Comment on "Effective confining potential of quantum states in disordered media"
11 Comment on “Effective ConfiningPotential of Quantum States inDisordered Media” [Phys. Rev. Lett. ,056602 (2016)]In the Letter [1], the inverse of the landscape function u ( x ) introduced in Ref. [2] was shown to play the role ofan effective potential. This leads to the following estima-tion of the integrated density of states (IDoS), in 1D, N ADJMF ( E ) = 1 π (cid:90) u ( x ) > /E d x (cid:112) E − /u ( x ) . (1)We consider here two disordered models for which weobtain the distribution of u ( x ) and argue that the precisespectral singularities are not reproduced by Eq. (1). Pieces model.—
We consider the Schr¨odinger Hamilto-nian H = − d / d x + (cid:80) n v n δ ( x − x n ), where the posi-tions of the δ potentials are independently and uniformlydistributed on [0 , L ] with mean density ρ . The land-scape function, which solves Hu ( x ) = 1, is thus parabolicon each free interval. In the limit v n → + ∞ (“piecesmodel”), intervals between impurities decouple and IDoSper unit length is N ( E ) = lim L →∞ (1 /L ) N ( E ) = ρ/ (cid:2) e πρ/ √ E − (cid:3) [3]. We compare it with (1). Assum-ing now ordered positions, x < x < · · · , we have u ( x ) = (1 / x − x n − )( x n − x ) for x ∈ [ x n − , x n ]. Wefirst study its distribution P ( u ) = (cid:104) δ ( u − u ( x ) (cid:105) . Thedisorder average can be replaced by a spatial average, P ( u ) = ρ (cid:82) ∞ d (cid:96) e − ρ(cid:96) (cid:82) (cid:96) d x δ ( u − x ( (cid:96) − x ) / P ( u ) = 4 ρ K ( ρ √ u ) , (2)where K ν ( z ) is the MacDonald function. Denoting by θ H ( x ) the Heaviside function, we can now deduce theestimate N ADJMF ( E ) = (1 /π ) (cid:104) (cid:112) E − /u θ H ( E − /u ) (cid:105) : N ADJMF ( k ) = kπ (cid:90) ∞ ξ d t (cid:112) t − ξ K ( t ) for ξ = ρ √ k (3)For k = √ E (cid:29) ρ , we get N ADJMF ( k ) (cid:39) k/π , as itshould. For low energy, k (cid:28) ρ , one gets N ADJMF ( k ) (cid:39) ( k/
2) exp {−√ ρ/k } , which is a rather poor approxima-tion of the Lifshitz tail N ( k ) (cid:39) ρ exp {− πρ/k } : the co-efficient in the exponential is underestimated and the pre-exponential function incorrect, thus overestimating theIDoS by an exponential factor. Supersymmetric quantum mechanics.—
We consider theHamiltonian [4] H = Q † Q , where Q = − ∂ x + m ( x ). Theanalysis is more simple for boundary conditions ψ (0) = 0& Qψ ( L ) = 0, leading to the Green’s function G ( x, y ) = (cid:104) x | H − | y (cid:105) = ψ ( x ) ψ ( y ) (cid:82) min( x,y )0 d z ψ ( z ) − , where ψ ( x ) = exp (cid:8) (cid:82) x d t m ( t ) (cid:9) . We study u ( x ) = (cid:82) L d y G ( x, y ) when m ( x ) is a Gaussian white noise with (cid:104) m ( x ) (cid:105) = µ g and (cid:104) m ( x ) m ( x (cid:48) ) (cid:105) c = g δ ( x − x (cid:48) ), thus B ( x ) = (cid:82) x d t m ( t ) is a Brownian motion (BM) with drift µ (in Ref. [5], the more regular case with m ( x ) being arandom telegraph process was considered, leading to thesame low energy properties). We have u ( x ) = e B ( x ) (cid:26) (cid:90) x d y e B ( y ) (cid:90) y d z e − B ( z ) + (cid:90) x d y e − B ( y ) (cid:90) Lx d z e B ( z ) (cid:27) ≡ u < ( x ) + u > ( x ) (4)The cases µ (cid:62) µ < u ( x ) growwith x for µ (cid:62) (cid:104) ln u ( x ) (cid:105) (cid:39) µ gx + cst for µ > µ <
0. We first discuss the term u > ( x ) = (cid:82) Lx d y G ( x, y ) of (4), which is the product of two inde-pendent exponential functionals of the BM u > ( x ) (law) =(4 /g ) Z ( − µ ) gx (cid:101) Z ( − µ ) g ( L − x ) / , where Z ( µ ) L = (cid:82) L d t e − µt +2 W ( t ) , W ( t ) being a Wiener process (a normalized BM with nodrift). The n th moment of Z ( µ ) L is ∼ e n ( n − µ ) L [6], thus (cid:104) u > ( x ) n (cid:105) ∼ exp (cid:8) n g ( L + 3 x ) + n µ g ( L + x ) (cid:9) , whichsuggests a log-normal tail. For µ (cid:62)
0, there is no limitlaw and u > ( x ) grows exponentially, hence the bound ofthe landscape approach is useless. For µ <
