incompressible quantum liquid versus quasi-two-dimensional electron solid

Physica B 184 (1993) 66-71 North-Holland PHYSICA

Incompressible quantum liquid versus quasi-two-dimensional electron solid

G u e n t h e r M e i s s n e r Department of Theoretical Physics, Universitiit des Saarlandes, Saarbriicken, Germany

A unified approach to study the nature of two condensed phases, an incompressible quantum liquid (IQL) of Bose condensed charge-vortex composites and a 2D quantum solid (QS) with a lattice-periodic structure of the guiding centers of Coulomb interacting electrons, is provided. By applying sum-rule techniques to the density and displacement fluctuations of the guiding centers at partial filling of the lowest Landau level and from calculations of ground-state energies and collective excitations, a competition between the two phases is found to possibly give rise to re-entrance behavior, if the mean number of magnetic flux quanta per electron exceeds a certain critical value.

1. Introduction

Strong correlat ions in a plane of interacting e lect rons are of significance in the limit of high magnet ic fields being applied perpendicular to that plane. Because of the well-known massive degeneracy associated with 2D free-electron mo- t ion of cyclot ron f requency w c, the e l ec t ron - e lec t ron interact ion within the lowest Landau level plays an impor tan t role. In actual systems [1] disorder potentials as e.g. at a hetero- junc- t ion surface, may also be essential. I f the disorder is small, however , the ensuing highly correlated mot ion of these electrons favors the forma- t ion of novel phases of condensed mat ter at t empera tu res (T) low enough compared to the cyclo t ron energy, i.e. k B T ~ h o ~ c, and at magnetic fields (B) sufficiently high such that the m e a n n u m b e r u - 1 = B/(4~one) of magnet ic flux quan ta ~b 0 = ch/e per electron exceeds one (n~: areal e lectron density, - e : electron charge) [2].

Thus , the many-body ground state of such 2D elec t ron systems with an odd number q = 3, 5, 7 . . . . o f flux quan ta per electron (i.e. the lowest L a n d a u level has rational filling factors u = 1/q)

Correspondence to: G. Meissner, Department of Theoretical Physics, Universit~t des Saarlandes, Bau 38, 6600 Saar- briicken, Germany.

turns out to be a corre la ted incompressible quantum liquid of Bose condensed charge-vortex composi tes [3]. The low-lying collective excita- t ions o f this I Q L - p h a s e exhibiting the fractional q u a n t u m Hall effect [4] (i.e. a quant ized Hall conduc tance o-12 = ve2/h accompanied by minima in the longitudinal conductance 0-11) are quadrupolar in nature , because of a peculiar noncom- mutivi ty of the density fluctuations of the guiding centers. This then implies a finite gap to exist exactly in the long-wavelength (k---~ 0) limit [5], in addi t ion to the so-called magne to - ro ton mini- m u m close to a reciprocal neares t -neighbor dis- tance, where the static s tructure factor has a max imum.

Below a certain critical filling factor Vc, a lat t ice-periodic structure of the guiding centers of the electrons may minimize repeatedly their C o u l o m b repulsion e2/erc at nonrat ional values giving rise to a 2D quan tum solid due to b roken magnet ic translat ional invariance [6] (e: back- g round dielectric constant ; r c = ( c h / e B ) 1/2 = (h / m*o~c)1/2: L a r m o r radius; m*: effective electron mass). In the limit u---~ 0, as the magnet ic field tends towards infinity, the ground-s ta te of that sort of a re-ent rant system may finally be identified with the tr iangular electron lattice fo rmed by a classical 2D Wigher crystal [7]. The low- lying collective mode of this QS-phase with a

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G. Meissner / Incompressible quantum liquid vs quasi-2D electron solid 67

nonvanishing shear modulus Ix > 0 is a Gold- stone mode restoring the broken magnetic trans- lational invariance. Therefore, it is gapless and can be identified with the magneto-phonon of the 2D Wigner crystal of the same dispersion k 3/2

in the limit k--)0 [8]. The liquid-solid transition may thus be expec-

ted to occur via softening of the magneto-roton minimum of the IQL-phase and the solid-liquid transition via softening of the shear modulus of the QS-phase, respectively. A unified approach to study the nature of the two condensed phases hence is provided by the dynamics of density and displacement fluctuations of the guiding centers of the interacting electrons. The single-mode approximation [9] in the IQL-phase and a self- consistent magneto-phonon approximation in the QS-phase have been shown to be the leading approximations of such an approach [10]. This actually reveals a possibility to relate our many- body theory to effective-field theories [11], similar to the familiar Ginzburg-Landau approach in phase transition problems.

