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Register a new userFinite size scaling of entanglement entropy at the Anderson transition with interactions
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We study the entanglement entropy(EE) of disordered one-dimensional spinless fermions with attractive interactions. With intensive numerical calculation of the EE using the density matrix renormalization group method, we find clear signatures of the transition between the localized and delocalized phase. In the delocalized phase, the fluctuations of the EE becomes minimum and independent of the system size. Meanwhile the EEs logarithmic scaling behavior is found to recover to that of a clean system. We present a general scheme of finite size scaling of the EE at the critical regime of the Anderson transition, from which we extract the critical parameters of the transition with good accuracy, including the critical exponent, critical point and a power-law divergent localization length.
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This chapter describes the progress made during the past three decades in the finite size scaling analysis of the critical phenomena of the Anderson transition. The scaling theory of localisation and the Anderson model of localisation are briefly sketched. The finite size scaling method is described. Recent results for the critical exponents of the different symmetry classes are summarised. The importance of corrections to scaling are emphasised. A comparison with experiment is made, and a direction for future work is suggested.
In Ref.1 (Physical Review B 80, 041304(R) (2009)), we reported an estimate of the critical exponent for the divergence of the localization length at the quantum Hall transition that is significantly larger than those reported in the previous published work of other authors. In this paper, we update our finite size scaling analysis of the Chalker-Coddington model and suggest the origin of the previous underestimate by other authors. We also compare our results with the predictions of Lutken and Ross (Physics Letters B 653, 363 (2007)).
We describe a new multifractal finite size scaling (MFSS) procedure and its application to the Anderson localization-delocalization transition. MFSS permits the simultaneous estimation of the critical parameters and the multifractal exponents. Simulations of system sizes up to L^3=120^3 and involving nearly 10^6 independent wavefunctions have yielded unprecedented precision for the critical disorder W_c=16.530 (16.524,16.536) and the critical exponent nu=1.590 (1.579,1.602). We find that the multifractal exponents Delta_q exhibit a previously predicted symmetry relation and we confirm the non-parabolic nature of their spectrum. We explain in detail the MFSS procedure first introduced in our Letter [Phys. Rev. Lett. 105, 046403 (2010)] and, in addition, we show how to take account of correlations in the simulation data. The MFSS procedure is applicable to any continuous phase transition exhibiting multifractal fluctuations in the vicinity of the critical point.
We consider scaling of the entanglement entropy across a topological quantum phase transition in one dimension. The change of the topology manifests itself in a sub-leading term, which scales as $L^{-1/alpha}$ with the size of the subsystem $L$, here $alpha$ is the R{e}nyi index. This term reveals the universal scaling function $h_alpha(L/xi)$, where $xi$ is the correlation length, which is sensitive to the topological index.
Two exact relations between mutlifractal exponents are shown to hold at the critical point of the Anderson localization transition. The first relation implies a symmetry of the multifractal spectrum linking the multifractal exponents with indices $q<1/2$ to those with $q>1/2$. The second relation connects the wave function multifractality to that of Wigner delay times in a system with a lead attached.
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