Table Of ContentMean-field Phase Diagram of Two-dimensional Electrons with Disorder in a Weak
Magnetic Field
Igor S. Burmistrov
Landau Institute for Theoretical Physics, Kosygina str. 2, 117940 Moscow, Russia
3
0 Mikhail A. Baranov∗
0 Institute for Theoretical Physics, University of Hannover, D-30167, Germany
2
We study two-dimensional interacting electrons in a weak perpendicular magnetic field with the
n fillingfactorν ≫1andinthepresenceofaquencheddisorder. IntheframeworkoftheHartree-Fock
a
approximation,weobtainthemean-fieldphasediagramforthepartiallyfilledhighestLandaulevel.
J
We find that the CDW state can exist if the Landau level broadening 1/2τ does not exceed the
0
critical value 1/2τ = 0.038ω . Our analysis of weak crystallization corrections to the mean-field
c H
1 resultsshowsthatthesecorrectionsareoftheorderof(1/ν)2/3 ≪1andthereforecanbeneglected.
]
l PACSnumbers: 72.10.-d,73.20.Dx,73.40.Hm
l
a
h
- I. INTRODUCTION temptsweremadetoderivesuchatheorymicroscopically
s
starting from the mean-field solution12,13. The effects of
e
m A two-dimensional electron gas (2DEG) in a perpen- a quenched disorder on the unidirectional CDW state
(stripe phase) were investigated in the framework of the
. dicular magnetic field was a subject of intensive studies,
t phenomenological elasticity theory14, and a rich variety
a both theoretical and experimental, for several decades.
m The behaviour of the system in a strong magnetic field ofdifferentregimes,whichdependonthestrengthofdis-
order, were found. However, to identify the phenomeno-
- whereonlythelowestLandaulevelisoccupied,hasbeen
d investigated in great details1. Several attempts2,3 were logicalparametersofthetheory,asuccessivemicroscopic
n theory should be developed.
made in order to incorporate the case with larger fill-
o
ing factors ν > 1 into the theory. Usually in these ap- At present, a thorough microscopic analysis of the ef-
c
[ proaches, the ratio of the characteristic Coulomb energy fects of disorder on the mean-field transition from the
(at distances of the order of the magnetic length) to the uniform state to the CDW one, as well as on the phase
1
cyclotron energy, has been assumed to be small. How- diagram of the mean-field CDW states, is absent. The
v
ever,inaweakmagneticfieldthisisnotthecase,andthe mainobjectiveofthepresentpaperistoinvestigatethese
1
4 characteristic Coulomb energy exceeds the cyclotron en- effects on the existence of the mean-field CDW states in
1 ergy. An attempt to investigate the situation with large 2DEG in a weak perpendicular magnetic field H (filling
1 Coulomb energy was made in Ref.4. factorν 1). For the consideredcaseofa largenumber
≫
0 The progressinunderstanding ofthe clean2DEGin a of the occupied Landau levels, the mean-field analysis is
3 weak magnetic field was achieved by Aleiner and Glaz- legitimate because the fluctuations of the order param-
0 man5. They have derived the low-energy effective the- eter are strongly suppressed11. On the other hand, the
/
t ory on the partially filled highest Landau level by using mean-field approachcannot be applied to the critical re-
a
m the small parameter 1/ν 1. By treating the effec- gion in the direct vicinity of the phase transition. This
tive interaction within the≪Hartree-Fock approximation, region is however small and does not lead to any signif-
-
d Koulakov, Fogler, and Shklovskii6 predicted a unidirec- icant uncertainty in our results for critical temperatures
n tional charge-density-wave (CDW) state (stripe phase) of the transitions.
o for the half-filled highest Landau level at zero temper- We assume the presence in the system of a weak
c
ature and in the absence of disorder. Moessner and quenched disorder, i.e. the elastic collisions time satis-
:
iv CanhdalkAenrd8eerxsotnen7dteodtthheeciadseeasofofaFpuakrutiyaallmyafi,llPedlathzimghaensnt ficyecslothtreoncofnredqituieonncyτ,0e≫thωeH−el1e,cwtrhoenrechωaHrge=,aenHd/mmtihsetehfe-
X
Landau level and showed the existence of the mean-field fective electron mass. Under this condition, the Landau
r
a CDW state on the half-filled Landau level below some levelbroadening1/2τ,whichisoftheorderof√ωHτ0/τ0,
temperature T . is much less than the spacing ω between them. This
0 H
Recently,theexistenceofcompressiblestatesnearhalf- case can be realized in high mobility samples which
filling with anisotropic transport properties was demon- were used for experimental studies of the anisotropic
strated experimentally for high Landau levels9,10. This magnetotransport9,10,15. Keeping in mind that the re-
stimulates an extensive study of the clean 2DEG in a lation T0 1/τ is usually hold, one expect a much pro-
∼
weakmagneticfieldandpinning ofstripes bydisorder11. nounced influence of the quenched disorder on the prop-
In the clean case, the properties of the CDW states erties of electrons on the partially filled highest Landau
can be described on the basis of the low energy effective level even for a small level broadening 1/2τ ωH.