In the IQL-phase, we have studied various renormalization effects on the guiding center density response evaluated with Laughlin's varia- tional ground-state wave function [12] in single- mode approximation. Results for the renormal- ized ground-state energy EL(/) ) and the magneto- roton dispersion tOE(k ) ar e compared with previous calculations [13] at rational filling factors v = 1/q for q = 3, 5, 7 and 9. In the QS-phase, effects of dispersive anharmonicities as well as of single-particle aspects on the ground-state energy Es(v ) and on the magneto-phonon dispersion tOs(k) have been investigated. Our estimates for the critical filling factor Pc from comparing total energy calculations in both phases, i.e. Es(v¢)= EL(Vc) , give values slightly higher than 1 [14]. Implications to be obtained, e.g., from the 1,- dependence of the shear modulus/x on a possible re-entrance behavior of the correlated electron system, are compared with experimental findings [15,16]. A discussion of modifications of this kind of a 2D Wigner crystal in high magnetic fields, to be expected from pinning due to disorder, must be deferred to a later publication, however.

2. Formal many-body theory

For the partially filled lowest Landau level, the wave-vector and frequency-dependent response function

Xaa(k, ~ = to + iv)

i f = g dt t), 0)1) (1) 0

of noncommuting density fluctuations

A(k) = ~ e x p ( - i k . X(l))

of the guiding centers X(l) of electrons with noncommuting Cartesian components is the fun- damental quantity in our unified many-body approach [5]. Thus, dispersion relations between frequencies w and wave vectors k of the intra- Landau-level collective excitations are to be determined from zeros of the real part of the inverse of that response function:

Re -1 Xaa(k, ~ = to(k))~-0. (2)

The ground-state energy as a function of the filling factor

E(~) = (9({X(O}))

= l f d2k 2 ~-~ v(k)(A(k)A(-k)) (3)

is determined by the static structure factor

(k) = < a ( k ) a ( - k) )

which at zero temperature can be obtained from the frequency integral

t t = X a(k,

( t

(4)

over the spectral function X~a(k, to) being given by the discontinuity of the complex response function Xaa(k, ~ ) at the real frequency axis.

68 G. Meissner / Incompressible quantum liquid vs quasi-2D electron solid

The bare 2D Coulomb repulsion 2rre2/elkl is modified in the interaction

v(k) = (2rrei/ek) 2 2 exp(- k rE~2 ) (5)

where s ~ denotes the antisymmetric tensor in 2D, the dynamics of density and displacement fluctuations of the guiding centers can now be explored.

of the guiding centers in eq. (3) by averaging over the fast cyclotron motion which also amounts to the relation between fluctuations of the averaged electron density

fi(k) = f d2r e-ik'~(r) = e-k2r[ /4A(k) (6)

and of the guiding center density A(k). It is this averaging procedure which casts the effective many-body Hamiltonian of N~ interacting electrons in a charge-compensating homogeneous background [17] into the form

( e2 ) / ~ = lhOOc -- - - ~ N e + ~ , ~ ( { X ( I ) } ) . (7)

EF L

A particular kind of particle-hole symmetry in the electron-electron interaction within the lowest Landau level finally allows to replace the averaged Coulomb repulsion v(k) in the interaction term V({X(/))) of eq. (7) rigorously by

r L Veff(k)= v(k)-~-~ dZk'eik'r'gv(k'), (8)

i.e. the effective interaction between the guiding centers Veff(k) < 0, if kr L > 1. This sort of competition between Coulomb repulsion and exchange attraction turns out to be a sufficient condition for the many-body system to become unstable against broken magnetic translation invariance at nonrational filling factors below a certain critical value ~,~ [6].

Together with the commutation relations of the density fluctuations

3. Rigorous results for the two phases

With the many-body approach of the previous chapter various rigorous results have been derived for both phases, the homogeneous IQL- phase exhibiting the fractional quantum Hall effect and the lattice-periodic QS-phase with a nonvanishing shear modulus [10].

