Correlación de Pearson

Recently, MBL has been studied mostly in spin chain systems, which exhibit the relevant physics in a simple setting. The one-dimensional spin-1/2 XXZ chain is an extensively studied model, which has the Hamiltonian:

HXXZ = L X n=1 h sx_nsx_n+1+sy_nsy_n+1+ ∆sz_nsz_n+1+h_nsz_ni, (1.16)

wheresµn=σµn/2is the operator for theµcomponent of the spin on siten,∆ is the Ising anisotropy,hnis a random on-site field drawn from a uniform distributionhn∈[−W, W], andLis the length of the chain. This expression corresponds to periodic boundary conditions, with siteL+1being equivalent to site1. (Note that this is by no means the only spin system in which MBL has been studied [41, 45, 49–53].) This model can be mapped to a system of spinless fermions on a one-dimensional lattice using the Jordan-Wigner transformation [54]: sz_{n ←→} c†_nc_n−1 2; s+ n ←→ c†neiπ Pn−1 j=1c†jcj_; (1.17) s_n− ←→ e−iπPnj=1−1c†jcj_c n,

wherec†nandcnare the creation and annihilation operators for a fermion on sitenrespectively. This gives an alternative representation of the Hamiltonian

(up to irrelevant constants): HXXZ= L X n=1 " 1 2 c†_nc_n+1+c†_n+1c_n+ ∆c_n†c_nc†_n+1c_n+1+hnc†ncn # . (1.18)

In this language it is clear that∆corresponds to the strength of a nearest- neighbour interaction. We will only discuss the physics of one-dimensional models, as these are the most thoroughly investigated and well understood systems, and they are the focus of the work in the following chapters.

Numerical studies of this system indicate that in the isotropic ‘Heisenberg’ limit (∆ = 1) the MBL transition occurs atWc '3.7, and suggest the exis- tence of a mobility edge at weaker disorder [55–57]. Fig. 1.1 [55] shows the phase diagram of this model, as determined by a finite-size scaling analysis of the energy spectrum and eigenstate properties on system sizes up toL= 22. In this study, the transition was determined by analysing the structure of the energy spectrum, the half-chain entanglement entropy and the size of its fluctuations, the Shannon entropy, and fluctuations in the half-chain mag- netisation. We will discuss some of these measures in the next section. These types of study, despite pushing the boundaries of what is achievable by exact numerical methods, are still far from the thermodynamic limit. The extracted critical exponents violate bounds derived under quite general assumptions (i.e. that the MBL transition separates a localised phase from a phase that satisfies the ETH) [58], suggesting that the scaling regime has not yet been reached, and the possibility of the existence of a many-body mobility edge has been questioned [59].

Recently, attention has turned to the delocalised region withW < Wc, where a variety of interesting physics may be studied [17]. Several numerical studies have observed anomalous subdiffusive spin transport in the XXZ model [60–62], most effectively by studying the scaling of the spin current with system length j ∼ L−γ (γ = 1 corresponds to diffusive transport, γ > 1 corresponds to subdiffusive transport, and γ diverges at the MBL

FIGURE 1.1: The phase diagram of the XXZ model (1.16) as a function of disorder strength hand energy density , showing the two phases and the mobility edge. The critical disorder strength was determined from an exact diagonali- sation study of system sizes up toL = 22. The location of

the MBL transition is determined from the spectral statis- tics (turquoise upwards-pointing triangles), entanglement entropy (red squares), fluctuations of the entanglement en- tropy (green circles), the decay of a long-wavelength spin density (blue downwards-pointing triangles), and the fluctu- ations of the half-chain magnetisation (yellow left-pointing triangles). The colour scheme indicates the scaling of the Shannon entropy with system size (D1in the notation used

transition). The transport is subdiffusive for all disorder strengths in the strongly interacting regime (∆ > 1), while a transition between diffusive and subdiffusive transport exists at finiteW in weakly interacting systems (∆ <1) [62]. This anomalous transport has been linked to long-tailed (i.e. non-Gaussian) distributions for the off-diagonal matrix elements of local spin operators in the ergodic phase [63]. It was found that these systems satisfy a modified version of the ETH in the subdiffusive regime, in which the scaling of the variance of the off-diagonal matrix elements with system size requires power-law corrections to the exponential in (1.14) [63]. Furthermore, it has been shown that energy transport is diffusive when the disorder is weak, and the system undergoes a transition to subdiffusive transport at a finite disorder strength (which depends on the strength of the interactions) before the MBL transition [64–66]. Subdiffusive spin transport has also been observed in a periodically driven Floquet version of the model, with associated long-tailed distributions of the matrix elements of local spin operators in the eigenstates of the Floquet unitary [67]. It is interesting to note that in this system, which does not conserve energy, the MBL phase is still present [68–72].