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arXiv:2107.09588 · Physical review. B./Physical review. B

Dynamical structure factor of the SU(3) Heisenberg chain: Variational Monte Carlo approach

Dániel Vörös, Karlo Penc

Published Nov 23, 2021• 9 citations
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Abstract

We compute the dynamical spin structure factor $S(k,\ensuremath{\omega})$ of the SU(3) Heisenberg chain variationally using a truncated Hilbert space spanned by the Gutzwiller projected particle-hole excitations of the Fermi sea, introduced by Dalla Piazza et al. [Nat. Phys. 11, 62 (2015)], with a modified importance sampling. We check the reliability of the method by comparing the $S(k,\ensuremath{\omega})$ to exact diagonalization results for 18 sites and to the two-soliton continuum of the Bethe ansatz for 72 sites. We get an excellent agreement in both cases. Detailed analysis of the finite-size effects shows that the method captures the critical Wess-Zumino-Witten SU(3)${}_{1}$ behavior and reproduces the correct exponent, with the exception of the size dependence of the weight of the bottom of the conformal tower. We also calculate the single-mode approximation for the $\mathrm{SU}(N)$ Heisenberg model and determine the velocity of excitations. Finally, we apply the method to the SU(3) Haldane-Shastry model and find that the variational method gives the exact wave function for the lowest excitation at $k=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}/3$.

DOI
10.1103/physrevb.104.184426

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We compute the dynamical spin structure factor S(k,\ensuremathω)S(k,\ensuremath{\omega})S(k,\ensuremathω) of the SU(3) Heisenberg chain variationally using a truncated Hilbert space spanned by the Gutzwiller projected particle-hole excitations of the Fermi sea, introduced by Dalla Piazza et al. [Nat. Phys. 11, 62 (2015)], with a modified importance sampling. We check the reliability of the method by comparing the S(k,\ensuremathω)S(k,\ensuremath{\omega})S(k,\ensuremathω) to exact diagonalization results for 18 sites and to the two-soliton continuum of the Bethe ansatz for 72 sites. We get an excellent agreement in both cases. Detailed analysis of the finite-size effects shows that the method captures the critical Wess-Zumino-Witten SU(3)1{}_{1}1​ behavior and reproduces the correct exponent, with the exception of the size dependence of the weight of the bottom of the conformal tower. We also calculate the single-mode approximation for the SU(N)\mathrm{SU}(N)SU(N) Heisenberg model and determine the velocity of excitations. Finally, we apply the method to the SU(3) Haldane-Shastry model and find that the variational method gives the exact wave function for the lowest excitation at k=\ifmmode±\else\textpm\fi2\ensuremathπ/3k=\ifmmode\pm\else\textpm\fi{}2\ensuremath{\pi}/3k=\ifmmode±\else\textpm\fi2\ensuremathπ/3.
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.
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We compute the momentum resolved dynamical spin structure factor S(k,ω)S(k,ω)S(k,ω) of the SU(4) Heisenberg model on the honeycomb lattice assuming the πππ-flux Dirac spin liquid ground state by two methods: (i) variationally using Gutzwiller projected particle-hole excitations of the πππ-flux Fermi sea and (ii) in the non-interacting parton mean-field picture. The two approaches produce qualitatively similar results. Based on this analogy, we argue that the energy spectrum of the projected excitations is a gapless continuum of fractional excitations. Quantitatively, the Gutzwiller projection shifts the weight from higher to lower energies, thus emphasizing the lower edge of the continuum. In the mean-field approach, we obtained the 1/distance41/\text{distance}^41/distance4 decay of the spin correlation function, and the local correlations show SMF33(ω)∝ω3S^{33}_{\text{MF}}(ω)\propto ω^3SMF33​(ω)∝ω3 behavior.
We compute the momentum-resolved dynamical spin structure factor $S(k,\ensuremath{\omega})$ of the SU(4) Heisenberg model on the honeycomb lattice assuming the $\ensuremath{\pi}$-flux Dirac spin liquid ground state by using two methods: (i) variationally using Gutzwiller projected particle-hole excitations of the $\ensuremath{\pi}$-flux Fermi sea, and (ii) in the noninteracting parton mean-field picture. The two approaches produce qualitatively similar results. Based on this analogy, we argue that the energy spectrum of the projected excitations is a gapless continuum of fractional excitations. Quantitatively, the Gutzwiller projection shifts the weight from higher to lower energies, thus emphasizing the lower edge of the continuum. In the mean-field approach, we obtained the 1/distance41/{\text{distance}}^{4}1/distance4 decay of the spin correlation function, and the local correlations show ${S}_{\text{MF}}^{33}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{3}$ behavior.
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