≪
theoryforsmooth“elastic”deformations11. Recently,at- One of the main results of our paper is that at zero
2
temperaturethemean-fieldCDWstateisdestroyedwhen r 1, and the magnetic field obeys the condition
s
≪
the Landau level broadening exceeds the critical value Nr 1, where N =[ν/2] is the integer part of ν/2. In
s
≫
1/2τ = 4T /π. At nonzero temperatures the quenched this case it is possible to construct an effective field the-
c 0
disorder leads to the decrease of the temperature of the oryfortheelectronsonthehighestpartiallyfilledLandau
CDW instability as compared to the clean case. The level by integrating out all other degrees of freedom5,18.
physicalreasonisthatthescatteringonimpuritiesbreaks We also assume that the electrons on the partially filled
the CDW correlations, and therefore results in the de- highestLandaulevelarespin-polarized. Thisassumption
struction of the coherent CDW state. This is somewhat is based on the calculations3,19 that shows the existence
similar to the suppression of the critical temperature in of fractional states, composite fermions, and skyrmions
conventional superconductors by magnetic impurities16 onlyonthe lowestandthe firstexcitedLandaulevels,as
orin anisotropicsuperconductorsby nonmagneticimpu- well as on the experimental observations.
rities17.
Inordertostudythetransitionfromtheuniformstate
The paper is organized as follows. In Sec. II we in-
totheCDWoneweemploythe Landauexpansionofthe
troduce the formalismthatallowsus to evaluatethe free freeenergyinpowersoftheCDWorderparameter∆(q ),
j
energy of the CDW state in the presence of disorder. In where the vectors q that characterize the CDW state,
j
Sec.IIIweinvestigatetheinstabilityoftheuniformstate havethesamelength7q =Q. Weperformtheexpansion
j
towards the formation of the CDW state, and present
uptotheforthorderintheCDWorderparameterunder
the mean-field phase diagrams at the half-filling and ar- the assumption Nr2 1. In this case the Hartree-Fock
bitrary temperature, and at zero temperature and arbi- approximation is wsel≫l justified8 because the corrections
trary filling. The weak crystallization corrections to the are small in the parameter a /l = 1/Nr2 1, where
mean-fieldsolutionare presentedin Sec. IV. Sec. V con- a = ε/me2 is the Bohr radiBus Hand l = s1/≪√mω the
B H H
tainsthecomparisonofthetheorywiththerecentexper-
magnetic length.
imental and numerical results. We end with conclusions
A. Formalism
in Sec. VI.
The thermodynamical potential of the spin-polarized
II. FREE ENERGY OF THE CDW STATES 2DEGprojectedontheNthLandaulevelinthepresence
ofthe randompotentialV (r)and the magnetic fieldis
dis
given by
We consider two-dimensional interacting electrons in
the presence of a weak quenched disorder and a weak
perpendicular magnetic field. The parameter that char- T
Ω= [ψ,ψ] [V ] [V ] exp( [ψ,ψ,V ]).
acterizes the strength of the Coulomb interaction is −N D D dis P dis S dis
r = √2e2/εv with v being the Fermi velocity and r Z Z (1)
s F F
ε the dielectric constant of a media. We assume that wheretheaction [ψ,ψ,V ]intheMatsubararepresen-
dis
S
the Coulomb interaction between the electrons is weak, tation has the form
α
= ψα (r) iω +µ V (r) ψα (r) T ψα (r)ψα (r)U (r,r′)ψα (r)ψα (r′) . (2)
S ZrXωn( ωn h n −H0− dis i ωn − 2 ωXm,νlZr′ ωn ωn−νl 0 ωm ωm+νl )
Hereψα (r)andψα (r)aretheannihilationandcreation case)
ωn ωn
operators of an electron on the Nth Landau level, T the
temperature, µ the chemical potential, ω =πT(2n+1)
n
the Matsubara fermionic frequency, and ν = 2πTn the
n
bosonic one. The free Hamiltonian for 2D elec-
H0 2πe2 1
trons with mass m in the perpendicular magnetic field U (q)= ,
Helec=trǫoanb-∂ealeAcbtriosnHi0nt=era(−ctii∇on−Ue0A(~r))2/o(n2mth).e TNhtehsLcraenednaedu 0 εq 1+ qa2B (cid:18)1− 6ωπHτ(cid:19) 1−J02(qRc)
level takes into account the effects of interactions with (cid:0) (cid:1)(3)
electronsonthe otherlevels,andhas the form(see Ref.5 where Rc = lH√ν is the cyclotron radius on the Nth
for the clean case and Ref.18 for the weakly disordered Landau level and 0(x) the Bessel function of the first
J
