The kernel polynomial method

Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed-matter physics. In this paper basic properties and recent developments of Chebyshev expansion based algorithms and the kernel polynomial method are reviewed. Characterized by a resource consumption that scales linearly with the problem dimension these methods enjoyed growing popularity over the last decade and found broad application not only in physics. Representative examples from the fields of disordered systems, strongly correlated electrons, electron-phonon interaction, and quantum spin systems are discussed in detail. In addition, an illustration on how the kernel polynomial method is successfully embedded into other numerical techniques, such as cluster perturbation theory or Monte Carlo simulation, is provided.

Figure 1

(Color online) Order $N=64$ expansions of $\delta \left(x\right)$ (left) and a step function (right) based on different kernels. Whereas the truncated series (Dirichlet kernel) strongly oscillate, the Jackson results smoothly converge to the expanded functions. The Lorentz kernel leads to relatively poor convergence at the boundaries $x=\pm 1$, but otherwise yields perfect Lorentz-broadened approximations.

Figure 2

(Color online) Standard (dashed) and typical density of states (solid line), $\rho \left(E\right)$ and ${\rho }_{\mathrm{typ}}\left(E\right)$, respectively, of the 3D Anderson model on a ${50}^3$-site cluster with periodic boundary conditions. For $\rho \left(E\right)$ we calculated $N=2048$ moments with $R=10$ start vectors and $S=240$ realizations of disorder, for ${\rho }_{\mathrm{typ}}\left(E\right)$ these numbers are $N=8192$, $R=32$, and $S=200$. The lower-right panel shows the phase diagram of the model obtained from ${\rho }_{\mathrm{typ}}\left(E\right)∕\rho \left(E\right)\to 0$ (mobility edge).

Figure 3

(Color online) Density of nonzero eigenvalues of the quantum double-exchange model with $S=3∕2$ (dashed line) and running average (dot-dashed), calculated for four electrons on an eight-site ring, compared to the classical result $S\to \infty$ (solid). Expansion parameters: $N=400$ moments and $R=100$ random vectors per $S^z$ sector.

Figure 4

(Color online) Nearest-neighbor $S^z-S^z$ correlations of the $XXZ$ model on a square lattice. Lines represent the KPM results with separation of low-lying eigenstates (bold solid and bold dashed) and without (thin dashed), open symbols denote exact results from a complete diagonalization of a $4\times 4$ system.

Figure 5

(Color online) One-particle spectral function and its integral for the Holstein model (a) on a ten-site ring with one electron, ${\epsilon }_p=g^2{\omega }_0=2.0t$, ${\omega }_0=0.4t$, and (b) on an eight-site ring, band filling $n=0.5$, ${\epsilon }_p=g^2{\omega }_0=1.6t$, ${\omega }_0=0.1t$. For comparison, in (a) the dashed lines represent quantum Monte Carlo data at $\beta t=8$ (46), and stars indicate the position of the polaron band in the infinite system (12). In (b) the curves denote results of dynamical DMRG for the same lattice size and $T=0$ (58).

Figure 6

(Color online) (a) The optical conductivity ${\sigma }^{\mathrm{reg}}\left(\omega \right)$ and its integral $S^{\mathrm{reg}}\left(\omega \right)$ for the Holstein Hubbard model at half filling with different ratios of the Coulomb interaction $U$ to the electron-lattice coupling ${\epsilon }_p=g^2{\omega }_0$, ${\omega }_0=0.1t$, and $g^2=7$. Black dotted lines denote excitations of the pure Hubbard model. (b) The one-particle spectral function at the transition point, i.e., for the same parameters as in the middle panel of (a). The system size is $L=8$.

Figure 7

Spin structure factor at $T=0$ calculated for the model (149) which aims at describing the magnetic compound ${\left(\mathrm{V}\mathrm{O}\right)}_2{\mathrm{P}}_2{\mathrm{O}}_7$. For more details see 108.

Figure 8

(Color online) The matrix element density $j(x,y)$ for the 3D Anderson model with disorder $W∕t=2$ and 12.

Figure 9

(Color online) Optical conductivity of the 3D Anderson model at disorder $W=12$ and for different chemical potentials $\mu$ and temperatures $\beta =1∕T$.

Figure 10

(Color online) Left: Schematic setup for the calculation of finite-temperature dynamical correlations for interacting quantum systems, which requires a separation into parts handled by exact diagonalization (ED), 1D Chebyshev expansion, and 2D Chebyshev expansion. Right: The lowest eigenvalues of the Holstein model on a six-site chain for different electron-phonon coupling ${\epsilon }_p$. The shaded region marks the lowest polaron band, which was handled separately when calculating the spectra in Fig. 11.

Figure 11

(Color online) Finite-temperature optical conductivity of a single electron coupled to the lattice via a Holstein-type interaction. Different colors illustrate how, in particular, the low-temperature spectra benefit from a separation of $C=0$, 1, or 6 low-energy states (91). The phonon frequency is ${\omega }_0∕t=0.4$.

Figure 12

(Color online) Magnetization as a function of temperature for the classical double-exchange model at doping $n=0.5$. We compare data obtained from the effective model $H_{\mathrm{eff}}$ (see text), from a hybrid Monte Carlo approach (3, the truncated polynomial expansion method (75, 76), and from a KPM based cluster Monte Carlo technique (109). $L$ denotes the size of the underlying three-dimensional cluster, i.e., $D=L^3$ is the dimension of the fermionic problem.

Figure 13

Spectral function for noninteracting tight-binding electrons. Based on the Lorentz kernel CPT exactly reproduces the infinite system result (left). The Jackson kernel does not have the correct analytical properties, therefore CPT cannot close the finite size gap at $k=\pi ∕2$ (right).

Figure 14

(Color online) Spectral function of the 1D Hubbard model for half-filling and $U=4t$. (a) CPT result with cluster size $L=16$ and expansion order $N=2048$. For similar data based on Lanczos recursion see 92. (b) Within the exact Bethe ansatz solution each electron separates into the sum of independent spinon (dashed) and holon excitations. The dots mark the energies of a 64-site chain. (c) CPT data compared to selected DDMRG results for a system with $L=128$ sites, open boundary conditions and a broadening of $ϵ=0.0625t$. Note that in DDMRG the momenta are approximate.

Figure 15

Spectral function $A^{+}(k,\omega )$ of a single electron in the Holstein model [corresponding to $N_{\mathrm{e}}=0$ in Eq. (145)]. For weak electron-phonon coupling the original band is still very pronounced (left), for intermediate-to-strong coupling many narrow polaron bands develop (right). The cluster size is $L=16$ (left) or $L=6$ (right) and the expansion order $N=2048$. See 45 for similar data based on Lanczos recursion.

Figure 16

(Color online) Comparison of a KPM and a MEM approximation to a spectrum consisting of five isolated $\delta$ peaks, and to a step function. The expansion order is $N=512$. Clearly, for the $\delta$ peaks MEM yields a higher resolution, but for the step function the Gibbs oscillations return.

Figure 17

(Color online) The optical conductivity of the Anderson model, Eq. (111), calculated with KPM and a projection method (49). The disorder is $W\approx 15$; temperature and chemical potential read $T=0$ and $\mu =0$.