Spontaneous Pseudospin Spiral Order in Bilayer Quantum Hall Systems

a r X i v :c o n d -m a t /0304417v 2 [c o n d -m a t .m e s -h a l l ] 30 J a n 2004

Spontaneous Pseudospin Spiral Order in the Bilayer Quantum Hall Systems

K.Park

Condensed Matter Theory Center,Department of Physics,University of Maryland,College Park,MD 20742-4111

(Dated:February 2,2008)

Using exact diagonalization of bilayer quantum Hall systems at total ?lling factor νT =1in the torus geometry,we show that there is a new long-range interlayer phase coherence due to spontaneous pseudospin spiral order at interlayer distances larger than the critical value at which the pseudospin ferromagnetic order is destroyed.We emphasize the distinction between the interlayer phase coherence and the pseudospin ferromagnetic order.

PACS numbers:73.43.-f,73.21.-b

I.INTRODUCTION

Bilayer quantum Hall systems at total ?lling factor νT =1exhibit one of the most interesting many-body correlation e?ects:spontaneous interlayer phase coher-ence,which is solely caused by the Coulomb interaction in the limit of zero interlayer tunneling.Recently this phenomenon has received drastic experimental support by Spielman et al.1,who discovered a strong enhance-ment in the zero-bias interlayer tunneling conductance for interlayer distance d

2 our predicted pseudospin spiral order may have been al-

ready observed experimentally.

II.HAMILTONIAN

Let us begin by writing the Hamiltonian for the bilayer

quantum Hall systems:

H=H t+?P LLL V Coul?P LLL(1)

where?P LLL is the lowest Landau level(LLL)projection

operator.V Coul represents the usual Coulomb interaction

between electrons:

V Coul

r ij

+ k

c/eB is the magnetic length,d is the inter-

layer distance,and r ij is the lateral separation between

the i-th and j-th electrons.In the above we have used a

pseudospin representation to distinguish the top(↑)and

the bottom(↓)layers.In general,we de?ne the pseu-

dospin operator as follows:

S≡

1

2 m c?m↑c m↓+c?m↓c m↑ =?tS x,(4)

where t is the single particle interlayer tunneling gap. Although Eq.(4)is valid for general t,we are inter-ested only in the limit of zero interlayer tunneling,i.e. t/(e2/?l0)→0,which is appropriate when considering spontaneous interlayer coherence(note that the t→0 limit is not the same as the t=0situation).We analyze the Hamiltonian in Eq.(1)by using exact diagonalization (via a modi?ed Lanczos method)in the torus geometry13.

III.SPONTANEOUS PSEUDOSPIN

MAGNETIZATION

As mentioned in the beginning,it will be shown that there is a new long-range interlayer phase coherence due to pseudospin spiral order at su?ciently large d.But,?rst,we should determine the critical interlayer distance, d c,at which pseudospin ferromagnetic order terminates. Of course,the most natural order parameter for the pseu-dospin ferromagnetic order is the pseudospin magnetiza-tion in the x-direction: S x .

It is tempting to apply exact diagonalization tech-niques to?nite systems with?xed number of electrons in each layer.But,S x changes the interlayer number dif-ference by unity(?S z=±1)so that the ground state ex-pectation value is precisely zero when computed naively without any explicit interlayer tunneling.This problem has been addressed in an ad hoc manner by introducing explicit interlayer tunneling11,which,however,severely obscures the e?ect of spontaneous phase coherence and may produce misleading results.The real solution is to realize that there is intrinsic,quantum-mechanical uncer-tainty in the layer index of electrons at small interlayer distance(even in the limit of zero tunneling)so that the true ground state|ψ is a linear combination of various states with di?erent S z:

|ψ = MλM|φM ,(5)

where|φM is the lowest energy state with S z=M,λM is a sharply peaked function of M with width O(

√

√

N(N+2)/2for N even(M?=0).Therefore, the pseudospin magnetization should be scaled as N+1 (

3

1

23

d/l 0

00.10.20.30.40.5

0.6

p e r p a r t i c l e

FIG.1:Pseudospin magnetization (S x )as a function of in-terlayer distance d/l 0for various numbers of electrons N .

magnetization S x is computed for the ?rst time in this article without any interlayer tunneling.And therefore d c is reliably determined without any ambiguity.