0, 1 /Z ( − µ ) ∞ is distributed by a Gamma law [6] and we get the exactdistribution of u > ( x ) for x & L − x → ∞ : P > ( u ) = 2 g − | µ | u − − | µ | / Γ( | µ | )Γ(2 | µ | ) K | µ | (cid:18) g √ u (cid:19) ∼ u →∞ u − −| µ | . (5) u < ( x ) = (cid:82) x d y G ( x, y ) should have the same statisticalproperties, as confirmed numerically. Although u > ( x )and u < ( x ) are correlated, the distribution of their sumis expected to present the same power law tail P ( u ) ∼ u − −| µ | , what we checked numerically.We now apply (1) : for µ (cid:62) u ( x ) has not limit lawwhen x & L − x → ∞ and the distribution of W = 1 /u ( x )converges to δ ( W ), hence N ADJMF ( E ) = √ E/π . For µ <
0, we get N ADJMF ( E ) = (1 /π ) (cid:82) ∞ /E d u P ( u ) (cid:112) E − /u ∼ E | µ | +1 / for E →
0, while the exact IDoS behaves as N ( E ) ∼ E | µ | [7]. Hence, Eq. (1) predicts a power lawwith an incorrect exponent, i.e. underestimates the IDoS.For boundary conditions ψ (0) = ψ ( L ) = 0, wehave also obtained P ( u ) ∼ u − −| µ | and N ADJMF ( E ) ∼ E | µ | +1 / , independently of the sign of µ in this case.Alain Comtet and Christophe Texier LPTMS,Universit´e Paris-Saclay, CNRS,F-91405 Orsay, France
Appendix (arXiv version) : numerics.—
The form(4) is appropriate for numerical simultation. In the insetof Fig. 1, we plot the result of a numerical simulationfor µ < (cid:104) ln u ( x ) (cid:105) , which is uniform in the bulk (while for a r X i v : . [ c ond - m a t . d i s - nn ] M a y µ >
0, it grows linearly, (cid:104) ln u ( x ) (cid:105) (cid:39) µ gx + cst). Then westudy its distribution and check the limiting behaviour P ( u ) ∼ u − −| µ | . ooooooooooo oo oo oo oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo o o o ⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳⊳ ×××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××××× u u ∞ P ( t ) ⅆ t x / L l n ( u ) Figure 1:
Cumulative distribution (cid:82) ∞ u d u (cid:48) P ( u (cid:48) ) of the land-scape function for drifts µ = − . , µ = − . , µ = − and µ = − . ( gL = 100 ; n s = 10 disorder realizations). Straightlines correspond to the power law u −| µ | . Inset : ln u ( x ) for gL = 200 and µ = − . (red line), and (cid:104) ln u ( x ) (cid:105) after aver-aging over n s = 50 000 realizations (blue line). [1] D. N. Arnold, G. David, D. Jerison, S. Mayboroda, andM. Filoche, Effective Confining Potential of Quantum States in Disordered Media, Phys. Rev. Lett. , 056602(2016).[2] M. Filoche and S. Mayboroda, Universal mechanism forAnderson and weak localization, Proc. Natl. Acad. Sci.U.S.A. (37), 14761 (2012).[3] Yu. A. Bychkov and A. M. Dykhne, Electron spectrum ina one-dimensional system with randomly arranged scat-tering centers, Pis’ma Zh. Eksp. Teor. Fiz. , 313 (1966) ;J. M. Luttinger and H. K. Sy, Low-lying energy spectrumof a one-dimensional disordered system, Phys. Rev. A ,701 (1973) ; C. Texier and C. Hagendorf, One-dimensionalclassical diffusion in a random force field with weakly con-centrated absorbers, Europhys. Lett. , 37011 (2009).[4] A. Comtet and C. Texier, One-dimensional disordered su-persymmetric quantum mechanics: a brief survey, in Su-persymmetry and Integrable Models , edited by H. Aratyn et al. , Lecture Notes in Physics, Vol. , pages 313–328,Springer, 1998 (available as arXiv:cond-mat/97 07 313).[5] A. Comtet, J. Desbois, and C. Monthus, Localizationproperties in one-dimensional disordered supersymmetricquantum mechanics, Ann. Phys. (N.Y.) , 312 (1995).[6] C. Monthus and A. Comtet, On the flux distribution ina one-dimensional disordered system, J. Phys. I (France) (6), 635 (1994) ; A. Comtet, C. Monthus, and M. Yor, Ex-ponential functionals of Brownian motion and disorderedsystems, J. Appl. Probab. , 255 (1998).[7] J.-P. Bouchaud, A. Comtet, A. Georges, and P. Le Dous-sal, Classical diffusion of a particle in a one-dimensionalrandom force field, Ann. Phys. (N.Y.)201