Matrix product state approach to a frustrated spin chain with long-range interactions

We make extensive simulations over a spin chain model that combines the frustrated $J_1-J_2$ spin chain and the long-range nonfrustrated ${(-1)}^{(r-1)}{r}^{-\alpha }$ decay interactions through the variational matrix product state method for both finite and infinite lengths. We study both the ground-state entanglement and phase diagram. We find that it is most entangled in the rotation invariant long-range ordered antiferromagnetic phase, where the entanglement scales approximately logarithmically. We determine the development of the Majudar-Ghosh point to a disorder line from entanglement. And we approximately determine the transition from the dimerized and incommensurate phase of the $J_1-J_2$ model to a decoupled phase by studying spin correlation and the dimerization order parameter. Some implications for entanglement in systems with long-range interactions are stated.

Figure 1

Ground-state phase diagram for Eq. (1). At $\alpha =\infty$, the model degenerates to the $J_1-J_2$ chain, whose phases undergo phase transitions under LRIs with decreasing $\alpha$: (i) the QLRO phase to a long-range order AFM phase [8, 9], (ii) the commensurate VBS phase terminating at a multicritical point at around $\alpha =1.7$ and $g=0.41$, and continued by a line of first-order transitions (thick solid line) [9], and (iii) the incommensurate VBS phase to a phase decoupled for odd and even sublattice.

Figure 2

Distribution of the half-chain entanglement entropy $S_{N/2}$ on the parameter domain $(\alpha ,g)\in [0.7,3.0]\times [0,1.0]$ divided into $24\times 50$ points for system size $N=60$.

Figure 3

Dependence of half-chain entanglement $S_{N/2}$ on $g$ for several values of $\alpha$ with $N=100$. Inset: Detail of the lines around $g=0.4$ for $\alpha =1.6,1.7$, and 1.8.

Figure 4

(a) Dependences of variance (open square) and middle chain entanglement (open circle) on $\alpha$ shown together for comparison for fixed $g=0.0$ and $N=100, D=520$. (b) Ground-state energy as a function of $g$ at $\alpha =1.0$ obtained from different algorithms. Open triangle: VMPS initiated from iDMRG or random state; open square: VMPS with proper initial state; cross: iDMRG. Inset: Extrapolation of position of peak of energy using $N$ ranging from 16 to 100. The fitting line is $g=1.0/N+0.339$. (c) Middle chain entanglement as a function of iDMRG steps at $\alpha =1.0,g=0.0$ with $D=1000$.

Figure 5

(a) Bipartite entanglement entropy $S$ as a function of subsystem length $L$ for each $\alpha$ with $g=0.0$. Only the bipartitions on even bonds (even $L$) are drawn. Solid lines are fitting to $S=\frac{c}{6}ln\left(L\right)+\text{const}$ for $1\le L\le 75$. Inset: The extracted $c$ vs $\alpha$. (b) Three groups of curves at $(1.0,0.0)$ (open circle), (1.0, 0.7) (open square), and (3.0, 0.0) (open triangle) each show dependence of $S$ on $L$ for different system sizes $N$ ranging from 32 to 200. $L$ is restricted to no larger than $\frac{3}{8}N$, i.e., not close to the chain center. Solid lines are fitting to the function as in (a). Inset: $c$ extracted from fitting of each group of curves as a function of $N$.

Figure 6

(a) $S(\alpha ,L,N)$ vs inverse chain length for different $\alpha$ and fixed $L=12$. Solid lines are fitting to $S(\alpha ,L,N)=-k/N+S(\alpha ,L,\infty )$ which is a linear function of $1/N$. (b) $S(\alpha ,L,N)$ vs inverse chain length for different $L$ and $\alpha =1.0$. Solid lines are fitting to the same function as in (a). The system sizes $N$ used are restricted to $[N_{min},200]$, with $N_{min}$ depending on $L$, such that $\frac{3}{8}{N}_{min}\ge L$, namely, $L$ should not be close to the chain center. (c) Extrapolated $S(\alpha ,L,N)$ value to $N=\infty$ as a function of $L$ ($L\le 64$) for several $\alpha$, obtained by curve fitting as above. The lines are a guide for the eye.

Figure 7

Absolute value of the intrasublattice correlation $C\left(r\right)$ for each $\alpha$ at $g=0.7$. $C\left(r\right)$ are obtained by evaluating Eq. (7) upon IMPS from iDMRG simulations with $D$ up to 960. Inset: $C\left(r\right)$ at $\alpha =1.4$ and $g=0.7$ multiplied by $\sqrt{r}{e}^{r/\xi }$ with $\xi =44.7$ using iDMRG and $D=640$.

Figure 8

Left and right plots show, respectively, dimerization $d$ and the negative of the intersublattice correlation for $r=1$, measured upon IMPS, as a function of iDMRG truncation dimension $D$ for several $(\alpha ,g)$ tuples.