IV.

PSEUDOSPIN SPIRAL INSTABILITY

We now address the nature of instability causing the destruction of pseudospin ferromagnetic order.This question is,in turn,directly related to the nature of the new ground state at d >d c .To be speci?c,we con-sider the lowest energy excitations

which are responsible for the ground

state instability.Fig.2shows the disper-sion curves of the lowest energy excitations at d/l 0=0.5and 1.5for a ?nite system with the total number of elec-trons N =13and the number of pseudospin-up electrons N ↑=7(or N ↓=6).There are several points to be em-phasized.

First,there is clearly a linearly dispersing Goldstone mode at small interlayer distances,such as d/l 0=0.5.However,withinnumericalaccuracy regarding the discreteness of ?nite-system wavevectors,the Goldstone mode seems to vanish for d d c ,which coincides with the destruction of pseudospin ferromagnetic order 14.Consis-tent with the conclusion from S x ,this shows that the ground state undergoes a phase transition around d =d c .Second,contrary to the prediction of the random phase approximation (RPA)theory 6,there is no softening of the dispersion curve (roton)at any ?nite momentum for any interlayer distance.Remember that the RPA theory predicts a roton around kl 0?1.4,which was believed to cause an instability toward charge density wave order.This already shows that previous theories have some se-rious ?aws in describing the phase transition at d =d c .We will show in the next paragraph that there are com-pletely di?erent low-energy excitations at large d/l 0.Third,several previous numerical studies 8,17were un-fortunately based on the exact diagonalization of N =8system,which,under careful investigation 18,can be shown to predict a Wigner crystal at large d rather than

00.150.30.450.6

?E /(e 2

/εl 0)

|k|l 0

00.030.060.090.12

?E /(e 2

/εl 0)

FIG.2:Dispersion curves of the lowest energy excitations at interlayer distance d/l 0=0.5and 1.5for a ?nite system with the total number of electrons N =13and the number of pseudospin-up electrons N ↑=7(or N ↓=6).It will be shown that the lowest energy state in the momentum channel k l 0=

2π/N (N ↓,0)(denoted

by arrows in Fig.2)begins to be ripped out of the well-de?ned dispersion curve at d/l 0?0.5,and it becomes completely separated from all the other excitations at d/l 0 1.5so that it becomes really the lowest energy excitation among all momentum channels.The same be-havior is found in all studied systems with di?erent values of N and N ↑,with exception of the N =8system due to a ?nite-size artifact as mentioned in the above.Note that there are three other degenerate excitations in the mo-mentum channels k l 0= 2π/N (0,N ↓),and

N ,which diverges in

the thermodynamic limit.We will show later that these peculiar properties are precisely the properties of pseu-dospin spiral state.

In order to understand why these excitations are the pseudospin spiral states,it would be best if we ?rst ex-

4

01

2

3

d/l 0

0.20.40.60.8

1O v e r l a p

FIG.3:Overlap between the pseudospin spiral wavefunction (described in the text)and the lowest energy state in the Hilbert space of momentum channel k l 0=

√

2πN at νT =1).We will show

later in Fig.3by explicitly computing the overlap be-tween |ψspiral (n =1) and the lowest-energy excitations (denoted by arrows in Fig.2)that the lowest-energy ex-citations are the pseudospin spiral states.But,?rst,we would like to emphasize that the pseudospin spiral state |ψspiral (n ) is the exact ground state in the presence of parallel magnetic ?eld 9.A surprising result of this ar-ticle is that,even without parallel magnetic ?eld,spiral states are the lowest energy excitations which cause the instability of pseudospin ferromagnetic order.

To elucidate the physical meaning of pseudospin spiral state,examine Eq.(8)which reveals that |ψspiral con-tains diagonal interlayer correlations between the states with p x =2π(m +n )/L and 2πm/L .Remember that the interlayer correlation between p x =2π(m +n )/L and 2πm/L is identical to the diagonal interlayer correla-tion between electrons with di?erent pseudospins which are separated laterally in the y -direction by nl 0

√

L

x

1

.(9)

Pseudospin spiral excitations (and therefore the diago-nal interlayer correlation)are schematically depicted in Fig.4.

