Many-body localization transition: Schmidt gap, entanglement length, and scaling

Abstract

Many-body localization has become an important phenomenon for illuminating a potential rift between nonequilibrium quantum systems and statistical mechanics. However, the nature of the transition between ergodic and localized phases in models displaying many-body localization is not yet well understood. Assuming that this is a continuous transition, analytic results show that the length scale should diverge with a critical exponent ν 2 in one-dimensional systems. Interestingly, this is in stark contrast with all exact numerical studies which find ν ∼ 1. We introduce the Schmidt gap, new in this context, which scales near the transition with an exponent ν > 2 compatible with the analytical bound. We attribute this to an insensitivity to certain finite-size fluctuations, which remain significant in other quantities at the sizes accessible to exact numerical methods. Additionally, we find that a physical manifestation of the diverging length scale is apparent in the entanglement length computed using the logarithmic negativity between disjoint blocks.

Probing many-body localization with neural networks

“…Figure 4(b) shows the absolute value of theh-derivative of the Schmidt gap for the same system. As the Schmidt gap is small in the ETH phase and large in the MBL phase, itsh-derivative is expected to attain its maximum value at the transition [50]. Comparing Figs.…”

“…(i) The "Schmidt gap" λ 1 (ρ A ) − λ 2 (ρ A ), where {λ j (ρ A ); λ j λ j +1 } denotes the spectrum of ρ A . Being the difference of the two largest eigenvalues of the density matrix, i.e., of the square of the two largest coefficients in the Schmidt decomposition of the system into A and B, it is nearly 0 for mixed ρ A , typical for the ETH regime, and approximates 1 for almost pure ρ A , characteristic of the MBL phase [50]. .…”

Gumbel statistics for entanglement spectra of many-body localized eigenstates

“…Entanglement entropies can be extracted from entanglement spectra [14,15]. An entanglement spectrum encodes statistics beyond the entanglement entropy [16], of which several have been studied in the context of MBL [17][18][19][20][21][22][23]. The physical information encoded in the entanglement spectrum of a many-body localized eigenstate is almost fully carried by the smallest few elements [19], indicating the potential physical significance of the extreme value statistics [24].…”

Schmidt gap in random spin chains

“…It was further employed in the characterization of 2D spin models in a region close to a topological spin liquid [25,26]. The time evolution of the Schmidt gap was analyzed in [27][28][29] for the dynamics after a quantum quench in homogeneous systems and in [30] for a quench to a many-body localized Hamiltonian. Whether or not the Schmidt gap can be applied as an instrument to detect criticality in random models is still an open question.…”