Spin-valley interplay in two-dimensional disordered electron liquid

0.3 MB·English

by  I. S. Burmistrov

Spin-valley interplay in two-dimensional disordered electron liquid
I.S.Burmistrov and N.M. Chtchelkatchev
L.D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117940 Moscow, Russia and
Department of Theoretical Physics, Moscow Institute of Physics and Technology, 141700 Moscow, Russia
(Dated: February 2, 2008)
We report the detailed study of the influence of the spin and valley splittings on such physical
observablesofthetwo-dimensionaldisorderedelectronliquidasresistivity,spinandvalleysuscepti-
bilities. Weexplain qualitatively thenonmonotonic dependenceof theresistivity with temperature
8
inthepresenceofaparallel magnetic field. Inthepresenceof eitherthespin splittingorthevalley
0
splitting we predict novel, with two maximum points, temperature dependenceof the resistivity.
0
2
PACSnumbers: 72.10.-d 71.30.+h, 73.43.Qt 11.10.Hi

n
a
J I. INTRODUCTION As is well known, in both Si-MOSFET1 and n-AlAs

4 quantum well11 2D electrons can populate two valleys.
1 Disordered two-dimensional (2D) electron systems Therefore, these systems offer the unique opportunity
have been in the focus of experimental and theoretical foranexperimentalinvestigationofaninterplaybetween
]
l researchforseveraldecades.1Recently,theinterestto2D the spinandvalleydegreesoffreedom. Recently,using a
l
a electronsystems has been renewedbecause of the exper- symmetry breaking strain to tune the valley occupation
h imental discovery of metal-insulator transition (MIT) in ofthe2Delectronsysteminthen-AlAsquantumwell,as
s- a high mobility silicon metal-oxide-semiconductor field- well as a parallelmagnetic field to adjust the spin polar-
e effect transistor (Si-MOSFET).2,3 Although, during last ization, the spin - valley interplay has been experimen-
m decade the behavior of resistivity similar to that of tally studied.12,13 However, the electron concentrations
. Ref. [2,3] has been found experimentally in a wide va- in the experiment were at least three times larger than
at riety of two-dimensional electron systems,4 the MIT in the critical one.11 Therefore, the spin - valley interplay
m 2D calls still for the theoretical explanation. has been studied in the region of a good metal very far
from the metal-insulator transition.
- Very likely, the most promising frameworkis provided
d
by the microscopic theory, initially developed by Finkel- In the present paper we report the detailed theoreti-
n
stein,thatcombinesdisorderandstrongelectron-electron cal results on the T-behavior of the 2D electron system
o
c interaction.5 PunnooseandFinkelstein6 haveshownpos- with two valleys in the MIT vicinity. In particular, we
[ sibility for the MIT existence in the special model of 2D study the effect ofa parallelmagnetic field and/ora val-

1 electron system with the infinite number of the spin and ley splitting (∆v) on the transport, and the spin and
valley degrees of freedom. The current theoretical re- valley susceptibilities. We find that in the presence of
v
9 sults7,8 do not support the MIT existence for electrons eitherthemagneticfieldorthevalleysplittingthemetal-
3 without the spin and valley degrees of freedom. There- lic behavior of the resistivity survives down to the zero
1 fore, it is natural to assume that the spin and valley de- temperature.14 For example, this result implies that at
2 grees of freedom play a crucial role for the MIT in the B =0 the metallic ρ(T)dependence canbe observedex-
1. 2D disordered electron systems. perimentally at temperatures T ≪∆v. Only if both the
0 Usually, in the MIT vicinity, from the metallic side, magnetic field and the valley splitting are present, then
8 i.e., for an electron density higher than the critical one, themetallicbehavioroftheresistivitycrossesovertothe
0 and at low temperatures T ≪ τ−1 the initial increase insulating one. Next, we predict novel, with two maxi-
v: of the resistivity (ρ) with loweritnrg temperature is re- mum points, T-behaviorofthe resistivityinthe presence
i placed by the decrease of ρ as T becomes lower than ofthe magnetic fieldand/orthe valley splitting. Finally,
X
somesamplespecific temperature.4 Here,τ denotesthe we find that as T vanishes the ratio of the valley sus-
tr
ar elastic scattering time. This nonmonotonic behavior of ceptibility (χv) to the spin one (χs) becomes sensitive
the resistivity has been predicted from the renormaliza- to the ratio of the valley splitting to the spin one. At
tion group (RG) analysis of the interplay between dis- high temperatures the ratio χv/χs is temperature inde-
order and electron-electron interaction in the 2D disor- pendent and canbe chosenequalunity. If the spin split-
dered electron systems.5,9 As a weak magnetic field B ting is larger (smaller) than the valley splitting, then at
is applied parallel to the 2D plane, decrease of the re- low temperatures the ratio χv/χs < (>)1. If the spin
sistivity is stopped at some temperature and ρ increases and valley splittings are equal each other, then the ratio
again.10 Further increase of B leads to the monotonic χv/χs =1 as temperature vanishes.
