Effect of Laughlin correlations on crystalline mean field solutions of the 2DEG in FQHE regime
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Effect of Laughlin correlations on crystallinemean field solutions of the 2DEG in FQHE regime
Alejandro Cabo
Grupo de F´ısica Te´orica, Instituto de Cibern´etica,Matem´atica y F´ısica, Calle E,No. 309, Vedado, La Habana, CubaandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,
F. Claro
Facultad de F´ısica, Pontificia Universidad Cat´olica de Chile,Vicu˜na Mackenna 4860, Santiago, ChileandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy,
Danny Martinez-Pedrera
Deutsches Elektronen-Synchrotron (DESY),Notkestrasse 85, D-22607, Hamburg, GermanyandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
The energy per particle of many body wavefunctions that mix Laughlin liquid with crystallinecorrelations for periodic samples in the Haldane-Rezayi configuration is numerically evaluated forperiodic samples. The Monte Carlo algorithm is employed and the wave functions are constructed insuch a way that have the same zeroes as the periodic Laughlin states. Results with up to 16 particlesshow that these trial wavefunctions have lower energy than the periodic Laughlin states for finitesamples even at ν = . Preliminary results for 36 particles suggest that this tendency could reach thethermodynamic limit. These results get relevance in view of the very recent experimental measuresthat indicate the presence of periodic structures in the 2DEG for extremely small temperatures andclean samples, inclusive at main FQHE filling fractions ν = , .PACS numbers: 73.43.Cd,73.43.Nq, 73.22.Gk Although the Fractional Quantum Hall Effect is de-scribed in its relevant aspects in terms of Jastrow-likemany-body wave functions, a link between this varia-tional approach and hamiltonian solutions is still desir-able. The states introduced by Laughlin cleanly incorpo-rate the tendency of particles to be as far away as possi-ble from each other. On the other hand the Hartree-Fockstates discussed in the literature (See [1] and the refer-ences therein) are approximate solutions of the Hamilto-nian and may have valuable information not taken care ofby the variational states. With this possibility in mindwe have constructed a new type of trial wave functionthat incorporates a part of each approach as describedbelow. Precisely in these days this construction gets rel-evance thanks to very recent experiments that had de-tected the presence of surprising periodic structures inthe 2DEG for highly clean samples subject to extremelysmall temperatures [2]. Assuming that the proposed herewavefunctions maintain their properties of showing lowerenergies than the so called periodic Laughlin states (inthe Haldane-Rezayi periodic scheme [3]) in the thermo-dynamic limit, they could have the opportunity of de-scribing the detected periodic structures. Qualitatively,the energy dependence on the sample size obtained heresuggests that the energy is lower in the limit of infinite size samples. However, this important conclusion needsfor more extensive calculations for its full confirmation,that are expected to be considered elsewhere.Analytic Hartree-Fock solutions in the lowest Landaulevel may be written down for filling fractions of the form ν = 1 /q . This unusual property was discovered afternumerical results showed that there is a self-consistentcharge density wave solution (CDW) to the Hartee-Fock(HF) equations that has a zero of order q − q − N e , where N e is the number of electrons.Let us begin by writing the explicit form of the counter-parts of the Laughlin wavefunction in the Haldane-Rezayischeme for implementing periodic boundary conditionsΨ L = exp( − X i =1 , ...N e y i r o ) n ϑ ( πL ( Z ∗ − R ∗ ) | − τ ∗ ) o q × Y i 1) zeroes of this kind, plus q generated by the center of mass factor. We can thusreplace the factors with spatially fixed zeroes in the HFsolution, by a proper Laughlin factor to obtain the sameshort range behavior (a zero of order q ) when any twoparticles approach each another. However, the presencenow of the determinantal function keeps the crystallineinformation associated with the optimization of the meanfield problem. Therefore, the proposed states have an apriori chance of lowering the energy per particle of theLaughlin states.Then, in the Landau gauge A = − B ( y, , L ( z ∗ , z ∗ , ...z ∗ N e ) D [ z ∗ , z ∗ , ...z ∗ N e ] exp( − X i =1 , ...N e y i r o ))Φ L ( z ∗ , z ∗ , ...z ∗ N e ) = exp( − iQ Z ∗ ) n ϑ ( πL ( Z ∗ − R ∗ ) | − τ ∗ ) o q − Y i 6. The number of particles in the re-gion is N e = N . The momenta allowed by the periodicboundary conditions are k = n L s + n L s , where the reciprocal lattice unit vectors and the normalvector are defined by s = − qr n × a , s = 1 qr n × a n = (0 , , , a i · s j = 2 πδ ij , while the allowed values of n and n are n ∈ {− N , − N , .., , ..., N − } ,n ∈ {− N , − N , .., , ..., N − } . The evaluations of the energy per particle of bothstates were done by employing the Monte-Carlo methodfor samples having a number of particles N e equal to4 and 16 for the case of the Laughlin states. As forthe trial wavefunctions investigated here the calculationswere done for 4, 16 and 36 particles. The results are illus-trated in the Tables I and II and in Fig. 1. As mentionedbefore, the parameter ξ is the one defining the probabilityof admission of new configurations as usually is needed todo in the Monte-Carlo algorithm. The margins of errorsreported correspond to the maximum deviation from the TABLE I: The results of the energy per particle for the coun-terparts of the Laughlin states in the Haldane-Rezayi periodicboundary conditions. The state for N e = 4 was evaluated twotimes for different values of the Monte Carlo new configura-tion admission coefficient ξ in order to check the independenceof its value. The calculations were done for N e = 4 , 16 parti-cles N e = N ξ ǫ . − . ± . . − . ± . . − . ± . N e = 4 , 16 were evaluated two times each one fordifferent values of ξ to check the independence of the resulton this constant. The evaluations were done for N e = 4 , N e = N ξ ǫ . − . ± . . − . ± . . − . ± . . − . ± . . − . ± . mean value of a set of the last 60 percent of the evaluatedenergies in the Monte-Carlo iterative process.The expectation value of the many-particle Hamilto-nian in the Laughlin state was evaluated using the MonteCarlo method for samples with N e = 4 , 16. For our trialwavefunctions calculations were done for 4, 16 and 36particles. Results are shown in Tables I and II. The pa-rameter ξ defines the probability of admission of newconfigurations in the Monte-Carlo algorithm. The errorsreported in the third column correspond to the meansquare of the fluctuations in the Monte Carlo output af-ter convergence was assured.Our results are plotted in Fig. 1. The error bars are notresolved at the scale of the plot.Note that a linear extrapolation to the thermodynam-ical limit N − > ∞ of the Laughlin state energy repro-duces former estimates of the energy of this state at ν = 1 / €€€€€€€€€€€€€€€€€€€€€ (cid:143)!!!!!!!!! Ne - - - - e FIG. 1: Energy per particle as a function of the inverse squareroot of the number of particles N = √ N e . The picture showsthat the introduction of correlations in the HF crystallinestates made their energies lower than the ones shown by theversions of the Laughlin state in the Haldane-Rezayi periodicscheme. Note also that, if the behavior in the large N limitis confirmed by more extensive evaluations, the results willimply the existence of a ground state with a slightly lowerenergy than the Laughlin one for macroscopic samples. Thecurve joining the points of evaluated energies for the new trialstate is a fitting of these three points to a quadratic polyno-mial in 1 / √ N e . The lower straight line with negative slope issimply a linear curve of 1 / √ N e minimizing the mean squaredeviations from the three measured energies. extrapolation curve in the variable 1 /N which join thethere three evaluated points of the energy N = 2 , N . However, being the number ofMonte Carlo method iterations for the 36 particles state( N = 6) yet limited, this indication is not yet conclusiveand further numerical evaluations will be done to give abetter foundation to this conclusion. Its validity, clearlyleads to the idea about that the recently detected peri-odic structures in extremely perfect 2DEG at very lowtemperatures, could be associated to the here proposedtranslation symmetry breaking states [2].In ending we would like to underline that the presentletter consider a particular HF state showing one electronper its periodicity unit cell. For fillings of the form 1 /q the HF solution produces a gap for all integers q . Theexperiment suggests, however, that even and odd q valuesare qualitatively different states. It has been shown inthe past that if only half electron is captured by the unitcrystalline cell this distinction is properly borne out [5].Future work will extend to cover such states, and shallbe reported elsewhere.This work was preparared during two visits to the Ab-dus Salam International Centre for Theoretical Physics,Trieste, Italy. Support from the Condensed Matter Sec-tion is gratefully acknowledged. One of us (A.C.) thanksthe Caribbean Network on Quantum Mechanics, Parti-cles and Fields (Net-35) of the ICTP Office of ExternalActivities (OEA). F.C. acknowledges partial support re-ceived from Fondecyt, Grants 1060650 and 7060650, andthe Catholic University of Chile. [1] A. Cabo, F. Claro, A. P´erez and J. Maz´e, Phys. Rev. B76 , 075308 (2007). [2] O. E. Dial, R. C. Ashoori, L. N. Pfeiffer and K. W. West, Nature , 566 (2010).[3] F.D.M. Haldane and E.H. Rezayi, Phys. Rev. B31 , 2529(1985).[4] A. Cabo, F. Claro, arXiv:cond-mat/0702251(2007).[5] F. Claro, Phys. Rev.