Multigrid Algorithms for Tensor Network States

The widely used density matrix renormalization group (DRMG) method often fails to converge in systems with multiple length scales, such as lattice discretizations of continuum models and dilute or weakly doped lattice models. The local optimization employed by DMRG to optimize the wave function is ineffective in updating large-scale features. Here we present a multigrid algorithm that solves these convergence problems by optimizing the wave function at different spatial resolutions. We demonstrate its effectiveness by simulating bosons in continuous space and study nonadiabaticity when ramping up the amplitude of an optical lattice. The algorithm can be generalized to tensor network methods and combined with the contractor renormalization group method to study dilute and weakly doped lattice models.

Figure 1

Multigrid DMRG illustrated for bosons in an optical lattice $V(x)$. The DMRG algorithm converges fast for the rather dense system at the coarsest grid (shown on the top), and this solution is then used to iteratively initialize DMRG calculations on finer grids, which substantially speeds up convergence. The filling of the circles illustrates the probability for a particle to be at that site.

Figure 2

Density profiles of a continuum system of bosons in an optical lattice consisting of $L=32$ unit cells. The top and middle panels shows the nonconverged results obtained for $N=32$ grid points per unit cell after 12 sweeps of the DMRG algorithm with two different initial states: initial state 1 is a random state, initial state 2 is obtained from an infinite-size growing procedure as implemented in the ALPS DMRG code [24]. The bottom panel shows the multigrid result.

Figure 3

The restriction of an MPS state reducing it from eight to four sites. The contraction along the indices $\beta$, ${\sigma }_1$, and ${\sigma }_2$ creates a new tensor ${\tilde{A}}_{{\alpha }_1{\alpha }_2}^{\tilde{\sigma }}$ with a new index $\tilde{\sigma }$ but keeping the old connections to the neighboring sites (${\alpha }_1$ and ${\alpha }_2$).

Figure 4

Prolongation operation doubling the number of sites: First, we transform the basis $\tilde{\sigma }$ of the initial matrix $\tilde{A}$ into two new local bases ${\sigma }_1$ and ${\sigma }_2$. With a singular value decomposition, we then split the rank-4 tensor ${\tilde{A}}^{\prime }$ into two matrices $A_1$ and $A_2$.

Figure 5

Comparison of the energies in a bosonic optical lattice ($V_0/E_r=6$, $1/\gamma =0.1$) with $L=32$ unit cells discretized with increasing discretization $N=16$, 32, 64, 128 obtained with different strategies: DMRG with initial state 1 optimizing an initial random state, DMRG with initial state 2 initializing the system with an infinite size procedure and linearly increasing the number of states (results obtained with the ALPS DMRG code [24]), MG DMRG, and MG DMRG combined with local optimization in the prolongations.

Figure 6

Heating due to finite ramping speeds in a system of length $L=16$ for $1/\gamma =0.08$ with $M=400$. Different lines show the energy difference to the ground state while ramping up to $V_0/E_r=0\to 4$. (inset) Ramping functions used in the time evolution: linear, $V_0(t)/E_r=4t/t_V$ (dashed); exponential, $V_0(t)/E_r=4[\exp (4t/t_V)-1]/[\exp (4)-1]$ (dotted); and $s$ like, $V_0(t)/E_r=4[3(t/t_V{)}^2-2(t/t_V{)}^3]$ (solid).

Figure 7

Coarse graining a ladder of spin-$1/2$ fermions into a system of hard-core bosons. Spin singlets are mapped to empty sites and hole pairs to hard-core bosons in the first step. Further restriction steps of the bosonic model are similar to dilute bosonic models discussed above.