Reptation quantum Monte Carlo algorithm for lattice Hamiltonians with a directed-update scheme

We provide an extension to lattice systems of the reptation quantum Monte Carlo algorithm, originally devised for continuous Hamiltonians. For systems affected by the sign problem, a method to systematically improve upon the so-called fixed-node approximation is also proposed. The generality of the method, which also takes advantage of a canonical worm algorithm scheme to measure off-diagonal observables, makes it applicable to a vast variety of quantum systems and eases the study of their ground-state and excited-state properties. As a case study, we investigate the quantum dynamics of the one-dimensional Heisenberg model and we provide accurate estimates of the ground-state energy of the two-dimensional fermionic Hubbard model.

Figure 1

(Color online) Pictorial representation of the sliding moves along the right imaginary-time direction. In the new configuration (bottom), a new head for the reptile is generated from the old configuration (top) and the tail is discarded.

Figure 2

(Color online) Pictorial representation of the sliding moves along the right imaginary-time direction when the worm operator sits at the tail of the reptile. In the new configuration (bottom), a new head for the reptile is generated from the old configuration (top), the old tail configuration is discarded and the worm discontinuity is moved to the “neck” of the reptile.

Figure 3

(Color online) Lowest-energy excitations as a function of the wave vector $q$ for an $L=20$ Heisenberg chain. The energies are extracted from the dynamical structure factor $S\left(q,\omega \right)$ and are compared to exact results by the Lanczos method.

Figure 4

(Color online) The same as in Fig. 3 for $L=80$. Exact results are given by the Bethe ansatz solution.

Figure 5

(Color online) Ground-state expectation value of the spin-spin correlation function $\mathcal{C}\left(d\right)$ for the Heisenberg model on an 80-site chain.

Figure 6

(Color online) Ground-state energy for the fermionic Hubbard model at half filling on an 18-site tilted-square lattice. The energy difference $\Delta E=E_{\text{exact}}-E$ is computed with distinct approximations described in the text.

Figure 7

(Color online) Relative efficiency of the directed-update scheme and the bounce algorithm. The measured quantity is the ground-state energy of the one-dimensional Heisenberg model on a chain of size $L=80$ sites.