growth of the resistivity as temperature is lowered, i.e., The presence of the parallel magnetic field and the
to an insulating behavior, in the whole T-range. These symmetry-breaking strain introduces new energy scales
experimental results suggest the significance of the elec- ∆ = g µ B and ∆ in the problem. Here, g and
s L B v L
tron spin for the existence of the metallic phase in the µ stand for the g-factor and the Bohr magneton, re-
B
2D disordered electron systems. spectively. Let us assume that the following conditions
2
hold: ∆ ≪ ∆ ≪ 1/τ . In addition, a magnetic field us assume that the wave functions ϕ(z)exp(±iQz/2)
v s tr
B & T/(De) is applied perpendicular to the 2D elec- are normalized and orthogonal with negligible overlap
⊥
tron system in order to suppress the Cooper channel. dzϕ2(z)exp(iQz). The vector Q = (0,0,Q) corre-
Here, e and D denote the electron charge and diffusion sponds to the shortest distance between the valley min-
coefficient, respectively. Due to the symmetry breaking, Rimainthereciprocalspace: Q∼a−1,witha beingthe
lat lat
the spin and valley splittings set the cut-off for a pole lattice constant.1,15
in the diffusion modes (“diffusons”) with opposite spin In the path-integral formulation 2D interacting elec-
and valley isospin projections. In the temperature range trons in the presence of the random potential V(r) are
∆ ≪T ≪τ−1 this cut-off is irrelevantand the 2D elec- described by the following grand partition function
s tr
tron system behaves as if no symmetry breaking terms
are applied. The temperature behavior of the resistiv- Z = D[ψ¯,ψ]eS[ψ¯,ψ], (2)
ity is governed by one singlet and 15 triplet diffusive
Z
modes.9 At low temperatures ∆ ≪ T ≪ ∆ , eight dif-
v s with the imaginary time action
fusive modes with opposite spin projections do not con-
tribute. Then, the ρ(T) dependence is determined by 1/T
the remaining one singlet and seven triplet modes. As S = dt −ψ¯τσ(r,t)∂tψτσ(r,t)−H0−Hdis−Hint . (3)
we shall demonstrate below the behavior of the resistiv- Z0 h i
ity can be either metallic or insulating. Surprisingly, we The one-particle Hamiltonian
foundthattheseventripletdiffusivemodesarenotequiv-
alent. They have to split into two groups of six and one H = drψ¯σ(r) − ∇2 −µ+ ∆sσ+ ∆vτ ψσ(r) (4)
modes for the spin susceptibility be T-independent. For 0 τ 2m 2 2 τ
temperatures T ≪ ∆ , next four diffusive modes with Z h e i
v
describes a 2D quasiparticle with mass m in the pres-
opposite isospin projections become ineffective. In this e
enceoftheparallelmagneticfieldandthevalleysplitting.
case,thetemperaturedependenceoftheresistivityisde-
Here, µ denotes the chemical potential. Next,
termined by one siglet and three triplet diffusive modes.
Although, the number of the remaining diffusive modes
correspondsformallyto single-valleyelectronswithspin, Hdis = drψ¯τσ1(r)Vτ1τ2(r)ψτσ2(r) (5)
the ρ(T) behavior is insulating. Z
The paper is organized as follows. In Section II we involves matrix elements of the random potential:
introduce the nonlinear sigma model that describes the
disordered interacting electron system. Then, we con- V (r)= dzV(R)ϕ2(z)ei(τ2−τ1)Qz/2. (6)
sider the short length scales at which the system has τ1τ2
Z
SU(4) symmetry in the combined spin and valley space
In general,the matrix elements V (r) induce both the
(Sec. III). The behavior of the system at the interme- τ1τ2
intravalley and intervalley scattering. We suppose that
diate and long length scales is studied in Sec. IV and
V(R) has the Gaussian distribution, and
Sec. V, respectively. We end the paper with discussions
of our results and with conclusions (Sec. VI).
hV(R)i=0, hV(R )V(R )i=W(|r −r |,|z −z |),
1 2 1 2 1 2
(7)
where the function W decays at a typical distance d. If
II. FORMALISM
d is larger than the effective width of the 2D electron
system,i.e.,d≫[ dzϕ4(z)]−1,thenonecanneglectthe
A. Microscopic Hamiltonian
z-dependenceofV(R)under the integralsigninEq.(6).
