Accurate determination of the Gaussian transition in spin-1 chains with single-ion anisotropy

The Gaussian transition in the spin-1 Heisenberg chain with single-ion anisotropy is extremely difficult to treat, both analytically and numerically. We introduce an improved density-matrix renormalization group procedure with strict error control, which we use to access very large systems. By considering the bulk entropy, we determine the Gaussian transition point to four-digit accuracy, $D_c/J=0.96845\left(8\right)$, resolving a long-standing debate in quantum magnetism. With this value, we obtain high-precision data for the critical behavior of quantities, including the ground-state energy, gap, and transverse string-order parameter, and for the critical exponent $\nu =1.472\left(2\right)$. Applying our improved technique at $J_z=0.5$ highlights essential differences in critical behavior along the Gaussian transition line.

Figure 1

Entropy $\mathcal{S}$ as a function of $D$ for $J_z=1$. (a) Calculations with $m=1000$. Open symbols are obtained for the lowest-energy level in Hilbert space $S_z^{\mathrm{tot}}=0$ with a range of $L$ values, and solid symbols for $S_z^{\mathrm{tot}}=1$. Inset: $\ln 2$ drop in $\mathcal{S}\left(L\right)$ for $D=0.92$. (b) Bulk entropy $\mathcal{S}\left(D\right)$ close to the Gaussian critical point, computed with $L=10000$ for a range of $m$ values. Insets: Fitting slopes $A_{L,1}$ and $A_{R,1}$ (left axis) and transition $D_c^{\mathrm{fit}}$ (right axis) obtained as functions of $m$.

Figure 2

(a) Gap as a function of $L$, computed for $D/J=0.95$ with several values of $m$. Solid symbols for $L\to \infty$ are extrapolated to $m\to \infty$ (inset), giving $\Delta \left(0.95\right)=0.00130\left(4\right)$. (b) Extrapolated gaps as a function of $\left|D-D_c\right|$.

Figure 3

(a) Finite-size extrapolation of the lowest two gaps at $D_c$. Fitting lines from CFT give $\Delta (S_z^{\mathrm{tot}}=0)=2\pi v/L$ with $v=2.564\left(2\right)J$ and $\Delta (S_z^{\mathrm{tot}}=1)=2\pi J\alpha /L$ with $\alpha =0.48516\left(1\right)$. (b) Extrapolated ground-state energy $e_g$ at $D_c$, with CFT fit $-0.86856650\left(4\right)J-\beta J/L^2$ and $\beta =1.3429\left(4\right)$. (c) Second derivative of extrapolated energy. (d) Extrapolated transverse string-order parameter (see text) of the Haldane phase, with fitting line $0.6036\left(4\right)|D-D_c{|}^{0.353\left(1\right)}$. Calculations for (a) and (b) performed with PBCs and $m=2000$, and for (c) and (d) with OBCs and $m=1000$.

Figure 4

$S\left(D\right)$ as in Fig. 1 for $J_z=0.5$. (a) Values of $L$ as indicated. Inset: $\ln 2$ drop in $\mathcal{S}\left(L\right)$ for $D=0.4$. (b) Values of $m$ as indicated.