kind. The range of the screened electron-electron inter-
action(3) is determined by the Bohr radius a . We also
B
assume the Gaussian distribution for the random poten-
3
tial V (r) The averaging over the random potential V (r) in
dis dis
Eq.(1) is straightforward and results in the following
1 1 quartic term
[V (~r)]= exp V2 (r) , (4)
P dis √πg −g dis
(cid:18) Zr (cid:19) αβ
g
ψα (r)ψα (r)ψβ (r)ψβ (r) (12)
swthaeterse,gan=d1in/tπrρoτd0u,cρeiNsrthreeptlhiceartmedodcyonpaiemsicoaflthdeenssyitsytemof 2ZrωXnωm ωn ωn ωm ωm
labeled by the replica indices α = 1,...,N in order to in the action. This term can be decoupled by means
r
average over the disorder. of the Hubbard-Stratonovichtransformation 21 with the
Hermitian matrix field variables22,23 Qαβ(~r)
nm
1
B. The Hartree-Fock decoupling and the average [Q]exp trQ2(~r)+iψ (~r)Q(~r)ψ(~r) , (13)
†
over disorder Z D Zr(cid:20)−2g (cid:21)
where the symbol tr denotes the matrix trace over the
The CDW ground state is characterized by the order Matsubaraandreplicaindices. Themeasureforthefunc-
parameter ∆(q) that is related to the electron density tional integral over the matrix field Q is defined as: the
integral (13) equals unity when the fermionic fields ψ
ρ(q) =L L n F (q)∆(q). (5) †
x y L N andψ vanish. NotealsothatinEq.(13)weintroducethe
h i
matrix notations according to
Here L L is the area of the 2DEG, n = 1/2πl2 the
x y L H
number of states on one Landau level, and the form- α,β
factor FN(q) is ψ†(···)ψ = ψωαn(···)αnmβψωβm. (14)
ωXn,ωm
q2l2 q2l2
F (q)=L H exp H , (6) After making all these steps, the action becomes
N N
2 − 4
(cid:18) (cid:19) (cid:18) (cid:19) N Ω 1
= r ∆ trQ2+
where LN(x) is the Laguerre polynomial. For the case S − T − 2g Zr
N 1, one can use the following asymptotic expression
for≫the form-factor (6) + ψ†(~r)(iω+µ 0+λ+iQ)ψ(~r), (15)
−H
Zr
R2 where ω is the frequency matrix (ω)αβ =ω δ δαβ.
F (q)= (qR ) , qR c =ν (7) nm n nm
N J0 c c ≪ l2
H
After the Hartree-Fockdecoupling20 of the interaction C. The saddle-point in the Q field
term in the action (2) we obtain
The Q matrix field can be naturally splitted into the
N Ω
r ∆ transverse V and the longitudinal P components as fol-
= +, (8)
S − T lows Q = V 1PV. The longitudinal component P has
−
α the block-diagonal structure in the Matsubara space,
+ ZrXωn ψωαn(r)hiωn+µ−H0−Vdis(r)+λ(r)iψωαn(r), tPinoαmnβ,∝andΘ(cnormre)s,pwonhdersetoΘm(xa)ssisivtehmeoHdeeasv.isTidhee tsrtaepnsfvuenrsce-
componentV isaunitaryrotationanddescribesmassless
(diffusive) modes (see Refs.24,25 for details).
n (L L )2
Ω = L x y U(q)∆(q)∆( q), (9) This decomposition of the variable Q into P and V is
∆
2 −
Zq motivatedbythesaddle-pointstructureoftheaction(15)
at zero temperature (ω 0) and in the absence of the
wherethepotentialλ(r)resultsfromtheperturbationof n →
potential λ(r). The corresponding saddle-point solution
the uniformelectrondensity by the chargedensity wave,
has the form Q =V 1P V, where the matrix P is
and is connected with the CDW order parameter as fol- sp − sp sp
lows (P )αβ =Pnδ δαβ (16)
sp nm sp nm
λ(q)=LxLyU(q)FN−1(q)∆(q), (10) with Psnp obeying the equation
πρτ Pn =iGn(r,r). (17)
andU(q)= n U (q)withtheHartree-Fockpotential 0 sp 0
L HF
−
UHF(q) given by This equation is equivalent to the self-consistent Born
approximationequation26. The Greenfunction Gn(r,r′)
U (q)=U (q)F2(q) e−iqplH2 U (p)F2(p). is determined as 0
HF 0 N −Zp nL 0 N Gn0(r,r′)= φ∗Nk(r)G0(ωn)φNk(r′), (18)
(11)
k
X
4
As a result, the thermodynamical potential can be writ-
ten as
G (ω )=[iω +µ ǫ +iPn] 1, (19)
0 n n − N sp −
whereǫ =ω (N+1/2)andφ (r)aretheeigenvalues
N H Nk
andeigenfunctionsofthe hamiltonian ,andk denotes
0
H T
pseudomomentum. Ω= ln [δP]I[δP]exp [δP,λ], (22)
In the case of a small disorder ωHτ0 1 the solution −Nr Z D S
≫
of equation (17) has the form26
signω ρ τ
Pn = n , τ =π 0 . (20)
sp 2τ rm√ωHτ0 where, following Ref.24, the integration measure I[δP] is
The fluctuations of the V field are responsible for the
localization corrections to the conductivity (in the weak
localization regime they correspond to the maximally
crosseddiagrams). However,intheconsideredcase,these
