Connecting distant ends of one-dimensional critical systems by a sine-square deformation

We study one-dimensional quantum critical spin systems with sine-square deformation, in which the energy scale in the Hamiltonian at position $x$ is modified by the function $f_x=\sin {}^2[\frac{\pi }{L}(x-\frac{1}{2})]$, where $L$ is the length of the system. By investigating the entanglement entropy, spin correlation functions, and wave-function overlap, we show that the sine-square deformation changes the topology of the geometrical connection of the ground state drastically: Although the system apparently has open edges, the sine-square deformation links those ends and realizes the periodic ground state at the level of the wave function. Our results propose a method to control the topology of quantum states by energy-scale deformation.

Figure 1

(a) Rescaling function $f_x$ of the SSD. (b) Bond strength $\langle S_j^x{S}_{j+1}^x\rangle$ for the $\mathit{XXZ}$ chain, Eq. (2), with $L=80$ and $(\Delta ,M)=(0.5,0)$. Squares and circles represent the data for the chain with SSD and the uniform open chain, respectively.

Figure 2

Entanglement entropy $\mathcal{S}(l)$ as a function of $g(l)=(L/\pi )\sin (\pi l/L)$ for $L=80$ and (a) an $\mathit{XXZ}$ chain with $(\Delta ,M)=(0.5,0)$, (b) an $\mathit{XXZ}$ chain with $(\Delta ,M)=(1.0,0.25)$, (c) a $J_1$-$J_2$ chain with $(J_2/J_1,M)=(0.5,0.125)$, (d) a $J_1$-$J_2$ chain with $(J_2/J_1,M)=(0.6,0.4)$, and (e) a two-leg ladder with $(J_{\perp }/J_{\parallel },M)=(1.0,0.25)$. (a–c, e) Central charge $c=1$; (d) $c=2$. Squares and circles represent data for the open system with SSD and the uniform open system, respectively. Solid and dotted lines show the slopes of $c/3$ and $c/6$, respectively. (f) Shape of subsystem $\Omega$ for which $\mathcal{S}(l)$ is calculated.

Figure 3

(a) Spin correlation functions $\langle S_j^{\alpha }{S}_{j^{\prime }}^{\alpha }\rangle$ ($\alpha =x,z$) in an $\mathit{XXZ}$ chain for $L=80$ and $(\Delta ,M)=(0.5,0)$ as a function of the distance $|j-j^{\prime }|$, where sites $(j,j^{\prime })$ are selected as $j=L/2-[r/2]$ and $j^{\prime }=L/2+[(r+1)/2]$. Squares and circles represent DMRG data for an open chain with SSD and a uniform open chain, respectively, while lines show the analytic result for a uniform periodic chain. (b) Schematic showing the relation between pairs $(j,j^{\prime })$ in the open chain with SSD and those in the periodic chain.

Figure 4

(a) Spin correlation function $(-1){}^r\langle {\mathbf{S}}_j·{\mathbf{S}}_{j^{\prime }}\rangle$, with $j^{\prime }=j+r$ (mod $L$), in an $\mathit{XXZ}$ chain for $L=24$ and $(\Delta ,M)=(1.0,0)$ as a function of $j$ and $r$. Symbols show data for an open chain with SSD: crosses represent correlations between sites $j$ and $j^{\prime }=j+r$ (pairs “within” the chain), while squares represent those between $j$ and $j^{\prime }=j+r-L$ (pairs “across” the edges). Lines show values of correlations in the uniform periodic chain. (b) Schematic showing the two sites $(j,j^{\prime })$ at a “distance” $r$.