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Quantum Percolation in Magnetic Fields

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Quantum Percolation in Magnetic Fields University of Pennsylvania ScholarlyCommons Department of Physics Papers Department of Physics 3-3-1986 Quantum Percolation in Magnetic Fields Yigal Meir Amnon Aharony A. Brooks Harris University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/physics_papers Part of the Quantum Physics Commons Recommended Citation Meir, Y., Aharony, A., & Harris, A. (1986). Quantum Percolation in Magnetic Fields. Physical Review Letters, 56 (9), 976-979. http://dx.doi.org/10.1103/PhysRevLett.56.976 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/311 For more information, please contact [email protected]. Quantum Percolation in Magnetic Fields Abstract A generalized average inverse participation ratio, for the one-electron wave functions of a dilute tight- binding model on a d-dimensional hypercubic lattice, is studied at finite magnetic fields. Extended vwa e functions appear above a quantum threshold bond concentration, pq. This threshold decreases at small magnetic fields, and shows a periodic dependence on the magnetic flux through a basic plaquette, with period φ0=ℏc/e. Extended states appear and disappear periodically even at d=2. Disciplines Physics | Quantum Physics This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/311 Vm, UME 56, NUMsER9 PHYSICAL REVIEW LETTERS 3 MARcH 1986 Quantum Percolation in Magnetic Fields Yigal Meir and Amnon Aharony School of Physics and Astronomy, Tel A vi v University, Tel A vi v 699?8, Israel A. Brooks Harris Department of Physics, University ofPennsylvania, Philadelphia, Pennsylvania, I 9104 (Received 8 August 1985) A generalized average inverse participation ratio, for the one-electron wave functions of a dilute tight-binding model on a d-dimensional hypercubic lattice, is studied at finite magnetic fields. Ex- tended wave functions appear above a quantum threshold bond concentration, p~, This threshold decreases at small magnetic fields, and shows a periodic dependence on the magnetic flux through a basic plaquette, with period Pp=ltcle Ext. ended states appear and disappear periodically even at 0 = 2. PACS numbers: 71.50.+ t The effects of magnetic fields on the metal-insulator The oscillation of pp with H implies a range of con- Anderson transition have been the subject of much centrations in which a periodic sequence of metal- speculation, some experimental work, and little de- insulator transitions should be observed as function of ' 3 tailed theoretical understanding. In the absence of H. Unfortunately, if the lattice constant is 1 A then the field, some wave functions become localized be- the period in H is of order 109 Oe. Reasonable periods cause of destructive interference of waves along paral- may be realized for larger lattices, e.g. , of granular lel routes. These cancellations are eliminated when units. In any case, the decrease in pq for small finite H the phases of the functions are modified by the mag- should be observable experimentally. netic flux through loops, and therefore one expects Although our results in d =2 are somewhat less ac- that weak fields may delocalize the wave functions, curate, we find there, too, finite regions in H (around yielding a negative magnetoresistance 'It . is also be- [email protected]) where pp(H) decreases (from 1) down to lieved that critical exponents may have different about 0.7. Hopefully, this is also observable (at least values for finite fields. 4 A strong magnetic field, on in computer experiments?). the other hand, may shrink the wave functions and The usual H =0 quantum percolation Hamiltonian thus decrease the conductivity. 2 is written as In an apparently different direction, much recent ef- fort has been invested in the understanding of the periodic dependence of the conductivity of a simple ring on the magnetic flux through it.' " Depending where ~i) is the state on the site i of the lattice, while on the size of the ring and on the method of averaging the nearest-neighbor transfer energy tjo (for the elec- over the randomness, physical properties of the ring tron to hop from i to j) is equal to 1 (with probability show periodicity in the flux with a period of either p) or to 0 (with probability 1 —p). Unlike the earlier '-" 'z '6 @o tcje or @—o—/2. work, we now introduce also very small random In the present paper we combine these iwo diagonal energies, e;, with zero average. These re- phenomena by studying the effects of a magnetic field, H, on quantum percolation. 