corrections are of the order of 1/N 1 and, therefore, 1 αβ
can be neglected. For this reason we≪simply put V =1. lnI[δP]=−(πρ)2 [1−Θ(nm)]δPnαnαδPmββm. (23)
Thepresenceofthe potentialλresultsinashiftofthe Z nm
X
saddle-pointvalue(20)due tothecouplingtothefluctu-
ations δP = P P of the P field. The corresponding
sp
−
effectiveactionfortheδP fieldfollowsfromEq. (9)after
integrating out fermions:
N Ω 1 The quadratic in δP part of the action (21) together
S[δP,λ] = trlnG−01− rT ∆ − 2g tr(Psp+δP)2+ with the contribution (23) from the integration measure
Zr Zr determine the propagator of the δP fields (see Ref18 for
+ trln 1+(iδP +λ)G−01 . (21) details)
Zr h i
gδ δ δαδδβγ 2[1 Θ(m m )] gδ δαβ gδ δδγ
δPαβ (q)δPγδ ( q) = m1m4 m2m3 − 1 3 m1m2 m3m4 , (24)
h m1m2 m3m4 − i 1+gπm1(m m ;q) − (πρ)2 1+gπm1(0;q)1+gπm3(0;q)
0 3− 1 0 0
where the bare polarization operator πm(n;q) is is the mean-field thermodynamical potential of the ho-
0
mogeneous state, and
πm(n;q)= n G (ω +ν )G (ω )F2(q). (25)
0 − L 0 m n 0 m N T
δΩ= ln [δP]exp ˜[δP,λ] (28)
−N D S
r Z
D. Thermodynamical potential takes into account the fluctuaftions of tfhe massive lon-
gitudinal field δP and their interaction with the CDW
order parameter (potential λ). The action ˜[δP,λ] has
To find the expansion of the thermodynamical poten-
S
tialΩ inpowersofthe CDW orderparameter∆(q), it is the form f
f
convenienttointroduceanewvariableδP =δP+iλand
expandtrlnintheaction(21)inpowersofthis newfield ˜= (2)[λ]+ int[δP,λ]+ (2)[δP]+ ∞ (n)[δP], (29)
S S S S S
δP. Thenthe thermodynamicalpotentifalcanbewritten nX=3
in the form f f f
with
f
N
Ω=Ω0+Ω∆+δΩ, (26) (2)[λ]= r λ(r)λ(r), (30)
S 2g
Xωn Zr
where
1 i
Ω0(µ)= trlnG−01− 2g trPs2p (27) Sint[δP,λ]=−g λ(r)trδP(r), (31)
Zr Zr Zr
f f
5
and
( i)n n
(n)[δP]= − tr δP(r )G (r r ), (32)
j 0 j j+1
S n
j=1Zrj
Y
f f
wherer =r . Note thatthe termsinthe action(29),
n+1 1
that are proportional to N2, are omitted because they
r FIG. 1: Second-order contribution to the thermodynamic
do not contribute to δΩ in the replica limit Nr 0. potential. Solid line denotes electron Green function, dashes
→
Another important observationis that the propagatorof are impurity lines and vertexesare λ(r)
the δP fields is the same as for the δP fields (24).
By using Eqs.(28)-(32) we can write
f
T T
δΩ= λ(r)λ(r) ln exp ˜ , (33)
int
−2g − N S
Xωn Zr r D E
where denotes the average over δP with respect
h···i
to the action ˜[δP,0]. This equation allows us to find
S FIG.2: Thethird-ordercontributiontothethermodynamical
the contributions to the thermodynamfical potential Ω
potential.
up to any orderfof the CDW order parameter ∆(q) =
F (q)U(q) 1λ(q)/L L .
N − x y
In this paper we will work only with the expansion
2. Third-order contribution
uptothefourthorderterm(theLandauexpansion). This
implies that our consideration is valid only close to the
transition point where the value of the order parameter ThecontributionofthethirdpoweroftheCDWorder
issmallandonecantruncatetheseries(28)afterseveral parameter to the thermodynamical potential δΩ(3) can
first terms. It should be mentioned, however, that we be written as
shouldavoidadirectvicinityofthephasetransition(the
critical region, for more details see Sec. IV) where the δΩ(3) = T S3 (c) = T S3 S(3) (c), (36)
fluctuations of the order parameter break the mean-field −3!Nr int δP −3!Nr int 0
approach. (cid:10) (cid:11) D E
f
where the superscript (c) indicates that only connected
diagrams are taken into account. Here we omit again
the terms that vanish in the replica limit N 0. Af-
r
1. Second-order contribution →
ter performing the averaging over δP with the help of
Eqs.(10),(24),(31), and (32), we obtain
Thesecondordercontributiontothethermodynamical
f
potential δΩ is
δΩ(3) T 3 U(q )∆(q )G (ω )
= (2π)2n j j 0 n
δΩ(2) =−2Tg Xωn Zrλ(r)λ(r)− 2NTr (cid:10)Si2nt(cid:11)0, (34) L3xL3y × δ(q1+Lq23+Xωqn3)jY=e1x"pZ2iqj q1x1q2y+−gπq1y0ωqn2x(0,.qj) (3#7)
(cid:16) (cid:17)
where standsfor the averageoverδP with respect
0
to theha·c·t·iion ˜(2)[δP]. We replace the average over the The contribution δΩ(3) corresponds to the diagram in
S Fig. 2.
