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

We compute the dynamical spin structure factor $S(k,\omega )$ 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 $S(k,\omega )$ 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 $\mathrm{SU}\left(N\right)$ 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 $k=\pm 2\pi /3$.

Figure 1

The filled Fermi sea of the fermions with three colors. The Fermi momentum is $k_F=\pi /3$; all the states between $-\pi /3$ and $\pi /3$ are occupied. At each $k$ the $f_{k,\mathsf{A}}^{†}{f}_{k,\mathsf{B}}^{†}{f}_{k,\mathsf{C}}^{†}|0\rangle$ form an SU(3) singlet, denoted by $\mathbf{1}$. The Fermi seas of the three colors are degenerate; they were shifted for visualization.

Figure 2

The structure factor for chain lengths from $L=12$ to $L=240$. Except for the singular peaks at $k=\pm 2\pi /3$, all the points for different sizes fall onto a single curve. The inset shows the power-law behavior of the $k=\pm 2\pi /3$ peaks. Fitting a function of the form ${a+b|k-2\pi /3|}^c+d(k-2\pi /3)$ to the points around the peak (but excluding the peak itself) gives an exponent $c=0.337$ (green line). The value of the exponent is sensitive to the fitting window; it fluctuates around the correct $\eta -1=1/3$ value. The error bars are smaller than the symbol sizes.

Figure 3

Finite size scaling of the $k=2\pi /3$ singular peak $S^{33}(2\pi /3)$, calculated from the Gutzwiller projected Fermi sea, plotted against $L^{-1/3}$. The straight line corroborates the nonanalytic contribution proportional to $L^{-1/3}$.

Figure 4

The energy of excitations in the single mode approximation, ${\varepsilon }_{\text{SMA}}\left(k\right)$, for the SU(3) Heisenberg model (red points) and the Haldane-Shastry model (blue points) calculated from the Gutzwiller projected Fermi sea for system sizes ranging from 12 to 240. The lines show the velocities of excitations, obtained from fitting the SMA in the $k\to 0$ limit. The error bars (not shown) are smaller than the symbol sizes.

Figure 5

Finite size scaling of the velocities $v\left(L\right)=\varepsilon \left(k_{\text{min}}\right)/k_{\text{min}}$. (a) The velocities of the SU(3) symmetric Heisenberg model obtained from exact diagonalization (ED), single mode approximation (SMA, calculated from the Gutzwiller projected Fermi sea used as an approximating ground state), and the variational dynamical structure factor calculated by using particle-hole excitations. The arrow represents the exact Bethe ansatz result in the thermodynamic limit, $v_{\text{BA}}^{\infty }=\pi /3$. (b) The single mode approximation using the Gutzwiller projected Fermi sea gives the exact value of the velocity for the Haldane-Shastry model, $v_{\text{HS}}^{\infty }=\pi /2$ for $L\to \infty$ (denoted by the arrow).

Figure 6

Finite size scaling of the ground state energy density, calculated for the approximating variational ground state $P_{\text{G}}|\text{FS}\rangle$ (variational and VMC denotes the points calculated numerically exactly and with Monte Carlo, respectively), and compared with exact diagonalization results (ED). The arrow shows the exact energy density in the thermodynamic limit obtained from Bethe Ansatz, Eq. (45).

Figure 7

The particle-hole excitation $|k,q_{\overline{\mathbf{3}}},\mathsf{A}\rangle =P_{\text{G}}{f}_{q_{\mathbf{3}},\mathsf{A}}^{†}{f}_{q_{\overline{\mathbf{3}}},\mathsf{A}}^{\phantom{†}}|\text{FS}\rangle$ with $k=q_{\mathbf{3}}-q_{\overline{\mathbf{3}}}$, $q_{\overline{\mathbf{3}}}\in \text{FS}$, and $q_{\mathbf{3}}\notin \text{FS}$. Such particle-hole excitations span the truncated Hilbert space. The states $|k,q_{\overline{\mathbf{3}}},\mathsf{A}\rangle$, $|k,q_{\overline{\mathbf{3}}},\mathsf{B}\rangle$, and $|k,q_{\overline{\mathbf{3}}},\mathsf{C}\rangle$ correspond to the three states in the center of the weight diagrams of $\mathbf{3}\otimes \overline{\mathbf{3}}=\mathbf{1}\oplus \mathbf{8}$ which has all three particles of different colors. These are therefore linear combinations of the singlet state $\mathbf{1}$ of the form $\frac{1}{\sqrt{3}}\left(\right|k,q_{\overline{\mathbf{3}}},\mathsf{A}\rangle +|k,q_{\overline{\mathbf{3}}},\mathsf{B}\rangle +|k,q_{\overline{\mathbf{3}}},\mathsf{C}\rangle )$ and the two states $\frac{1}{\sqrt{2}}\left(\right|k,q_{\overline{\mathbf{3}}},\mathsf{A}\rangle -|k,q_{\overline{\mathbf{3}}},\mathsf{B}\rangle )$ and $\frac{1}{\sqrt{6}}\left(\right|k,q_{\overline{\mathbf{3}}},\mathsf{A}\rangle +|k,q_{\overline{\mathbf{3}}},\mathsf{B}\rangle -2|k,q_{\overline{\mathbf{3}}},\mathsf{C}\rangle )$ belonging to the adjoint representation $\mathbf{8}$. The latter ones are used for the calculation of $S^{33}(k,\omega )$ and $S^{88}(k,\omega )$. The three dispersions are degenerate but shifted for visualization.

