Probability distribution of the entanglement across a cut at an infinite-randomness fixed point

We calculate the probability distribution of entanglement entropy $S$ across a cut of a finite one-dimensional spin chain of length $L$ at an infinite-randomness fixed point using Fisher's strong randomness renormalization group (RG). Using the random transverse-field Ising model as an example, the distribution is shown to take the form $p\left(S\right|L)\sim L^{-\psi \left(k\right)}$, where $k\equiv S/ln\left[L/L_0\right]$, the large deviation function $\psi \left(k\right)$ is found explicitly, and $L_0$ is a nonuniversal microscopic length. We discuss the implications of such a distribution on numerical techniques that rely on entanglement, such as matrix-product-state-based techniques. Our results are verified with numerical RG simulations, as well as the actual entanglement entropy distribution for the random transverse-field Ising model which we calculate for large $L$ via a mapping to Majorana fermions.

Figure 1

Illustration of how entanglement is created across a cut in the TFIM. The bond crossing that cut must first be decimated ($1\to 2$), resulting in the two spins on either side being fully correlated, but not yet entangled. The cut now lies “within” that combined spin. When the field on this site is decimated ($2\to 3$), the state of that spin is put into an equal superposition of the two possible correlated states, thus becoming entangled. The site is frozen out, there is now one new bit of entanglement across the cut, and the cut once again lies on a bond. Thus, the amount of entanglement $S$ across the cut is half the number $N$ of decimations across it, $S=N/2$.

Figure 2

Plot of the function $\varphi \left(x\right)$ [Eq. (11)].

Figure 3

Probability distribution of $N$ from numerical RG simulations obtained by (top) running the RG until a fixed logarithmic energy cutoff $\mathrm{\Gamma }$, and (bottom) by running the RG to completion for a fixed system size $L$. Dashed lines show the normalized analytical results [Eq. (10)] with $\ell =ln\mathrm{\Gamma }$ in the fixed $\mathrm{\Gamma }$ case, and $\ell =\frac{1}{2}lnL$ in the case with fixed $L$. There is excellent agreement between the analytical and numerical results.

Figure 4

Distribution of entanglement entropy $S$ sampled from cuts of random excited eigenstates of the random transverse-field Ising model of varying $L$. The random fields and couplings are taken from the fixed point distribution [Eq. (2)] with $\mathrm{\Gamma }=1$, corresponding to a flat distribution. With $\ell =\frac{1}{2}lnL/L_0$ for $L_0\approx 6\times {10}^{-4}$, all these curves collapse well towards a single curve, with the slight exception of the smallest $L=20$, due to finite size effects. The dashed line shows the analytic prediction [Eq. (19)] with a fixed constant shift to align with the numerical data.