'z '6 Considering a dilute tight-binding one-electron model on a d-dimensional hypercubic lattice, with a concentration p of nonzero nearest-neighbor off-diagonal transfer energy ele- ments, we calculate the quantum threshold pp(H) above which extended states begin to appear. Our results, shown in Fig. 1 for d =3, indicate a periodic dependence of p~ on P, with period go. Here and below fields H are measured by their flux p= Haz through a unit cell, where a is the lattice constant. I I I I I Although the results are not periodic with period P,/2, .2 .6 .8 l there is a significant increase in also for half-integral pp FIG. 1. The quantum threshold p~ as function of the values of $/$o, when the magnetic phase factors are magnetic flux $/$p, for d = 3. The graph repeats periodical- real. ly, with period 1. 1986 The American Physical Society VoLUME 56, NUMaER 9 PHYSICAL REVIEW LETTERS 3 M&RcH 1986 move the degeneracies of the eigenstates, and thus tonian (1) on finite clusters, with up to eleven bonds, simplify the choice of the inverse participation ratio and find the eigenvalues E and the (normalized) (see below). They have no other effect on the results eigenfunctions tliF (i ) (which are now nondegenerate). [except in the close vicinity of $/fo —0, —, , where These are then used to calculate the inverse participa- e, = sin(2n @/@,) l. tion ratio, defined as Earlier work on (1)' ' showed that the electronic wave functions are all localized for p & pq, and some extended state appeared above the quantum threshold for the cluster I . If the eigenstate is p~ (which is higher than the geometrical one, p, ). In extended, particular, no extended states were found for any ilia (i) —1/JN (N is the number of sites on I ), then —Wz. — p & 1 at d =2. In the present work we present a re- y(l ) For a localized state, QF(i) 8;, and finement of Ref. 16, in which the problem with degen- y(I ) —N. We averaged y(I ) over many realizations erate eigenstates is removed by the diagonal e s. We of the random (small) e, 's, and found a rather fast calculate a modified average inverse participation ratio convergence to a sharp average value. We next as a power series in p, and show that it diverges for weighted the average y (I ) by p '(1 —p) ', where Nb — p p~ as (p~ p) ~. For H=0, our new results and N~ are the numbers of bonds inside and adjacent agree qualitatively with the previous ones, and we find to I', and find the average X(p), which is expected to —Nz. ' pq 0 35+002 0 22 0 16 0 13 0 11 and 009 (with diverge if y (I ) errors of + 0.01) and y = 2.2+ 0.2, 1.4, 1.2, 1.1, 1.0, For H = 0 we found eleven terms in the new series and 1.0 (with errors of + 0.1) for d = 3, 4, 5, 6, 7, and for X(p), and analyzed them using inhomogeneous 'o 8, respectively. The values of p~ agree with those of differential approximants. In the (L,M, N) approxi- Refs. 15 and 17. mant we solved the linear equation The magnetic field is introduced into the Hamiltoni- '9 P2 d X/dp + P t X = Po, (5) an (1) via phase factors on t,, Pp, Pt, and P2 being polynomials of degrees L, M, and iV. With the assumption that X —(p~ —p) ~, the ratio where A is the magnetic vector potential. In three P„/P2 is an approximant for y/(p~ —p). The esti- dimensions, with the magnetic field H along the z mates for 3 & d & 8 were quoted above. We found no direction, and in the Landau gauge, this becomes indication of a singularity below p=1 at d=2. We = also note that our results are inconclusive as to the i,, t, oexp(iHay/ito) (3) identification of the upper critical dimension, at which ' if r;~ is along the x direction, and lj = tj otherwise. y becomes equal to unity (this could happen anywhere In two dimensions we take the field to be perpendicu- for d ~ 6). lar to the xy plane. It is not trivial to generalize these We now turn to nonzero fields. We first note that the rules to d & 3. For the purpose of the present calcula- phase factors (3) do not affect any physical property of tion we chose to use the algorithm (3) for all d, i.e., to clusters which contain no loops. One can always project loops on a single (xy ) plane. choose new phases for ilia(i ), which will absorb those As explained in Ref. 16, we now solve the Hamil- arising from the fields, and ~tlirt(i ) ~ will not be affect- FIG. 2. The inverse participation ratio y(I') for the clus- FIG. 3. An example of different clusters which have the ter shown in the inset. same topology. VOLUME 56, NUMsER 9 PHYSICAL REVIEW LETTERS 3 M~RcH 1986 TABLE I.
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