Figure 8

Graphical representation of the $\mathbf{3}\otimes \overline{\mathbf{3}}=\mathbf{1}\oplus \mathbf{8}$ decomposition of the one particle-hole excitations using the weight diagrams. The irreducible representation of a single particle is the three-dimensional $\mathbf{3}$, represented by a Young tableau with a single box. The weight diagram shows the three states, the values of the diagonal $T^3$ and $T^8$ operators for an added $\mathsf{A}$ fermion are $(T^3,T^8)=(1/2,1/2\sqrt{3})$—these are the coordinates of the point labeled by the red $\mathsf{A}$. The coordinates of the added $\mathsf{B}$ fermion are $(-1/2,1/2\sqrt{3})$, and of the $\mathsf{C}$ are $(0,1/\sqrt{3})$. The irreducible representation of a hole is $\overline{\mathbf{3}}$ since the two remaining fermions anti-symmetrize, this is denoted by two vertical boxes in the Young-tableau. The hole $\overline{\mathsf{A}}$ (colored by teal), with coordinates $(-1/2,-1/2\sqrt{3})$, is the antisymmetrical combination of $\mathsf{B}$ and $\mathsf{C}$ fermions (see also Fig. 7). The product of $\mathbf{3}\otimes \overline{\mathbf{3}}$ contains a singlet, Eq. (49), and the eight states of the $\mathbf{8}$. The two states of Eqs. (56) and (57) are at the center $(T^3,T^8)=(0,0)$ of the weight diagram for $\mathbf{8}$. These constitute the final states in the $S^{33}(\mathbf{k},\omega )$ and $S^{88}(\mathbf{k},\omega )$ structure factor when the initial state is a singlet $\mathbf{1}$.

Figure 9

Comparing $S^{33}(k,\omega )$ for $L=18$ calculated using Gutzwiller projected one particle-hole excitations (red) and by ED (blue). The area of the circles is proportional to the matrix element squared.

Figure 10

$S^{33}(k,\omega )$ and $S^{88}(k,\omega )$ for $L=72$ calculated by VMC. The area of the circles is proportional to the matrix element squared. The green background shows the two soliton continuum $(k,\omega )=(q_{\mathbf{3}\overline{\mathbf{3}}},{\varepsilon }_{\mathbf{3}\overline{\mathbf{3}}})$ of the Bethe ansatz solution, Eqs. (64) and (65), in the thermodynamic limit. For this plot, the total number of uncorrelated measurements was ${10}^8$.

Figure 11

The finite-size structure of a tower at low energies. $S^{(0,0)}$ denotes the weight of the lowest energy peak in the tower, with energy $E_{0,0}$ and momentum $k_{0,0}$, the $S^{(i,i^{\prime })}$ are the peaks in the tower following the notation in Eqs. (66). The momenta are measured from $k_0$, the momentum of the lowest energy peak in the tower. We expect the ratio between the weights of $S^{(1,0)}$ and $S^{(0,0)}$ peaks to give the exponent ${\eta }^{+}$, similarly ${\eta }^{-}=S^{(0,1)}/S^{(0,0)}$.

Figure 12

Finite size gap at the $k=2\pi /3$ multiplied by the system size $L$, as a function of $1/L$. The blue circles show the ED gap; the VMC results are shown by black crosses. The arrow points to $\pi \eta v_{\text{BA}}=4{\pi }^2/9$, the known value in the thermodynamic limit using $\eta =4/3$ and $v_{\text{BA}}=\pi /3$, the velocity from the Bethe ansatz, Eq. (36).

Figure 13

The ratios of the weights of the low energy peaks. The ratios provide information about the exponents, see Eqs. (72) and (73).

Figure 14

Scaling of the lowest peak from ED, variational, and VMC methods for the Heisenberg model and the exact values for the Haldane-Shastry model, together with the asymptotic $L^{-1/3}$ power-law behavior (cyan line). The $L^{-1/4}$ magenta line is a guide to the eye. Note the slight downward bending of the ED data which suggests that the exponent is in fact smaller than $-1/4$, tending toward $-1/3$.

Figure 15

The configurations forming the ‘arc'—the lowest energy excitations for momenta from $k=2\pi /L$ (a) to $k=2\pi /3$ (e)—in the Haldane-Shastry model for $L=15$. The Gutzwiller projected wave function of these particle-hole excitations are exact eigenstates of the Hamiltonian Eq. (22).