-Stratonovich field, along with a spin-field-dependent steepest descent treatment of the charge part. We test this method by applying it to one- or two-site approximations to the single-band Hubbard model with different band structures, and comparing the results to quantum Monte Carlo and simplified exact diagonalization calculations. The resulting electron self-energies, densities of states, and magnetic transition temperatures show reasonable agreement with the quantum Monte Carlo simulation over wide parameter ranges, suggesting that the semiclassical method is useful for obtaining a reasonable picture of the physics in situations where other techniques are too expensive.

Figure 1

Example of the effective potential for $\phi$, $V\left(\phi \right)$ given in Eq. (16), in the square-lattice half-filled Hubbard model with $U∕t=6$. Solid line, $T∕t=0.4$ (about 14% above $T_N\sim 0.35t$); broken line, $T∕t=0.3$ (about 14% below $T_N$). At both temperatures, $V$ is far from a simple parabola, and regions far from the local minima have appreciable occupation probability.

Figure 2

Comparison of the electron self-energy computed by evaluating the static average and integrating over two static fields $\phi$ and $x$ with the action Eq. (14) (2-field) and the semiclassical approximation, which is defined in this section, for the infinite-dimensional fcc lattice Eq. (3). Energy is scaled by the variance of the free DOS $(v=1)$. Squares (circles) are real (imaginary) parts of the self-energy, and filled and open symbols are obtained by the two-field approximation and the semiclassical approximation, respectively.

Figure 3

Comparison of the many-body density of states computed for the paramagnetic phase of the infinite-dimensional fcc lattice [Eq. (3)] by the “full” semiclassical approximation (solid line) to that computed using an additional approximation in which $\xi$ is assumed to be independent of $\phi$ (broken line). Energy is scaled by the variance of the free DOS $(v=1)$. The three-peak structure seen in the broken line is unphysical; the solid line is in good agreement with QMC results (not shown). Note that $T_C\sim 0.073$ for these parameters; we have presented data at $T=0.05\simeq 3T_C∕4$ (chosen to reveal the three-peak structure clearly), and have artificially suppressed ferromagnetism.

Figure 4

Example of the effective potential $V$ (upper panel) and charge field $\xi$ (lower panel) as functions of $\phi$ in the metallic Hubbard model on an infinite-dimensional fcc lattice [the free DOS is given in Eq. (3)] with $U=4$ and $n=0.5$. Energy is scaled by the variance of free DOS $(v=1)$. Solid lines, $T=0.08$ (about 14% above $T_C\sim 0.073$); broken lines, $T=0.06$ (about 18% below $T_C$). A constant term $(=-U)$ has been subtracted from $\xi$. At both temperatures, the potential is seen to differ markedly from a simple parabola. The $\phi$ dependence of $\xi$ indicates strong coupling between the two fields.

Figure 5

Ratio of the classical Gaussian approximation [Eq. (24)] to the full Gaussian [Eq. (23)] mean square fluctuation of the spin Hubbard-Stratonovich field $R_{\chi }={⟨{\phi }^2⟩}_{class}∕⟨{\phi }^2⟩$ for infinite-dimensional fcc lattice at densities $n=0.5$ (upper panel) and 0.75 (lower panel). Energy is scaled by the variance of the free DOS $(v=1)$. The curves cross because of the temperature dependence of the critical interaction strength (evident from the $U$ at which the ratio approaches unity).

Figure 6

Frequency dependence of full Gaussian self-energy [Eq. (26)] (filled circles) and classical Gaussian approximation [Eq. (27)] (open circles), for infinite-dimensional fcc lattice with $n=0.75,T=0.04$, and $U$ values indicated. Energy is scaled by the variance of the free DOS $(v=1)$.

Figure 7

Frequency dependence of imaginary part of full Gaussian self-energy [Eq. (26)] (filled circles) and classical Gaussian approximation [Eq. (27)] (open circles) for $n=0.75,U=0.5$, and $T$ values indicated. Energy is scaled by the variance of the free DOS $(v=1)$.