fullaction ˜[δP,0]bytheaverageoverthefquadraticpart
S
˜(2)[δP] only becaufse the higher order in δP terms lead
S
to the contrifbutions that are proportional to N2, and
r
thereffore vanish in the replica limit Nr f0. 3. Forth-order contribution
→
With the help of Eqs.(10),(24), and (31), we obtain
The forth order contribution δΩ(4) is
δΩ(2) T U2(q)G2(ω )
=n 0 n ∆(q)∆( q). (35)
L2xL2y L2 Xωn Zq 1+gπ0ωn(0,q) − δΩ(4) = −4!TN Si4nt (c)
r
(cid:10) (cid:11) (c)
The corresponding diagram in the usual “cross tech- T 1
= S4 S(4)+ (S(3))2 , (38)
nique” is shown in Fig. 1. −4!N int 2
r (cid:28) h i(cid:29)0
6
where isthefreeenergyofthenormal(homogeneous)
0
F
state, N the total number of electrons, µ and µ the
e 0
chemical potentials of the CDW state and the normal
state respectively.
In order to find the free energy of the CDW state to
the forth order in the CDW order parameter we expand
Ω (µ )aroundthe pointµtothe secondorderinµ µ .
FIG.3: Theforth-ordercontributiontothethermodynamical 0 0 − 0
This results in
potential.
where again only terms which is proportional to Nr are = +Ω(µ) Ω (µ) 1(µ µ )2∂2Ω0. (41)
kept. By using Eqs.(10),(24),(31), and (32), we find F F0 − 0 − 2 − 0 ∂2µ0
δΩ(4) T 4 U(q )∆(q )G (ω )
L4xL4y = (2π)2nL4 Xωn jY=1"Zqj 1+j gπ0ωnj(0,0qj)n # TChDeWdiaffnedretnhceenµor−mµa0l sotfattehseischemical potentials in the
1 gπωn(0, q +q )
δ(q +q +q +q ) − 0 | 1 2|
× 1 2 3 4 1+gπωn(0, q +q )
0 | 1 2| 1
i i ∂δΩ ∂Ne −
exp qxqy qyqx exp qxqy qyqx .(39) µ µ0 = , (42)
× 2 1 2 − 1 2 2 3 4 − 3 4 − ∂Ne (cid:18) ∂µ (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
Intheusual“crosstechnique”thecontributionδΩ(4) cor-
responds to the diagram shown in Fig. 3. and from Eq.(26) we obtain
E. Free energy 1 ∂δΩ(2) 2 ∂Ne −1
= +Ω +δΩ+ . (43)
0 ∆
F F 2 ∂µ ∂µ
The free energy of the CDW state can be written in (cid:18) (cid:19) (cid:18) (cid:19)
the form
= +Ω(µ) Ω (µ )+(µ µ )N , (40) With the expression (35) for δΩ(2) this gives
0 0 0 0 e
F F − −
2
n (L L )3 U2(q)G3(ω ) 1
= +Ω +δΩ+ L x y T 0 n ∆(q)∆( q) T G2(ω ) − . (44)
F F0 ∆ 2 " Xωn Zq [1+gπ0ωn(0,q)]2 − # h Xωn 0 n i
F. Free energy of the triangular CDW state expansion are as follows
T (Q) 1
The CDW order parameter for the triangular lattice a =3 1 0 , (47)
symmetry (bubble phase) can be written in the form7 2 h − π2T Xn ξn2 +γ2(Q)i
(2π)2 3 where
∆(q)= ∆(Q) δ(q Q )+δ(q+Q ) , (45)
L L − j j
x y Xj=1h i ξ =n+ 1 + 1 i µ , γ(Q)= FN(Q) (48)
n
2 4πTτ − 2πT 4πTτ
where the vectors Q have the angle 2π/3 between each
j
other and obey the condition Q +Q +Q =0.
