Feature: Schmidt gap

[tanlin2013]

Characterizing the MBLT

The main quantity we compute, new in the context of MBL, is the Schmidt gap. For two chain halves (or as close to for odd L), A and B, as shown in Fig. 1(a), an eigenvector’s reduced density matrix is ρA,k = TrB(|Ek〉 〈Ek|) for a particular sample of the random fields. The disorder-averaged Schmidt gap is then defined as ∆ = 〈λk1 − λk2〉k, where λk1, λk2 refer to the largest eigenvalues of the reduced density matrix ρA,k, 〈·〉k denotes average over eigenstates and · denotes the average over many samples. The Schmidt gap has previously been shown to act as an order parameter for quantum phase transitions [34, 36]. We explore the possibility of using it for characterizing the MBLT. Unlike entanglement entropy, the Schmidt gap ignores most of the spectrum of ρA,k, describing only the relationship between the two dominant states across the A − B cut. This is pertinent in light of the recent finding that while the Schmidt values decay polynomially in the MBL phase [22], finite size corrections are stronger for small Schmidt values. In the ergodic phase we expect strong entanglement to produce multiple, equally likely orthogonal states, thus ∆ ∼ 0. In the MBL phase, however, a single dominant state should appear on either side of the cut, with ∆ rising towards 1 as h → ∞, implying a tensor product. This behaviour is shown in Fig. 2(a) and becomes becomes sharper with increasing L. To see this more vividly, we plot the derivative of ∆ with respect to h in Fig. 2(b). The derivative has a peak at h = ̃ hc, which not only becomes more pronounced but also shifts to the right with L. We infer this to be the finite size precursor to the transition point, which suggests that in the thermodynamic limit, L → ∞, the derivative of the Schmidt gap diverges at the MBLT and ̃ hc asymptotically approaches the transition point hc.

Difficulties

The problem for tsdrg approach is that it's hard to access reduced density matrix in the center of the system.
One may consider to use bipartite fluctuations as a probe to the entanglement.

If it's possible to derive the Schmidt gap in terms of the cumulants of fluctuations.

Reference

http://arxiv.org/abs/1704.00738
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.85.035409