# Partial Transpose in Gapped Time-reversal Symmetric Spin Chains

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

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

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.

EDIT: I since have found a couple of references. The earliest one seems to be Pollman and Turner who did the above calculation. It also shows up in this more comprehensive account by Ryu et al..

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.

EDIT: As was shown by Fannes et al, the translation invariant pure states are $$w*$$-dense in the set of all translation invariant states. It seems that this notion of convergence will not be tight enough, at least so far i could only check that $$Z$$ is a continuous on Hilbert space with its norm given by the inner product. I would expect a stronger convergence to hold for exponentially clustering states.

• 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. – Anyon Apr 11 at 12:41
• Do I understand correctly that a special case would be the "normal" partial transpose, with a wavefunction with real coefficients? – Norbert Schuch Apr 11 at 13:30
• A real structure is just a way to formalize what is meant when we say a wave function has real coefficients, so yes. – Lorenz Mayer Apr 11 at 13:43
• 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. – Lorenz Mayer Apr 11 at 14:07
• 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. – Lorenz Mayer Apr 11 at 14:14