1 2 3 and T (Q)=U(Q)/4,
0
By using Eqs.(35),(37),(39) and (44), we obtain the
following expression for the free energy of the triangular
T2(Q) √3Q2 ξ3
CDW state a =i8 0 cos n , (49)
3 π3T2 4 3
L L ! n ξ2 +γ2(Q)
t = +4 x yT (Q) a ∆2+a ∆3+a ∆4 . (46) X n
F F0 2πl2 0 2 3 4 h i
H
Here the three coefficientsha , a , and a of theiLandau and
1 2 3
7
24T3(Q) 1 ξ4 √3Q2 1
a = 0 n 3D (0)+ 1+cos D (Q)+D (√3Q) + D (2Q)
4 π4T3 2 4 n 2 n n 2 n
( Xn ξn2 +γ2(Q) " (cid:16) (cid:17)(cid:16) (cid:17) #
h 2 i
ξ 1
+ 3 n ξ 2 − , (50)
2 n−
"Xn ξn2 +γ2(Q) # hXn i )
h i
with the free energy of the unidirectional CDW state reads
ξ2 γ2(Q)
D (Q)= n− (51)
n ξn2 +γ2(Q) u = +4LxLyT (Q) b ∆2+b ∆4 . (53)
F F0 2πl2 0 2 4
H
h i
G. Free energy of the unidirectional CDW state
Here the coefficients b and b of the Landau expansion
2 4
The CDW order parameter of the unidirectional state are
(stripe phase) is6,7,8
(2π)2 a2 T0(Q) 1
∆(q)= ∆(Q) δ(q Q)+δ(q Q) , (52) b2 = 3 = 1− π2T ξ2 +γ2(Q) (54)
LxLy h − − i h Xn n i
where the vector Q is oriented along the spontaneously
chosen direction, and from Eqs.(35),(37),(39) and (44), and
2
4T3(Q) ξ4 1 ξ 1
b = 0 n D (0)+ D (2Q) +2 n ξ 2 − . (55)
4 π4T3 4 n 2 n 2 n−
(Xn ξn2 +γ2(Q) " # "Xn ξn2 +γ2(Q) # hXn i )
h i h i
We note that in the limit 1/τ 0 evaluation of fre- tion
→
quency sums in expressions for free energy of the trian-
T 1 1
gular and unidirectional CDW states (46)-(55) leads to = (56)
the results obtained in Refs.7,8 in the clean case. T0(Q) π2 n ξn2 +γ2(Q)
X
for the instability line. The solution T(Q) of this equa-
tiondependsonthemodulusQof thevectorthatcharac-
terizestheCDWstate. ThetemperatureT ofthesecond
III. MEAN-FIELD PHASE DIAGRAM 2
orderphase transitioncorresponds to the maximalvalue
of T(Q):
A. Instability line
T =maxT(Q) (57)
2
Q
The vanishing of the coefficient in front of the
quadratic term in the Landau expansion of the free en- andthecorrespondingvalueQ ,T =T(Q ),determines
0 2 0
ergy signals about the instability of the normal state to- the period of the CDW state. The Hartree-Fock poten-
wards the formation of the CDW. This instability corre- tial (11) has minima at those vectors Q for which the
k
spondsto the secondorderphasetransitionfromthe ho- form-factor F (Q ) vanishes. In the clean case this cor-
N k
mogeneousstatetotheCDWstate. Asusual,thespecific responds to Q =minQ =r /R , where r 2.4 is the
0 k 0 c 0
parametersoftheformingCDWstatearedeterminedby firstzeroof the Besselfunction ofthe firstkin≈d 6. It can
the high order terms in the Landau expansion. be seen from Eq.(56) that a weak disorder does not shift
FromEqs. (47)and(54)weobtainthefollowingequa- the vectorQ (seeAppendix). Thusthe equationforthe
0
8
temperatureofthesecondorderphasetransitionintothe
CDW state reads
1.0 π τ
T 2 ′ 1 1 µ
= ψ + +i (58) π/8T0τ = 0.0
T π2ℜ 2 4πTτ 2πT
0 (cid:18) (cid:19) π/8T0τ = 0.1
ttwrhhaeenrrseeitaiψlon′p(azinr)t,tihsaenthdcleeTad0ne≡rciavsTae0t.i(vQe0)ofthdeigtaemmmpearafutunrcetioofn,thℜe , T / Tature0 00..68 π//88TT00τ == 00..59
Eq.(58) contains the chemical potential µ that, to- er
p
gether with the temperature T and the broadening of m
e
Landau levels 1/2τ, determines the filling factor ν = t
N 0.4
ν 2N of the partiallyfilled highest Landaulevel. How-
−
ever, in order to find this relation one needs to know
the density of states in the system. This question about
0.2
the density of states is a very subtle27 and beyond the
scope of the present paper. For this we use the chemical
potential µ rather than filling factor ν .
N 0.0
Eq.(58) can be solved analytically in the two extreme 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
µ π
cases: when temperature T is closed to the temperature
chemical potential, / 2 T
T ofinstabilityintheabsenceofdisorder,andwhenthe
0 FIG.4: Thespinodallinesobtained from Eq.(58) areshown
temperature T is close to zero.
for different values of dimensionless parameter π/8T0τ.
In the first case, the broadening of the Landau level
1/2τ andthe chemicalpotentialµaresmallcomparedto
the temperature T of the instability in the clean case,
0 1.5
and, therefore, the leading order expansion in powers of
1.4
1/T τ and µ/T is legitimate. It appears that the pres- T = 150 mK
0 0
1.3 T = 25 mK
ence of disorder decreases the temperature of instability
1.2
linearly:
1.1
Uniform phase
T 7ζ(3) µ2 1 1.0
=1 , ,µ 2πT. (59) T0
T0 − π3T0τ − 4T02 2τ ≪ e, T/ 0.9
ur 0.8
anIdnEtqh.e(5o8p)proesdituececsasteo T → 0, one has 1/2τ,µ ≫ 2πT, perat 0.7
m 0.6
e
π 1 π4T2 T 0.5
= 1 1 12µ2τ2 . (60)
8T τ 1+4µ2τ2 − − 48T2 0.4
0 (cid:20) (cid:16) (cid:17) 0 (cid:21) 0.3 Unidirectional
WeseefromEq.(60)thatatzerotemperaturethesecond
0.2 CDW phase
orderphasetransitioncanoccuronlywhen the broaden-
0.1
ingoftheLandaulevelissmallerthansomecriticalvalue,
0.0
1/2τ 1/2τ =4T /π.