In the IQL-phase of the uniform system of interacting electrons the exact expression

- o~ (k) + ~ ( k ) x ; ] ( k , O)

( do Faa(k, oJ) O) L ( k ) o~(k ) J 7 7 --

- 0 (11)

for the dispersions relation WL(k ) of magneto- rotons was obtained by applying sum-rule techniques. The spectral width function Fan/> 0 in eq. (11) has the same symmetry properties as the

tt spectral function Xaa. For the inverse of the Xaa(k, 0), in eq. static (g2 = 0) susceptibility, -1

(11) the rigorous sum rule

o~(k) ~(k) f d---w-w~r Faz(k' w)

-~ (12)

holds. Due to the quadrupolar nature of these collective excitations, both the first (n = 1) and third (n = 3) frequency moment

[A(k), A(k ' ) ] = - 2 i s in(k s s~,k'~r~/2)A(k + k ' ) , (9)

to be derived from those of the guiding centers [18],

[X~(I), X~(I')] =" 2 6 l rLEa/3 l l ' , (10)

+c¢

~,n(k) = f o~"x'~(k, o,) do~/~ : C.k* + . . .

- oc

(n = 1, 3) behave like k 4 in the limit k---~ 0 giving rise to a gap at k = 0 in the excitation spectrum ~OL(k = 0) ~ 0. Mathematically, this result holds


at least as long as the to-integral over Faa in eq. • - 4 • (12) for k--*0 diverges slower than k , if at all.

Combined with the sum rule (12) this then provides a rigorous proof that also the static susceptibility Xaa(k, O) = k 4, i.e. the long-wavelength density fluctuations are strongly suppressed in that IQL-phase.

In the QS-phase with a lattice-periodic structure of the expectation values R(1) =- (X(I) ) of the guiding centers of the electrons, an exact expression

-itoctos(k)G ~ + M ~ ( k , O )

- to~(k) f dto r ~ ( k , to)/to to - tos(k)

~ 0 (13)

for the dispersion relation tos(k) of the magneto- phonons was obtained. The static self-energy of the magneto-phonons

M.~(k,O) = b ~ ( k ) -

°

j dto y~t~(k, to)/to~ (14)

is exactly given by the difference between a generalized dynamical matrix

G~(k)

= e ~ r e ~ ( [ [ u ~ ( k ) , ~ ' ] , u ~ ( - k ) ] ) m * t o ~ h -2 , (15)

and the to-integral over matrix elements of the spectral width function 7~(k , to)>i O. The Car- tesian components of the guiding center displace- ments u~( l )= X ~ ( I ) - R~(I) in the fluctuations

Inserting eq. (16) into eq. (13) we readily obtain for the long-wavelength limit of the gapless Goldstone mode, being isotropic for a triangular lattice, the exact expression

to2(k) = detlM~, (k , 0)ltoc 2

2,rr e 2 - - _ _ , 2 2 q / zk3 + °(k4) • (17)

r r t (.I) c Iz

The isothermal shear modulus

I ~ = m*neZl122 (18)

in eq. (17) is rigorously related to the second derivative of the free energy with respect to displacement deformations according to a generalized elastic sum rule [19]. The nonvanishing shear modulus /~ > 0 in the dispersion relation tos(k)~l , t k 3/2 of the Goldstone mode, therefore, reveals the crystalline nature of that phase rather generally• In the limit u--~ 0, as the magnetic field tends towards infinity, the ground- state of the QS-phase becomes identical with that of a triangular electron lattice formed by a classical 2D Wigner crystal. Therefore, the shear modulus and the ground-state energy are found

• c l 2 3 / 2 - 1 to be exactly given as/x 0 = 0.25406e ne e and c l 1 / 2 2 • E o / N ~ = - 0 . 7 8 2 1 3 u (e /erE), respectively [7]. All of the foregoing results are exact. The

single-mode approximation [9] in the IQS-phase and a self-consistent magneto-phonon approximation in the QS-phase have been shown to be leading approximations of such a many-body approach [10].

4. Transitions between liquid and solid phases

u,~(k) = N e I/2 ~ G,(l) exp(ik. R(l)) l

obey the commutation relations (10). In the long-wavelength limit k ~ 0 we then find for this static self-energy

M.~(k, O) = 2,rre2nek~k~/(kem *)

+ Z ,~k~ ,k~ + o . /3 (k 4) . (16)

For the question whether sufficiently high magnetic fields may induce a transition from the quantum-liquid to the quantum-solid phase, possibly exhibiting re-entrance behavior in passing through rational values of the filling factor u, it is important to notice that there is a continuous symmetry in our correlated electron system, since the underlying Hamiltonian is invariant under infinitesimal magnetic translations [6]. Thus, the generator of magnetic translations is

70 G. Meissner / Incompressible quantum liquid vs quasi-2D electron solid

where pt=(h/i)a/Oxz denotes the canonical momentum operator of the lth electron with the position operator x t in the vector potential A(xt). If the expectation values R(1)= (X(l)) of the guiding center coordinates form some kind of lattice, however, this symmetry is broken sponta- neously. Since the expectation value (IS(r)) of the averaged electron density is no longer homogeneous in that case, we may conclude

([t~(r) , / i ts]) = h_ O(~(r)) # 0 (19) i

with the Cartesian components

• 2

D~ =V, + le~t3r~/2r L

of D apparently being related to Laughlin's quasi-particle and quasi-hole operators, i.e.

shows that a nontrivial solution, D (fi(r)) # 0, of eq. (20) requires the existence of a singular solution of eq. (21) at vanishing frequency S2 = 0.

