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arXiv:0801.3565 · Physical Review B

Density-matrix renormalization-group simulation of the SU(3) antiferromagnetic Heisenberg model

M. Aguado, M. Asorey, Elisa Ercolessi, F. Ortolani, Stefano Pasini

Published Jan 26, 2009• 15 citations
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Abstract

We analyze the antiferromagnetic $\mathit{SU}(3)$ Heisenberg chain by means of the density-matrix renormalization group. The results confirm that the model is critical and the computation of its central charge and the scaling dimensions of the first-excited states show that the underlying low-energy conformal field theory is the $\mathit{SU}{(3)}_{1}$ Wess-Zumino-Novikov-Witten model.

DOI
10.1103/physrevb.79.012408

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We analyze the antiferromagnetic SU(3)\mathit{SU}(3)SU(3) Heisenberg chain by means of the density-matrix renormalization group. The results confirm that the model is critical and the computation of its central charge and the scaling dimensions of the first-excited states show that the underlying low-energy conformal field theory is the SU(3)1\mathit{SU}{(3)}_{1}SU(3)1​ Wess-Zumino-Novikov-Witten model.
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We develop an efficient method to perform density matrix renormalization group simulations of the SU(NNN) Heisenberg chain with open boundary conditions taking full advantage of the SU(NNN) symmetry of the problem. This method is an extension of the method previously developed for exact diagonalizations and relies on a systematic use of the basis of standard Young tableaux. Concentrating on the model with the fundamental representation at each site (i.e., one particle per site in the fermionic formulation), we have benchmarked our results for the ground-state energy up to N=8N=8N=8 and up to 420 sites by comparing them with Bethe ansatz results on open chains, for which we have derived and solved the Bethe ansatz equations. The agreement for the ground-state energy is excellent for SU(3) (12 digits). It decreases with NNN, but it is still satisfactory for N=8N=8N=8 (six digits). Central charges ccc are also extracted from the entanglement entropy using the Calabrese-Cardy formula and agree with the theoretical values expected from the SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ Wess-Zumino-Witten conformal field theories.
Using the density-matrix renormalization-group technique we study the long-wavelength properties of the spin S=3/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2S=3/2 nearest-neighbor Heisenberg chain. We obtain an accurate value for the spin velocity $v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3.87\ifmmode\pm\else\textpm\fi{}0.02$, in agreement with experiment. Our results show conclusively that the model belongs to the same universality class as the S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 Heisenberg chain, with a conformal central charge c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 and critical exponent $\ensuremath{\eta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$.
We have developed an efficient method for performing density matrix renormalization group (DMRG) simulations of the SU(NNN) Fermi-Hubbard chain, fully leveraging the SU(NNN) symmetry of the problem. This method extends a previously developed approach for the SU(NNN) Heisenberg model and relies on the systematic use of the semistandard Young tableaux (SSYT) basis in a DMRG algorithm ``\`a la White.'' Specifically, the method aligns the site-by-site growth process of the infinite-size part of the DMRG, in its original formulation, with the site-by-site construction of the SSYT (or Gelfand-like) basis, based on the chain of unitary subgroups $U(1)\ensuremath{\subset}U(2)\ensuremath{\subset}U(3)\ensuremath{\subset}U(4)\ensuremath{\cdots}$. We give special emphasis to the calculation of the symmetry-resolved reduced matrix elements of the hopping terms between the left and the right block, which makes direct use of the basis of SSYT and of the Gelfand-Tsetlin coefficients, offering a computational advantage in scaling with NNN compared to alternative methods that rely on summing over Clebsch-Gordan coefficients. Focusing on the model with homogeneous hopping between nearest neighbors, we have calculated the ground-state energy as a function of UUU, i.e., the atom-atom interaction amplitude, up to N=6N=6N=6 for filling 1/N1/N1/N (one particle per site in average), and for one atom (respectively, hole) away from filling 1/N1/N1/N. It allows us to compute the charge gaps and to estimate in the thermodynamical limit, the critical value Uc{U}_{c}Uc​ (separating the Mott insulator from the metallic phase), which is proved to be finite (i.e., Uc>0{U}_{c}>0Uc​>0) and to increase with NNN. Central charges ccc are also extracted from the entanglement entropy using the Calabrese-Cardy formula and are consistent with the theoretical predictions: $c=N\ensuremath{-}1$, expected from the SU(N)1\mathrm{SU}{(N)}_{1}SU(N)1​ Wess-Zumino-Witten conformal field theory in the spin sector for the Mott phase, and c=Nc=Nc=N in the metallic phase, reflecting the presence of one additional (charge) gapless critical mode.