≤ c 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
ForothercasesEq.(58)canbesolvednumerically,and
Disorder, p / 8 T0 t
the corresponding instability (spinodal) line is shown in
Fig. 4. FIG. 5: Phase diagram at νN = 1/2. The spinodal line
obtainedfromEq.(61)isshownbythesolidline. Thetriangles
and rhombi are the experimental data after Ref. 9. The
Landau level index N =2,3,4,5 increases from left to right.
B. Half-filled Landau level (ν =1/2)
N
We now consider the case of the half-filled Nth Lan-
daulevel(νN =1/2),thatisrelatedtotherecentexperi- where ζ(2,z) = ∞m=0(m+z)−2 is the generalized Rie-
mann zeta function. The analytical solutions of this
ments9. Inthiscasethechemicalpotentialiszero,µ=0,
P
equation in the cases of high and low temperature can
provided the density of states is symmetric around the
beobtainedfromEqs.(59)and(60)byputtingµtozero.
centeroftheNthLandaulevel. AsfollowsfromEq.(58),
Theentirebehaviorofthespinodalline,obtainednumer-
the temperature of the second order phase transition for
ically from Eq.(61), is shown in Fig. 5.
this case can be found from the equation
We mention that at ν = 1/2 the coefficient a van-
N 3
T 2 1 1 ishes due to the particle-hole symmetry. It means that
= ζ 2, + , (61)
T π2 2 4πTτ the transitionfromthe normalstateinto the CDWstate
0 (cid:18) (cid:19)
9
is close to its critical value 1/2τ = 4T /π. Under these
c 0
1.00
conditions, the CDW order parameter ∆ is small, and
onecanuse the Landauexpansions(46)and(53)atzero
Uniform temperature. The coefficients of these expansions are
phase 8T τ 8T τ 2
0 0
a =3 1 H (µτ) , a =2π H (µτ),
t8 T 00.95 2 (cid:16) − π 1 (cid:17) 3 (cid:18) π (cid:19) 2 (65)
p / and
er, 8T τ 3
d a =3π2 0 H (µτ,QR ) (66)
or 4 π 3 c
s (cid:18) (cid:19)
Di 0.90
for the triangular CDW state, and
Unidirectional Triangular a π2 8T τ 3
2 0
CDW phase CDW phase b2 = , b4 = H4(µτ,QRc) (67)
3 2 π
(cid:18) (cid:19)
fortheunidirectionalCDWstate. Hereweintroducefour
0.85
functions H (z) as
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 i
Chemical potential, m t 1
H (z)= , H (z)=4zH (z), (68)
1 1+4z2 2 1
FIG. 6: Phase diagram at zero temperature near ν = 1/2.
N
The solid line is obtained from Eq.(72), the dashes are the
spinodal line and thedots are obtained from Eq.(74). 1 108z2 5
H (z,r) = − +3R (z,r)+2R (z,r)
3 2(1+4z2)3 0 1
1
isofthesecondorderforbothcasesofunidirectionaland + 2R2(z,r)+2R√3(z,r), (69)
triangularlatticesymmetry. Therefore,tofindthestruc-
and
ture of the CDW state, one has to take into account the
fourth order terms in the the Landau expansion. In the 28z2 1
H (z,r)= − +2R (z,r)+R (z,r), (70)
vicinity of the spinodalline, it followsfromEqs.(50) and 4 (1+4z2)3 0 2
(55) with µ=0 that
where
12T3 2H (z) 1 2 (ar)
a4 = π4T03 −7ζ(4,u)+12Φ0(u)+6Φ2(u)+8Φ√3(u) Ra(z,r)= 2(1ar) − 3(ar)arctan1+4zJ20 2(ar).
h (6i2) J0 J0 −J0 (71)
and
These expressions result in the following equation on
the line of the first order transition from the uniform to
2T3
b = 0 3ζ(4,u)+4Φ (u)+2Φ (u) , (63) the triangular CDW state
4 π4T3 − 0 2
h i π H2(µτ)
where we introduce the new variable u = 1/2+1/4πTτ =H1(µτ)+ 2 . (72)
8T τ 9H (µτ,QR )
and the new function 0 3 c
As before, the maximum of the solutions 1/τ(Q) of
1 1 1 Eq.(72) with respect to Q should be found. It appears
Φ +z = ζ 2, +z (64)
a(cid:18)2 (cid:19) z2J02(ar0)" (cid:18) 2 (cid:19) cthleaatntchaesem8,axbiumtuamt siosmneotsheixftaecdtlvyalauteQQ=+QδQ0 awsitinh tthhee
0
1 1 shift δQ = 0.02(µτ)2R 1 for small µτ 1. The ex-
− zJ0(ar0)ℑψ(cid:18)2 +z+izJ0(ar0)(cid:19)# istence of th−e shift is a c−feature of the di≪sordered case.
Below in the limit µτ 1, we will neglect this shift. In
≪
with beingtheimaginarypart. Withtheseexpressions this case Eq.(72) can be written as
ℑ
weminimized t,uwithrespecttotheorderparameter∆ π
F =1 2.94(µτ)2. (73)
and found that the unidirectional CDW state has lower 8T τ −
0
free energy.
