Universal Geometric Entanglement Close to Quantum Phase Transitions

Under successive renormalization group transformations applied to a quantum state $|\Psi ⟩$ of finite correlation length $\xi$, there is typically a loss of entanglement after each iteration. How good it is then to replace $|\Psi ⟩$ by a product state at every step of the process? In this Letter we give a quantitative answer to this question by providing first analytical and general proofs that, for translationally invariant quantum systems in one spatial dimension, the global geometric entanglement per region of size $L\gg \xi$ diverges with the correlation length as $(c/12)\log (\xi /ϵ)$ close to a quantum critical point with central charge $c$, where $ϵ$ is a cutoff at short distances. Moreover, the situation at criticality is also discussed and an upper bound on the critical global geometric entanglement is provided in terms of a logarithmic function of $L$.

Figure 1

An infinite one-dimensional quantum system is divided into different parties corresponding to contiguous blocks of size $L$.

Figure 2

Diagrammatic representation of the components $A(L{)}_{(\alpha {\alpha }^{\prime }),(\beta {\beta }^{\prime })}$ of matrix $A(L)$ for a block of size $L$. In the diagram, the violet diamonds correspond to the components ${\sqrt{D}}_{\alpha \beta }$ of matrix $\sqrt{D}$ and the half-ellipses correspond to quantum states $|{\tau }_{\alpha \beta }⟩$. The scalar product between two quantum states is represented by two half-ellipses together, one of them representing the bra $⟨|$ and the other the ket $|⟩$. The different emergent lines correspond to the Greek indices $\alpha$, $\beta \dots$, where common legs between objects correspond to shared summed indices, and free legs correspond to free indices.

Figure 3

The repeated multiplication of matrix $A(L)$ for a block of size $L$ produces the same matrix but for blocks of larger size, i.e., $A(kL)=A^k(L)$, for $k=2,3\dots$.