R
In this case, the intravalley scattering survives only:
To start out, we consider 2D interacting electrons
with two valleys in the presence of a quenched disor- hVτ1τ2(r1)Vτ3τ4(r2)i=W(|r1−r2|,0)δτ1τ2δτ3τ4. (8)
der and a parallel magnetic field at low temperatures
T ≪ τ−1. We assume that the perpendicular magnetic In the opposite case, d≪[ dzϕ4(z)]−1, one finds16
tr
field B & T/(De) is applied in order to suppress the
Cooper⊥channel. Usingoneelectronorbitalfunctions,we hVτ1τ2(r1)Vτ3τ4(r2)i= δRτ1τ2δτ3τ4W˜(|r1−r2|,0) (9)
write an electron annihilation operator as h
+δ δ δ W˜(|r −r |,2Qτ ) dzϕ4(z),
τ1τ4 τ2,τ3 τ1,−τ2 1 2 2
ψσ(R)= ψτσ(r)ϕ(z)eiτzQ/2, (1) iZ
τX=± where W˜(r,Q) = dzW(r,z)exp(iQz). The other cor-
where z denotes the coordinate perpendicular to the 2D relationfunctions,e.g. withτ =−τ andτ =τ ,vanish
1 2 3 4
R
plane,rthein-planecoordinatevector,andR=r+zez. due to integration over (z1+z2)/2 coordinate. It is the
The subscript τ enumerates two valleys and ψσ is the last term in Eq. (9) that contributes to the intervalley
τ
annihilation operator of an electron with the spin and scattering rate 1/τ . Assuming Q−1 ≪ d, one can ne-
v
isospin projections equal σ/2 and τ/2, respectively. Let glect the intervalley scattering rate in comparison with
3
the intravalley scattering rate 1/τ ∼ W˜(r,0). At last, where S represents the free electron part20
i σ
allowing for a low electron concentration n in 2D elec-
e σ
tron systems, we consider the case when the following S =− xx Tr(∇Q)2. (17)
σ
inequality holds, n d2 ≪1. Then, both Eqs. (8) and (9) 32
e
read Here, σ denotes the mean-field conductivity in units
xx
e2/h. The symbol Tr stands for the trace over replica,
1
hVτ1τ2(r1)Vτ3τ4(r2)i= 2πντ δτ1τ2δτ3τ4δ(r1−r2), (10) theMatsubarafrequencies,spinandvalleyindicesaswell
i as integration over space coordinates.
1
=2πν d2rdz dz W(r,|z −z |)ϕ2(z )ϕ2(z ), The term5
1 2 1 2 1 2
τ
i Z
Q−1 ≪d, [ ϕ4(z)dz]−1 ≪n−1/2. (11) SF =4πTzTrη(Q−Λ)+πTΓ d2r trInαQtrI−αnQ
e
Z αn
Z X
Here, ν is the thermodynamic density of states. Under −πTΓ d2r (trIαQ)⊗(trIα Q). (18)
2 n −n
conditions (11), the interaction part of the Hamiltonian Z αn
X
is invariant under global SU(4) rotations of the electron
operator ψσ in the combined spin-valley space: involves the electron-electron interaction amplitudes
τ which describe the scattering on small (Γ) and large
Hint = e22ǫZ dr1dr2 ψ¯τσ11(r1)ψτσ11|(rr11−)ψ¯rτσ222|(r2)ψτσ22(r2). Fr(Γeinn2o)kreamlsntagelliieznsa5taiwnondh.i2tc1hheTiqshueraenisntpitoteynraszicbtolieoringfoinaramltlphyleiitnustpdreeocsdiufiaccreedherbaeyt-
(12)
latedwiththe standardFermiliquidparametersas5,17,19
A dielectric constant of a substrate is denoted as ǫ. The
Γ = −zFσ/(1 + Fσ), 4Γ = Γ + zFρ/(1 + Fρ), and
low energy part of H can be written as5,17,18,19 2 0 0 2 0 0
int z = πν⋆/2 where ν⋆ = m⋆/(2π) with m⋆ being the ef-
1 fective mass. The caseof the Coulombinteractioncorre-
Hint = dr1dr2 ρ(r1)Γs(r1−r2)ρ(r2) (13) sponds to the so-called “unitary” limit,22 Fρ →∞.