By comparing the free energies of the triangular and
the unidirectional CDW states, we can find the line of
the first order transition between them
C. Phase diagram at zero temperature
π H2[2H +H ][3H + H2+2H H ]
=H 2 3 4 4 4 3 4 ,
In this section we analyze the zero temperature phase 8T τ 1− 2H (2H 3H )2
0 3 3−p 4
diagram in the case where the Landau level broadeding (74)
10
For the case µτ 1 Eq.(74) can be simplified as The constants β are given by
i
≪
8Tπ0τ =1−18.44(µτ)2. (75) β1 = TT00′((QQ00)) ≈2.58 , β2 =(J0′(Q0))2 ≈0.27, (80)
Forothervaluesofµτ Eqs.(72)andEq.(74)weresolved and function Φ is defined by Eq.(65).
a
numerically, and the results are shown in Fig. 6. When We mention that the function π2g(z)/2ζ(2,1/2+ z)
the parameter 8/πT0τ decreases at a fixed value of the decreases monotonically from the value 0.35 at z = 0
chemicalpotential,theCDWorderparametergrows,and to zero at z . Therefore, we obtain the following
hence, we go beyond the applicability of to the Landau inequality for→th∞e shift δT of the mean-field transition
expansion. temperature T
2/3
δT πr
3 0 N 2/3, N 1 (81)
IV. WEAK CRYSTALLIZATION T ≤ 16√β − ≫
CORRECTIONS (cid:18) 1(cid:19)
(the equality corresponds to the clean case.
The CDW order parameter ∆(r) introduced in Eq.(5) The appearance of a noninteger powers in Eq.(76) re-
can be thought of as a saddle-point solution for the sultsfromthefactthatthemomentumdependenceofthe
plasmon field that appears in the Hubbard-Stratonovich correlationfunction for the order parameter fluctuations
transformationofthe electron-electroninteractioninthe contains (Q Q0)2 rather than Q2 (see Ref.28).
−
action(2). The Landauexpansions(46) and (53) for the Eq.(76) was derived under the assumption that the
free energy of the CDW states were derived under the main contribution in the momentum space comes from
assumption that one can neglect the fluctuations of the the region Q Q0. This assumption is justified under
≈
CDW order parameter. This is legitimate for N 1 the following condition28
≫
and not very close to the transition (outside the critical
region). However, when one approaches the instability 1
g
line, the fluctuations of the CDW order parameter in- 4πTτ
(cid:18) (cid:19) N2/3 (82)
crease. To analyse the effects of the order parameter 1 ≪
r2f
fluctuation, we introduce, following the original ideas of 0 4πTτ
Brazovski28, the fluctuations of the CDW order parame- (cid:18) (cid:19)
ter ∆(r) ∆(r)+δ(r) in the Landau expansion of the The combination of functions in the left hand side of
→
free energy and average over the fluctuations δ(r). We inequality (82) decreases monotonically from 0.023 to 0
present below the results of the corresponding analysis while z increases from zero to infinity and, hence, the
only for the most interesting case of the half-filled Lan- condition (82) is hold.
dau level. According to Eq.(81), the fluctuations reduce the
We find that the transition from the uniform to the transition temperature by the amount of the order of
unidirectionalCDWstatebecomesofthefirstorder,and N 2/3 1 and, therefore, in the considered case of the
−
≪
takes place at the lower temperature that can be found weak magnetic field (N 1) their effects can be ne-
≫
fromthe followingequation(see Eq.(61 forcomparison) glected. Theseresultsindicatethatthecriticalregionfor
the considered transition is indeed small, and the mean-
field approachgives a good approximation for N 1.
T 2 1 1 1 ≫
= ζ 2, + g N 2/3, (76)
−
T π2 2 4πTτ − 4πTτ
0 (cid:18) (cid:19) (cid:18) (cid:19)
V. DISCUSSIONS
Here function g(z) is defined as
A. Comparison with experimental results
3πr 2/3 λ2(z) 2/3 2λ (z)+λ (z) 1/3
g(z)=3 0 0 0 2
(cid:20) 16 (cid:21) (cid:20) f(z) (cid:21) (cid:20)4λ0(z)−λ2(z)(cid:21) Nowwediscuss the possibleapplicationsofourtheory
(77) to the recent experiments. Although our mean-field the-
where we introduce the following three functions orywasderivedforthecaseofalargenumberoftheoccu-
piedLandaulevelN 1,andneglectscorrectionsofthe
≫
2 1 1 order of 1/N, while experimentally one has N = 2,3,4,
f(z)= β ζ 2, +z +β z2ζ 4, +z , (78)
π2 1 2 2 2 we however expect that Eq.(58) gives a good estimation
" (cid:18) (cid:19) (cid:18) (cid:19)#
for the temperature of the transition from the uniform
to the CDW state, even for N = 2,3,4. We have com-
plementary assurance that it can really be the case be-
2 1 1
λ (z)= ζ 4, +z +2Φ 4, +z , (79) cause Eq.(58) can be obtained without introducing the
a π4"− (cid:18) 2 (cid:19) a(cid:18) 2 (cid:19)# CDW order parameter and considering the mean-field
