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arXiv:cond-mat/9706169 · Physical Review Letters

Comment on “One-Dimensional Disordered Bosonic Hubbard Model: A Density-Matrix Renormalization Group Study”

Nikolay V. Prokof'ev, Boris V. Svistunov

Published May 11, 1998• 50 citations
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Abstract

We present the phase diagram of the system obtained by continuous-time worldline Monte Carlo simulations, and demonstrate that the actual phase diagram is in sharp contrast with that found in Phys. Rev. Lett., 76 (1996) 2937.
DOI
10.1103/physrevlett.80.4355

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Figure 2 of the above paper contained a small numerical error. The correct version is shown below. Figure 2. The phase diagram of the extended Hubbard model. The phase boundary obtained in the present scheme (solid line) is compared with that obtained in the FS scheme (dotted line) and the Monte Carlo results. The dashed line U = 2V gives the phase boundary in the atomic limit (t = 0) between a singly occupied paramagnetic phase and a CDW region. No other part of the paper is affected by this error.

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We use the density-matrix renormalization group to study the one-dimensional bosonic Hubbard model, with and without disorder. We obtain the gaps in all phases, certain correlation functions, and the superfluid density. A finite-size-scaling analysis enables us to obtain an accurate phase diagram and the $\ensuremath{\beta}$ function at the superfluid--Mott insulator transition and the Kosterlitz-Thouless essential singularity in the pure case. We also obtain coupling constants used in effective field theories for this system.
By means of the density matrix renormalization group technique, we accurately determine the zero-temperature phase diagram of the one-dimensional extended Bose–Hubbard model with on-site and nearest-neighbor interactions. We analyze the scaling of the charge and of the neutral ground-state energy gaps, as well as of various order parameters. In this way we come to an accurate location of the boundaries between the superfluid and the insulating phases. In this last region, we are able to distinguish between the conventional Mott insulating and density-wave phases and the Haldane insulator phase displaying long-range string ordering, as originally predicted by Dalla Torre et al (2006 Phys. Rev. Lett. 97 260401).
We use the density-matrix renormalization group to calculate the phase diagram of the one-dimensional extended Hubbard chain at half filling. We find that for one specific U,U,U, the critical value of Vc{V}_{c}Vc​ for the charge-density-wave versus spin-density-wave phase transition is smaller than that predicted from the Monte Carlo simulation. Our results are consistent with the strong- and weak-coupling limit results, whereas the previous results from the Monte Carlo simulations deviate from the weak-coupling limit result. As UUU increases, we find that the first-order phase transition develops gradually. The tricritical point is estimated to occur at Uc=4−−6,{U}_{c}=4--6,Uc​=4−−6, not Uc=3{U}_{c}=3Uc​=3 as previously predicted from the Monte Carlo simulation. Due to the topological difference, we observe that with the periodic boundary condition, the phase transition is more abrupt for finite chains of length N=4MN=4MN=4M than for chains of N=4M+2,N=4M+2,N=4M+2, where MMM is an integer, for the same UUU and V.V.V. The difference will diminish in the thermodynamic limit.
Figure 2 of the above paper contained a small numerical error. The correct version is shown below. Figure 2. The phase diagram of the extended Hubbard model. The phase boundary obtained in the present scheme (solid line) is compared with that obtained in the FS scheme (dotted line) and the Monte Carlo results. The dashed line U = 2V gives the phase boundary in the atomic limit (t = 0) between a singly occupied paramagnetic phase and a CDW region. No other part of the paper is affected by this error.