Spectral functions with the density matrix renormalization group: Krylov-space approach for correction vectors

Frequency-dependent correlations, such as the spectral function and the dynamical structure factor, help illustrate condensed matter experiments. Within the density matrix renormalization group (DMRG) framework, an accurate method for calculating spectral functions directly in frequency is the correction-vector method. The correction vector can be computed by solving a linear equation or by minimizing a functional. This paper proposes an alternative to calculate the correction vector: to use the Krylov-space approach. This paper then studies the accuracy and performance of the Krylov-space approach, when applied to the Heisenberg, the t-J, and the Hubbard models. The cases studied indicate that the Krylov-space approach can be more accurate and efficient than the conjugate gradient, and that the error of the former integrates best when a Krylov-space decomposition is also used for ground state DMRG.

Figure 1

$S(k,\omega )$ for an antiferromagnetic Heisenberg chain of $L=64$ sites calculated with the Krylov-space approach. We have used $\eta =0.075, m_{\text{min}}=64$, and $m=1000$ DMRG states, with a truncation error kept at ${10}^{-8}$. Solid (red) lines indicate the lower boundary of the Bethe ansatz spin excitation continuum, ${\omega }^l\left(k\right)$; see Eq. (15). The dash-dotted (green) line indicates the upper boundary of the spin excitation continuum, ${\omega }^u\left(k\right)$.

Figure 2

(a) [(b)] $S(\pi ,\omega ) \left[S\right(\pi /2,\omega \left)\right]$ for a Heisenberg chain for different system sizes. (c) $\max \left[S\right(\pi ,\omega \left)\right]$ extracted from panel (a) as a function of $Lln{\left(L\right)}^{1/2}$. (d) $\max \left[S\right(\pi /2,\omega \left)\right]$ extracted from panel (b) as a function of ${[Lln\left(L\right)]}^{1/2}$.

Figure 3

(a)–(d) $S(q,\omega )$ calculated with the Krylov-space approach (a),(b), and the conjugate gradient method (c), (d) for a Heisenberg (a)–(c) and a Hubbard (b)–(d) chain of $L=48$ sites, using $\eta =0.1$ and $m=800$ DMRG states ($m_{\text{min}}=64$). (e), (f) Cuts at $k=\pi$ and $\pi /2$ of the spectra in panels (a)–(c) and (b)–(d) as a function of frequency. In the conjugate gradient method, a maximum of 1000 iteration steps has been imposed, regardless of the error.

Figure 4

(a)–(c) $A(k,\omega )$ for a t-J chain of $L=48$ sites with Krylov-space approach for $m=100$ and $\eta =0.1$ (a) and $m=400$ and $\eta =0.02$ (c). (b)–(d) Same as panels (a), (c) but with the conjugate gradient method. The chemical potential $\mu \simeq 0.44$ is indicated with a solid black line.

Figure 5

(a) $S(k_x,0,\omega )$ component for the two leg ladder t-J model with $L=48\times 2$ sites at $t_x=t_y=J=1.0$ and filling $n=0.8125$. $m=600$ states are kept in the DMRG simulations. (b) $S(k_x,\pi ,\omega )$ component of the spectrum for the same parameter values of panel (a).

Figure 6

(a) CPU times for a DMRG run as a function of the frequency $\omega$ performed with Krylov-space and conjugate gradient methods for a Heisenberg model. (b) Same as panel (a) but for the t-J chain model investigated in Sec. 3c.

Figure 7

CPU times for a DMRG run performed with Krylov and conjugate gradient methods for a Heisenberg (a), Hubbard (b), and t-J model (c). In panels (a), (b), the gray and red bars indicate the CPU times for $\omega =0$ and 2, respectively. In panel (c), the gray and red bars indicate the CPU times for $\omega -\mu =0$ and 1, respectively.