A simple way to avoid metastable configurations in the density-matrix renormalization-group algorithms

A simple way to avoid metastable configurations in the density-matrix renormalization-group algorithms

J. C. Xavier

Instituto de Física, Universidade Federal de Uberlândia, Caixa Postal 593, 38400-902 Uberlândia, MG, Brazil

ABSTRACT

We use the spin-1 Heisenberg chain with periodic boundary conditions to ilustrate that the systems get stuck in metastable configurations only when the density-matrix renormalization-group algorithm start with small number of states m. We also show that the convergence of the energies have a huge improvement if we start the algorithm with a large number of states m.

Keywords: spin model, DMRG, spin-1

The density-matrix renormalization-group [1, 2] (DMRG) is one of the most appropriate techniques to study static properties of the one-dimensional systems at zero temperature (for a review see, for example, Refs. 3 and 4). It is also possible to calculate dynamic properties [3-7] and work at finite-temperature through the DMRG [8-10]. The main advantage of DMRG, compared with the Lanczos exact diag-onalization [11], is its capability to obtain the ground-state properties of very large systems in a well controlled way. Note that it is also possible to investigate large systems by Monte Carlo methods. However, the Monte Carlo technique is not appropriated to study frustrated/fermionic systems due to the "sign" problem.

Although the DMRG algorithm was developed for one-dimensional systems, it has been used to treat two-dimensional systems [12-17]. The procedure consists in mapping the low-dimensional model on an one-dimensional model with long range interactions. As first point out by Liang and Pang [12], the energies of two-dimensional systems, converge slower than the ones of one-dimensional systems with short range interactions. The number of states needed to keep a fixed accuracy seems to increase exponentially with the width of the system.

A similar effect also appears when we study one-dimensional systems with periodic boundary condition (PBC) [1, 2]. Since the DMRG was developed, it was observed that the ground state energy (as a function of the number of states m kept in the truncation process) converge faster for the system with open boundary condition (OBC) than the one with PBC. Although it is not completely understood, it seems that everytime that an operator that acts in the left block is directly connected with an operator that acts in the right block (see Fig. 1), the ground state energy convergence is slower. This has been observed for one-dimensional systems as well for the two-dimensional systems.

Another difficulty also appears when the left and right blocks are directly connected. Some times, in the simulations, the system gets stuck in some local minimum of energy (see Fig. 3(a)), even working with large values of m [18, 19]. This is a serious problem. If the energies do not change increasing m, we may think naively that the true ground state energy was reached. But in fact, the energy found is far from the true ground state.

Few years ago [19], White proposed a variation of the density-matrix renormalization-group algorithm with a single center site. This new algorithm has the advantage of (i) decreasing the computation time as well as the memory used, and more important, (ii) to avoid that the system gets stuck in metastable configurations. As discussed by White [19], the main reason that the simulations get stuck in metastable configurations is due to the fact that some important fluctuations between the system and environment are absent. This happens because the environment blocks lost some relevant states in the truncation process [20]. In order to incorporate the missing states for those fluctuations, White suggests that we have to add in the density-matrix ρ small terms like [19]

for each operator A_i that is connected directly with the environment. The constant a is a small free parameter, the magnitude of a is chosen to vary from a ~ 10^-2 — 10^-4 [19].

In fact, using the one-dimensional spin-1 Heisenberg model with PBC as a test, White found that single-site DMRG method with the correction to the density-matrix, gets lower energy than the standard two-sites DMRG method [19].

It is very important to note that, in the standard DMRG algorithms, White shown [1,2] that the new blocks built in the renormalization process are better represented by the following transformation: . Where the rows of O are the m eigenvectors of the density-matrix p that are associated with the largest eigenvalues. So, if we add in the density matrix the term Δρ, the matrix O will not have anymore all the optimal states. Actually, it is quite surprising that the DMRG with the correction in the density matrix gets a lower energy.

In this article, we present a simple procedure to avoid that the system gets stuck in metastable configurations using the standard DMRG algorithm with two center sites. Note that the DMRG method with two center sites demand more CPU time and memory. But as we see, it has the advantage of obtaining lower energy than the single-site DMRG method with correction in the density-matrix. Besides that, it is free of non-controlled parameters, and more important, the transformation matrix O is built considering only the optimal states.

As a benchmark test, in this work, we consider the one-dimensional spin-1 Heisenberg model

where the coupling J was set to unity to fix the energy scale. We investigated the model above with DMRG technique with OBC and PBC. We use the finite-size algorithm for sizes up to L = 100 and keeping up to m = 4000 states per block in the final sweep. The discarded weight, 1 — P_m, was typically about 10^-6— 10^-13 in the final sweep (see Fig. 4(b)).

