Violation of the area law and long-range correlations in infinite-dimensional-matrix product states

We propose to construct a family of states of one-dimensional spin chains by replacing the finite-dimensional matrices in matrix product states by infinite-dimensional operators constructed from bosonic annihilation and creation operators. The resulting states are demonstrated to violate the area law and to exhibit long-range correlations. In addition, we propose an efficient way to prepare the states experimentally, in which the spins interact sequentially with an ancilla system.

Figure 1

(a) Graphical representation of MPS as a tensor network. The tensors $A_{s_i}^{(i)}$ are drawn as rectangles with two horizontal legs corresponding to the two virtual indices (running from $1$ to $D$) and one vertical leg representing the physical index $s_i$. The input and output states are drawn as rectangles with one leg, and the combination of two legs into a line joining two tensors means contraction of the corresponding indices. (b) In the iMPS construction, the three-legged tensors are replaced by pieces of a cylinder, the vacuum state is represented by a half sphere, and the combination of two pieces corresponds to multiplication. The modes of the operators $V_{s_i}(z_i)$ correspond to vibrations of the cylinder pieces at different frequencies, and removing modes corresponds to reducing the number of frequencies at which the cylinder pieces can vibrate.

Figure 2

Renyi entropy $S_L^{(2)}$ of a subchain of length $L$ for $N=4000$, $\alpha =0.15$, various values of $M$, and (a) $\delta =1$ and (b) $\delta =0$ obtained from Monte Carlo simulations. The faint dots on both sides of the curves for $M=1,2,5,10,20,50$ indicate the uncertainty in the numerical estimates. These curves are computed by averaging the right-hand side $r$ of (6) over the outcome of 50 or more Monte Carlo trajectories with different initial conditions, and the dots show $-\ln [\mathit{mean}(r)\pm \sqrt{\mathit{var}(r)}/2]$, where $\mathit{mean}(r)$ and $\mathit{var}(r)$ are the sample mean and the sample variance of $r$, respectively. We note that we use the same Monte Carlo trajectories to estimate $S_L$ for all values of $L$.

Figure 3

The same data as in Fig. 2a, but plotted as a function of $\sin (\pi L/N)$. The black solid lines are fits of the form $S_L^{(2)}=a[\sin (\pi L/N)]{}^b+c$ to the numerical data (gray crosses), where $a$, $b$, and $c$ are fitting parameters. The inset shows the exponent $b$ (crosses) as a function of $M$ ($b=0.0093$ for $M\to \infty$), and the dotted line is the fit $b=0.967\exp (-0.246M)$.

Figure 4

Schematic view of the scaling of the entropy of a subchain with the length of the subchain for various models. For MPS, the entropy is bounded by the logarithm of the virtual dimension, which is independent of $L$. For the critical case, the entropy scales with $\ln [N\sin (\pi L/N)/\pi ]$, which is on the boundary between constant behavior and a power law, and for the iMPS considered in this paper, there is a transition from the critical case to a power-law scaling of the entropy with $\sin (\pi L/N)$ as $M$ decreases toward one.

Figure 5

Renyi entropy $S_L^{(2)}$ of a subchain of length $L$ for $N=4000$, $M=1$, $\delta =0$, and various values of $\alpha$ as indicated. The gray crosses are obtained from Monte Carlo simulations, and the black solid lines show the values predicted by the approximate relation in Eq. (7).

Figure 6

The correlation function (8) as a function of the separation $k$ between the two spins for $N=500$, $\alpha =0.15$, and (a) $\delta =1$ and (b) $\delta =0$. The gray crosses show results obtained from Monte Carlo simulations, and the solid black lines are the approximation in Eq. (9). (Note that the point $k=0$, for which the correlation function is unity, is not included in the plots.)

Figure 7

To prepare the state (5) experimentally, we initialize $N$ spins, which are here encoded in atoms, in the state $(|+1\rangle +|-1\rangle ){}^{\otimes N}$, and for each mode included in the vertex operators, we let the spins interact one by one with a cavity field, which is subsequently subjected to a measurement. The inset shows the level structure of the atoms and the interaction with the cavity field and a classical field as described further in the text.

Figure 8

Probability of successfully projecting all the $M+\delta$ cavity modes on the vacuum state after the interactions with the atoms as a function of the number of atoms for $\alpha =0.15$ and (a) $\delta =1$ and (b) $\delta =0$. The gray crosses are numerical results, and the black solid lines represent the approximate relation in Eq. (20).

Figure 9

Renyi entropy $S_L^{(2)}$ as a function of the length of the subchain for $N=4000$, $\alpha =0.15$, and various values of squeezing of the cavity field. The gray crosses are results from Monte Carlo simulations, and the black solid lines are the analytical approximation derived in Appendix B.

Figure 10

The correlation function (8) as a function of the separation between the spins for $N=500$, $\alpha =0.15$, and various values of the squeezing of the cavity field. The gray crosses are results from Monte Carlo simulations, and the black solid lines are the analytical approximation in Eq. (23).

Figure 11

Renyi entropy $S_L^{(2)}$ obtained from Monte Carlo simulations with $N=4000$, $\alpha =0.15$, and various amounts of leakage.

Figure 12

The correlation function $\langle {\sigma }_z(0){\sigma }_z(k)\rangle$ obtained from Monte Carlo simulations with $N=4000$, $\alpha =0.15$, and various amounts of leakage.

Figure 13

Success probability as a function of the number of spins for $\alpha =0.15$ and various values of $g$. The solid line almost following the points for $g=0.05\times \pi /2$ is the fit $P_{\mathrm{pr}}=\exp (-0.0704N^{0.8227}-0.6043)$.