Entanglement entropy from DMRG for XXZ model

[sou27]

As the title indicates, I was playing around with the DMRG module for the (OBC) XXZ model, shown below

H = \sum \sigma^x_i\sigma^x_{i+1} + \sigma^y_i\sigma^y_{i+1} - \Delta\sigma^z_i\sigma^z_{i+1}

In particular, when I take 0 > \Delta > -1, and plot the entanglement entropy in the chain, there appears to be a weird result. For example, taking \Delta = -0.8, I get the following for the entanglement entropy

There appears to be a modulo 2 behaviour in the chain. To my understanding, the model should be described the C=1 compactified boson CFT in this regime, and the above plot should be following the Cardy-Calabrese formula. While the general structure seems to agree with the formula, I can’t understand the apparent splitting of the line. Is this an issue with my implementation of DMRG? Or perhaps is this the expected result?

[miles]

Hi Soumil,
I’m not 100% sure, but I think this odd-even effect is a result of having a finite-size system with open boundaries. The Cardy-Calabrese theory in its simplest form probably does not account for arbitrary/general boundary conditions. There may either be some special boundary conditions that could remove this effect, or you might have to consider periodic systems or very large systems. Or more simply, you could just fit the formula to only the even or odd numbered sites and ignore the other sites.

[adityab999]

Hi,

As Miles correctly guessed, this is due to a parity effect (L odd or even for a fixed partition site, or partition site odd or even for a fixed L) in a finite sized system in OBC.. see e.g. [1010.5796] Universal parity effects in the entanglement entropy of XX chains with open boundary conditions and [cond-mat/0512475] Boundary effects in the critical scaling of entanglement entropy in 1D systems .. see also [0911.4660] Parity effects in the scaling of block entanglement in gapless spin chains for a parity-effect in PBC.