Now we compute the momentum of pseudospin spi-ral states by directly applying the magnetic translation operator onto |ψspiral (n ) .The explicit algebra

shows

(a)

(c)

(b)

FIG.4:Schematic diagram for (a)the pesudospin ferromag-netic state (note the vertical interlayer correlation),(b)the pseudospin spiral state with clockwise spiraling direction,and (c)the pseudospin spiral state with anti-clockwise spiraling direction.Ellipses indicate the Coulomb correlation of neu-tral pairs between electron in one layer and hole in the other layer.It is shown in the text that there is an interlayer co-herence due to pseudospin spiral order at d >d c ,where the ground state is conjectured to be a bound state of two pseu-dospin spiral states with opposite winding direction,i.e.a bound state of (b)and (c).

that the momentum of ψspiral (n )is |k |l 0=nN ↑

N/l 0.Therefore,the pseudospin spiral states with two di?erent long spiraling periods do not occur in adjacent momenta.

It is important to note that |ψspiral (n =1) is the pseu-dospin spiral excitation in the limit of long spiraling pe-riod (=L ),which initiates the destruction of pseudospin ferromagnetic order.Therefore,it may be indicative of a second order phase transition 19.

As mentioned earlier,Fig.3shows the overlap be-tween the pseudospin spiral state |ψspiral (n =1) in Eq.(8)and the lowest-energy excited state with k l 0=

5

01

234

d/l 0

0.005

0.01

0.0150.02

>/N

2

FIG.5:Ground state expectation value of the new order

parameter (T 2

n with n =1)measuring the diagonal interlayer correlation.Note that the diagonal interlayer correlation is physically equivalent to the pseudospin spiral order.

It should be noted that,though the overlap is lowered at larger d/l 0,the low energy excitation is in general adi-abatically connected to the pseudospin spiral state.In fact,the reason for lower overlaps at larger d/l 0is that our trial wavefunction in Eq.(8)is not very accurate any-more for larger d/l 0since it is based on the assumption that the ground state is the (1,1,1)state.Of course,the ground state itself is a?ected by the pseudospin spi-ral instability which,after all,destroys the pseudospin ferromagnetic order at the critical interlayer distance.Therefore,it would be satisfactory if,in addition to the evidence due to the low energy excitation,one can prove directly that the ground state has the pseudospin spiral order at d ?d c .The next section will be devoted to the pseudospin spiral order in the ground state.

V.

PSEUDOSPIN SPIRAL ORDER IN THE

GROUND STATE

It is conjectured that the ground state at large d is in fact a bound state of two pseudospin spiral states with opposite winding direction,i.e.a bound state of (b)and (c)in Fig.4,which requires a complicated interaction be-tween the two spiral states (note that this conjecture is consistent with the fact that the ground state itself does not break translational symmetry).So instead of con-structing a trial wavefunction we take a more transpar-ent approach by de?ning a new order parameter which measures the pseudospin spiral order (or equivalently the diagonal interlayer correlation):

T n ≡

1

6

destroyed at d=d c.On the other hand,we have shown that there is a new interlayer phase coherence due to the pseudospin spiral order for d>d c which,we speculate, causes the longitudinal drag anomaly because of remain-ing interlayer correlation.Regarding the Hall resistance, our pseudospin spiral state is expected to be incompress-ible,as shown in the dispersion curve.However,the low-est energy gap is estimated to be very small so that the Hall plateau will be di?cult to be observed in current temperature ranges and impurity concentrations.

The author is very grateful to S.Das Sarma for his careful reading of manuscript and his constant support throughout this work.The author is also indebted to E. Demler,S.M.Girvin,A.Kaminski and V.W.Scarola for their insightful comments.This work was supported by ARDA.

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19In terms of the viewpoint that the interlayer coherent state is a macroscopic condensate of neutral electron-hole pairs, the pseusospin spiral instability with long spiral period can be interpreted as a gradual change in the nature of neutral pairs from tightly bound s-wave-like pairing to a pairing with extended size and higher symmetry.And this contin-uous change in the nature of pairing may indicate a second order phase transition which is consistent with the view-point from the pseusospin spiral instability with long spiral period.

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