Figure 8

Self-energies for a square-lattice half-filled Hubbard model computed from the semiclassical approximation (filled circles) and QMC calculations (open circles) for interactions and temperatures indicated.

Figure 9

Density of states of square-lattice half-filled Hubbard model computed using semiclassical (heavy lines) and QMC methods (light lines) for paramagnetic [(a) and (c)] and antiferromagnetic [(b) and (d), $T\sim 0.8T_N$] phases at interactions and temperatures indicated. In the antiferromagnetic phase, the majority spin density of states is shown by a solid line, the minority spin by a broken line.

Figure 10

Néel temperature of square-lattice half-filled Hubbard model as functions of $U$ computed from semiclassical approximation (filled symbols) and QMC simulation (open symbols), HF (light solid line), and large-$U$ limit $(T_N=4t^2∕U)$ (light broken line).

Figure 11

Comparison of the density of states of infinite-dimensional FCC Hubbard model with $U=4$ and temperatures chosen so that $⟨m⟩=0.4$ in each case. Energy is scaled by the variance of free DOS $(v=1)$. Heavy lines: results of the semiclassical approximation at $n=0.6$ and $T=0.07$ ($\sim 13\%$ below $T_C\sim 0.08$). Light lines: results of QMC at $n=0.58$ and $T=0.04$ ($\sim 20\%$ below $T_C\sim 0.05$) taken from Ref. 43. Solid and broken lines are for the majority and minority spin.

Figure 12

Ferromagnetic Curie temperature of single-band Hubbard model on an infinite-dimensional fcc lattice as a function of electron density for $U=2$ (squares) and 4 (circles). Energy is scaled by the variance of the free DOS $(v=1)$. Filled symbols, semiclassical approximation; open symbols, QMC simulation (Ref. 43). Curie temperatures $(\times {10}^{-1})$ computed by HF approximation to the same model with $U=2\left(4\right)$ are shown as a light solid (broken) line. Two-site DMFT results for the transition temperatures are not shown, but representative two-site results for $n=0.5$ are $T_C\sim 0.17\left(0.26\right)$ for $U=2\left(4\right)$. Phase boundary at $T=0$ for $U=2\left(4\right)$ computed by the two-site DMFT is shown as a filled (open) triangle.

Figure 13

Upper panel: Phase diagram of single-band Hubbard model on a three-dimensional fcc lattice as a function of electron density and temperature for interaction $U=6$. Energy is scaled by the variance of the free DOS $(v=1)$. Squares and circles are the Curie temperature and Néel temperature for the layer-type antiferromagnetic state, respectively. Filled symbols, results of the semiclassical approximation; open symbols, QMC results taken from Ref. 43. Curie temperature $(\times {10}^{-1})$ computed by HF approximation to the same model is shown as a light solid line. Lower panel: Expansion of the phase diagram, showing the region near $n=1$ enclosed by a broken line in the upper panel. The semiclassical results for $T_C$ and $T_N$ indicate the instability of ferromagnetic and antiferromagnetic states, respectively, toward the paramagnetic state.

Figure 14

Nearest-neighbor spin correlations $-⟨{\sigma }_{1z}{\sigma }_{2z}⟩$ of a square-lattice Hubbard model $(U∕t=20)$ as functions of $T$ computed using two-site DCA (open circles) and the FI method (open squares), compared to QMC simulation (filled symbols) and high-temperature series expansion (light line).

Figure 15

Distribution of the spin fields ${\phi }_1$ and ${\phi }_2,P({\phi }_1,{\phi }_2)$, computed by the two-impurity DCA for the Hubbard modelon a square lattice. $P({\phi }_1,{\phi }_2)$ is defined by $P({\phi }_1,{\phi }_2)=\exp \{-\beta V({\phi }_1,{\phi }_2)\}∕Z_{approx}$. Parameters are $U∕t=20$ and $T∕t=0.5$. Larger peaks at $({\phi }_1,{\phi }_2)\simeq (\pm 20t,\mp 20t)$ than at $({\phi }_1,{\phi }_2)\simeq (\pm 20t,\pm 20t)$ indicate antiferromagnetic correlation.