Suppose you have a one-dimensional quantum spin system with on-site Hilbert spaces $\mathcal{S}$. Suppose there is an anti-unitary, anti-linear operator $C$ on $\mathcal{S}$ inducing an anti-linear, anti-unitary operator $C_X$ on any $\mathcal{H}_{X} := \bigotimes_{x \in X} \mathcal{S}$.

In this situation one can define a partial transpose; namely consider disjoint subsets $X_1,X_2 \subset \mathbb{Z}$ and let $A = A_1 \otimes A_2$ be a operator on $\mathcal{H}_{X_1} \otimes \mathcal{H}_{X_2}$. Then define the partial transpose to be the $\mathbb{C}$-linear extension of

$$ (A_1 \otimes A_2)^{T_1} = (C_{X_1} A_1^* C_{X_1}) \otimes A_2 \ .$$

Assume $\Omega$ is a injective translation invariant matrix product state symmetric under $C_{\mathbb{Z}}$. Consider two adjacent disjoint intervals $X_1,X_2$ and $X = X_1 \cup X_2$ and let $L = \min(|X_1|,|X_2|)$. Then

$$ \lim_{L \rightarrow \infty} \text{Tr}(\rho_X^{T_1} \rho_X) = \pm \lim_{L \rightarrow \infty} \text{Tr}(\rho_X^2)^{\frac{3}{2}} \ . $$

Here, if $\mathcal{C}$ implements $C$ on the auxiliary space, the sign is $+1$ if $\mathcal{C}$ is a real structure and $-1$ if $\mathcal{C}$ is quaternionic.

1) Are some references to this? Is this known? I know that people have calculated some things with partial transposes in critical systems, but for gapped systems? There is of course the work by Shinsei Ryu et al, but they work with fermionic systems (which is my goal as well) and they don't seem to give proofs.

I want to conclude: since MPS states are dense in Hilbert space, the above then holds for all $C$-invariant states.

2) In going from the statement about MPS to general states: what could go wrong? For example, there is the problem of frustration, which i think plays no role here because i am considering pure states in the thermodynamic limit.

This post imported from StackExchange Physics at 2019-04-13 07:45 (UTC), posted by SE-user Lorenz Mayer