# Partial Transpose in Gapped Time-reversal Symmetric Spin Chains

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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
I don't know any references on this, but in 1D you could presumably use a Jordan-Wigner transformation and use the fermionic results. By the way, it'd be good if you'd add link to Ryu's work in the post.

This post imported from StackExchange Physics at 2019-04-13 07:45 (UTC), posted by SE-user Anyon
Do I understand correctly that a special case would be the "normal" partial transpose, with a wavefunction with real coefficients?

This post imported from StackExchange Physics at 2019-04-13 07:45 (UTC), posted by SE-user Norbert Schuch
A real structure is just a way to formalize what is meant when we say a wave function has real coefficients, so yes.

This post imported from StackExchange Physics at 2019-04-13 07:45 (UTC), posted by SE-user Lorenz Mayer
I did it on the AKLT state... for MPS i am also pretty confident because one can do the calculation (its just a bit lengthy). I am more wondering what could go wrong in going from a statement about MPS to a statement about all states.

This post imported from StackExchange Physics at 2019-04-13 07:45 (UTC), posted by SE-user Lorenz Mayer
I have a calculation for MPS, and i want to conclude from there a statement about all pure translation invariant exponentially clustering $C$-symmetric ground states by taking limits.

This post imported from StackExchange Physics at 2019-04-13 07:45 (UTC), posted by SE-user Lorenz Mayer
@lcv Basically, yes, there is a normalization "hidden". In terms of the entanglement spectrum $\Lambda$ (of the half-chain), the LHS is $\mathrm{tr}(\Lambda^2)^3$ and the trace in the RHS is $\mathrm{tr}(\Lambda^2)^2\,\mathrm{tr}(\Lambda)^2$, which is homogeneous in $\Lambda$ (and thus in the state). Normalization implies $\mathrm{tr}\Lambda=1$, which then yields the relation above between the two traces.
@NorbertSchuch, great thanks! All is good under the sun. (The issue it's not entirely 'accademic'. We may decide to normalize probabilities to $\lambda$. Indeed in some communities they pick $\lambda=100$). Have a good day
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