Before presenting the results, let us first briefly describe the DMRG algorithm for finite-size systems, which is crucial in our discussion. We can divide the finite-size DMRG algorithm in three parts, as illustrated in Fig. 2. In the first part, we use the standard infinite system density-matrix algorithm until reach the desired lattice size. This first part of the algorithm is used only once to obtain the environment blocks that will be used in the second part of the algorithm. As we see later, the fact that the system gets stuck in some metastable configurations is related with this part of the algorithm. Once we reach the desired size we start to sweep, as illustrated in Fig. 2. Note that in the third part of the algorithm the environment blocks are those blocks built in the second part of the algorithm. Usually, all the measurements are done in last iteration of the third part of the algorithm (the symmetric configuration).

In general, if we want the ground state energy E₀(L, m) of a system with lattice size L and m = m_final states per block, we start the first part of the algorithm with m₀< m_final (typically is used m₀ ~ 50 — 100). We increase at each sweep the states kept in the truncation process until we reach the final value m_final.

In Fig. 3(a), we present the error in the energy, defined as ΔE = E₀(L, 4000) — E₀(L, m) [21], as function of the number of sweeps for the one-dimensional spin-1 Heisenberg model with size L = 100 under PBC. We have done for each value of m five sweeps [22]. As we can see in this figure, for the sweeps with m between 100-400, there is no improvement in the convergence [23]. This is a serious problem. For instance, if we had defined the error in the energy as ΔE = E₀(L,400) — E₀(L,m), we could be thought that we hadreachedthe groundstates energy withhighaccuracy. But in fact, the energy is still far from the true ground state. In order to avoid this serious problem, as we mentioned before, White proposed to add in the density-matrix the small term Δρ defined in Eq. 1. By using this new approach, White was able to improve the convergence up to two orders of magnitude [19].

In this article, we show another simple procedure to avoid that the system to get stuck in metastable configurations. Our main result is presented in Fig. 3(b). In this figure, we show the error in the energy as function of m for different values of m₀. We clearly see that if we start the first part of the DMRG algorithm with a small value of m₀ (filled triangles down), the convergence is much worse than the one that started with m₀ = m_final (filled circles). From this observation, we conclude that a simple way to gain a huge improvement in the convergence is to start the algorithm with large values of m₀. And more important, if we start with m₀ = m_final the system does not get stuck in metastable configurations. We emphasize that the system gets stuck in metastable configurations, as we observe in Fig. 3(b), due to the fact that some important states were missed in the truncation process in the first part of the algorithm.

It is also important to observe that the difference of the energies obtained with m₀ = m_final and m₀ = m_final/2 is very small. In particular for m = m_final = 800 this difference is 2x10^-7. So, we can save CPU time starting the DMRG algorithm with m₀ = m_final/2, and even so, we still obtain results with a very high accuracy.

In Table 1, we show some energies obtained using the standard DMRG algorithm with two center sites (standard two-sites) and m₀ = m_final /2 for two values of m_final .We also present the energies (taken from Ref. 19) obtained with the single-site DMRG method with the correction to the density-matrix (corrected single site). Note that the energies obtained by the standard DMRG with two center sites are slightly lower than the ones acquired by White in Ref. 19.

Finally, we point out that indeed the convergences of the energies have a significant improvement if we use m₀ = m_final (or m₀ = m_final /2) instead of m₀ = 50 for the model under PBC. However, the energies obtained with PBC still converge much slower than the ones with OBC, as we can observed in Fig. 4(a). As we can see in this figure, if we use OBC and m = 400 we basically obtain the ground state energy with the precision of the computer. Similar behavior is also found in the truncation error 1 - P_m, as presented in Fig. 4(b).

In conclusion, in this work we study the one-dimensional spin-1 Heisenberg model with periodic/free boundary conditions. We shown that the standard DMRG algorithm with two center sites gets stuck in metastable configurations only when few states are selected in the grow process (the first part of the algorithm, where is used the infinity size algorithm). A simple way to obtain a huge improvement in the convergence is to start the algorithm with the same numbers of states of the final sweep. Even using this procceding, we observe that the convergence of the ground state energy for the system under periodic boundary condition is slower than the one with open boundary condition. Finally, our results suggest that it may be possible also that a similar improvement can be reached with this approach if it is consider the single-site DMRG algorithm without the correction to the density-matrix.

Acknowledgments

This work was supported by Brazilian agencies FAPEMIG and CNPq.

[20] Note that some trials, such as adding extra randomstates, were tested. However did not work very well (see Ref. 19).

[21] We use m₀ = 2000, in the first part of the algorithm, to obtain the energy E(L,4000).

[22] For m > 2000 we did two sweeps.

[23] Similar results were obtained by White in Ref. 19, although for m = 600 our error in the energy seems to be one order of the magnitude smaller than the one presented by White.

(Received on 3 June, 2009)

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Publication Dates

• Publication in this collection
  15 Oct 2009
• Date of issue
  Sept 2009

History

• Received
  03 June 2009