This then first provides the critical equation

)~-1(rlr2; 0; TM) -- ~ ( r , r2 ;0 ; TM)--=0 (22)

for the relation between the melting temperature T M and the filling factor u in terms of a generalized polarization operator )~ and a generalized particle-hole interaction ~,. By analytic continua- tion to finite frequencies 12 # 0 this comparison, however, also quite generally reveals the possibility for the liquid-solid transition to occur via softening of the magneto-roton excitations tOL(k ) at a certain finite wave vector k = G # 0, becom- ing a reciprocal lattice vector G of the QS-phase.

5. D i s c u s s i o n and c o n c l u s i o n s

D~ = V~rL(D 1 +_ iD2) .

This actually reveals a possibility to relate our many-body theory to effective-field theories [11], similar to the familiar Ginzburg-Landau approach in phase transition problems, consider- ing, e.g., the IQL-phase as Bose condensed charge-vortex composites.

The broken magnetic translation invariance may then be used to derive the homogeneous integral equation

{~ ~(r,r2;O )-qj(rarz;O)}or2(~(r2))=-O. (20)

A comparison of this equation with

{)~-1(rlr2; a ) -- qJ(rlr2; g 2 ) } x ( r z r ; ; g2)

= t~(rl, r ; ) , (21)

which could be anticipated to hold for the response function of the averaged electron density

Since space limitation does not allow to exploit this new many-body approach in detail, we would rather like to conclude with a few re- marks.

Quite generally, so far a comparison between theory and experiments [15,16] seems to be rather promising. First, it is interesting to note that the familiar mean-field result [20], k B TM/(e2/ erE) = 0 . 5 5 7 u ( 1 - u), is easily recovered from eq. (22) in approximating the polarization operator by X0-GoGo using the free-electron Green's functions G O of the lowest Landau level and approximating the effective interaction g'0 by the effective potential Oef f of eq. (8), as shown previously already [21]. To take care of the strong correlations, however, a different starting point for )~ is necessary. Since the magneto-roton excitations in the single-mode approximation [9] exhibit a pronounced magneto-roton minimum, where the static structure factor So(k ) has a maximum [9,10], due to Feymman's relation

x(rlr:;/2 = to + b/)

= ~ dtem/{[P(r l o

, t ) , p ( r 2, 0 ) ] ) ,

toOL(k) h~ 2 = - - rk (k)/So(k),

we have investigated renormalization effects


using for the polarization operator the single- mode approximation, i.e.

- o3(k))~ol(k, 12) = 12 2 - O~Xol(k, 0 ) .

Since this amounts to a renormalization of the magneto-rotons of the form

~o~(k) = oJ~L(k)[1 + x0(k, 0)Ueff(k)] ,

we could find indications of re-entrance from a comparison of results obtained for the ground- state energy EL(v ) and for the magneto-roton dispersion wL(k ) with previous calculations [13] at rational filling factors v = 1/q for q = 3, 5, 7 and 9. Such effects are even further enhanced by including the interaction of magneto-rotons in g,. In the QS-phase a similar tendency was found from studying effects of dispersive anharmonicities as well as of single-particle aspects on the ground-state energy Es(v ), on the magneto- phonon dispersion a,s(k ) and on the shear modulus/z. However , a detailed analysis of all these investigations, and the inclusion of modifications to be expected from pinning due to disorder, e.g., must be deferred to a later publication [22].

In conclusion, therefore, we would just like to mention that our estimate for the critical filling factor u c from comparing total energy calculations in both phases, i.e. Es (vc)= EL(Uc), gives values slightly higher than 1 [14] and thus seems to be rather consistent with experimental findings, too.

Acknowledgements

I have benefitted from conversations with W. Apel, J. Hajdu, K. von Klitzing and G. Land- wehr. This work was supported in part by the Deutsche Forschungsgemeinschaft.

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incompressible quantum liquid versus quasi-two-dimensional electron solid

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