Signatures of coherent electronic quasiparticles in the paramagnetic Mott insulator

We study the Mott insulating state of the half-filled paramagnetic Hubbard model within dynamical mean field theory using a recently formulated stochastic and nonperturbative quantum impurity solver. The method is based on calculating the impurity self energy as a sample average over a representative distribution of impurity models solved by exact diagonalization. Due to the natural parallelization of the method, millions of poles are readily generated for the self energy which allows us to work with very small pole-broadening $\eta$. Solutions at small and large $\eta$ are qualitatively different; solutions at large $\eta$ show featureless Hubbard bands whereas solutions at small $\eta \le 0.001$ (in units of half bare band width) show a band of electronic quasiparticles with very small quasiparticle weight at the inner edge of the Hubbard bands. The validity of the results are supported by agreement within statistical error ${\sigma }_{\text{QMC}}\sim {10}^{-4}$ on the imaginary frequency axis with calculations using a continuous time quantum Monte Carlo solver. Nevertheless, convergence with respect to finite size of the stochastic exact diagonalization solver remains to be rigourously established.

Figure 1

Converged solutions at $U=4$ $\left(n_s=5\right)$ for large $\left(\eta =0.1\right)$ and small $\left(\eta =0.001\right)$ broadening, showing the upper and lower Hubbard bands as well as the very sharp peak at the inner edge. (Bottom) Corresponding real and imaginary parts of the self energy. (The pole at $\omega =0$ is not shown in $\mathrm{Im}\Sigma$.) The $\eta =0.001$ calculation is an average of $6·{10}^6$ samples with a corresponding self energy consisting of $>{10}^8$ poles.

Figure 2

Closeup of the quasiparticle at $\epsilon =-1$ (not integrated over band energies), together with $\mathrm{Im}\Sigma$ and $\omega -\mathrm{Re}\Sigma$, showing that this is an actual quasiparticle that solves $\omega -\epsilon -\mathrm{Re}\Sigma =0$ close to the bare band edge $\epsilon =-1$ with a large lifetime ${\left(\mathrm{Im}\Sigma \right)}^{-1}$. The tangent at the crossing correspond to the inverse quasiparticle weight $Z^{-1}=1-{\partial }_{\omega }\mathrm{Re}\Sigma \approx 25$.

Figure 3

Spectral function $A(\epsilon ,\omega )$ resolved with respect to band energy $\epsilon$. Showing the broad incoherent Hubbard bands together with the very narrow band of gap edge quasiparticles.

Figure 4

Dependence on $\eta$, broadening of poles, for the imaginary part of the self energy, for $\eta ={10}^{-1},{10}^{-2},{10}^{-3},{10}^{-4}$. To resolve any peak structure requires a small $\eta \le {10}^{-2}$, with convergence for $\eta \le {10}^{-3}$.

Figure 5

DMFT iterations at $\eta =0.001$, showing $\omega -\mathrm{Re}\Sigma \left(\omega \right)$ (lower panel) and corresponding $A\left(\omega \right)$, close to inner band edge. The iterations are started from a (featureless) converged solution at $\eta =0.1$ and show that the quasiparticle peak is not an explicit finite size effect but rather a feature that is iteratively enhanced in the DMFT cycle.

Figure 6

Upper Hubbard band $A\left(\omega \right)$, Dist-ED converged at various $n_s$ (with $\eta =0.001)$ and various $\eta$ (with $n_s=7)$ at $U=4$ and compared to results using D-DMRG (Ref. [15]).

Figure 7

Energy scale ${\epsilon }_{\text{off}}$ of separation between the peak in $A\left(\omega \right)$ and $\mathrm{Im}\Sigma \left(\omega \right)$, and dependence on $n_s$. This energy scale must be well resolved in order to observe the quasiparticles. (The numbers are estimates with accuracy limited primarily by the broadness of the peaks.)

Figure 8

Upper Hubbard band for various $U=2.6,4,8,20,50$, shifted to same center (left). Inner edge (right) showing the gradual disappearance (arbitrary offset) of the peak structure for large $U$ and compared to the exact result to lowest order in $1/U$ (Ref. [12]). At the largest $U$ there are no actual quasiparticles.

Figure 9

Dist-ED for various various $n_s$ (using $\eta =0.001)$ and various $\eta$ (using $n_s=7)$, compared to CT-QMC calculations at inverse temperature $\beta =200$ and $U=4$ on the Matsubara frequencies $i{\omega }_n=\frac{\pi }{\beta }(2n+1)$, with ${\sigma }_{\text{QMC}}$ the calculated statistical error of the QMC data.

Figure 10

Schematic of the sampling of the continuous impurity-bath Greens function $G_0=\frac{a_0}{z}+g_0\left(z\right)$ in terms of a five level $\left(n_s=5\right)$ system $G_0^{\nu }=\frac{a_0}{z}+{\sum }_{i=1}^4\frac{a_i^{\nu }}{z-b_i^{\nu }}$. The location of the satellite poles at finite $\omega$ are generated stochastically based on the distribution $-\mathrm{Im}{g}_0\left(\omega \right)$ with random (normalized) residues.