2 0
Z h The symbol tr involves the same operations as in Tr
+ma(r1)Γt(r1−r2)ma(r2) (14) excepttheintegrationoverspacecoordinates,andtrA⊗
i trB = Anαnα;;τσ11τσ22Bmββm;σ;τ22στ11. The matrices Λ, η and Ikγ are
where given as
ρ(r)= ψ¯σ(r)ψσ(r), (15) Λαβ;ζ1ζ2 =sign(ω )δ δαβδζ1ζ2,
τ τ nm n nm
Xστ ηαβ;ζ1ζ2 =nδ δαβδζ1ζ2, (19)
ma(r)= ψ¯σ1(r)(ta)σ1σ2ψσ2(r). nm nm
τ1 τ1τ2 τ2 (Iγ)αβ;ζ1ζ2 =δ δαγδβγδζ1ζ2.
σ1σX2;τ1τ2 k nm n−m,k
Here, Γ (q) = U(q)+Fρ/(4ν) involves the long-range In the absence of ∆s and ∆v, the action Sσ + SF
part of sthe Coulomb inte0raction U(q) = 2πe2/(qǫ) and is invariant under the global rotations Qαnmβ;;στ11στ22(r) →
Γt(q) = F0σ/(4ν). Quantities F0σ and F0ρ are the stan- uστ11τσ33Qαnmβ;;στ33στ44(r)[u−1]στ44τσ22 in the combined spin-valley
dard Fermi liquid interaction parameters in the singlet space for u ∈ SU(4). The presence of the parallel mag-
and triplet channels, respectively. The matrices ta with netic field and the valley splitting generates the symme-
a = 1,...15 are the non-trivial generators of the SU(4) try breaking terms:
group.
S =iz ∆ Trσ Q, S =iz ∆ Trτ Q, (20)
sb s s z vb v v z
where σ and τ are Pauli matrices in the spin and val-
B. Nonlinear sigma model z z
ley spaces, respectively. The Q-independent part of the
action reads5,23
At low temperatures, Tτ ≪1, the effective quantum
tr
theory of 2D disordered interacting electrons described S =−2πTzTrηΛ+ Nr d2r χ0∆2+χ0∆2 . (21)
bytheHamiltonian(3)isgivenintermsofthenon-linear 0 2T s s v v
Z
σ-model. This theory involves unitary matrix field vari- (cid:0) (cid:1)
ables Qmα1nα;2τ1;στ12σ2(r) which obey the nonlinear constraint withχ0s,v =2zs,v/πbeingabarevalueofthespin(valley)
Q2(r) = 1. The integers α = 1,2,...,N denote the susceptibility.
j r
replica indices. The integers m,n correspond to the dis-
crete set of the Matsubara frequencies ω =πT(2n+1).
n C. F-algebra
The integers σ = ±1 and τ = ±1 are spin and valley
j j
indices, respectively. The effective action is
The action (16) involves the matrices which are for-
S =S +S +S +S +S , (16) mally defined in the infinite Matsubara frequency space.
σ F sb vb 0
4
In order to operate with them we have to introduce a III. SU(4) SYMMETRIC CASE
cut-off for the Matsubara frequencies. Then, the set of
rules which is called F-algebra can be established.23 At
A. F-invariance
the end of all calculations one should tend the cut-off to
infinity.
The global rotations of Q with the matrix exp(iχˆ) At short length scales L ≪ L , L where L =
s v s,v
where χˆ = χαIα play the important role.23,24 For
α,n n n σxx/(16zs,v∆s,v), the symmetry breaking terms Ssb
example, F-algebra allows us to establish the following
andS canbeomittedandtheeffective theorybecomes
P vb
relations p
SU(4)invariantinthecombinedspin-valleyspace. Then,
Eqs.(17)and(18)shouldbesupplementedbytheimpor-
spIαeiχˆQe−iχˆ = spIαQ+2inχα ,
n n −n tantconstraintthatthecombinationz+Γ −4Γremains
2
trηeiχˆQe−iχˆ = trηQ+ in(χα)σ1σ2spIαQσ2σ1 constantin the courseof the RG flow. Physically,it cor-
n τ1τ2 n τ2τ1
αn responds to the conservation of the particle number in
X
− n2(χα)σ1σ2(χα )σ2σ1, (22) the system.5 In the special case of the Coulomb or other
n τ1τ2 −n τ2τ1 long-ranged interactions which are of the main interest
αn
X for us in the paper the relation
where sp stands for the trace over replica and the Mat-
subara frequencies.
z+Γ −4Γ=0 (26)
2
D. Physical observables
holds. With the help of Eqs. (22), one can check that
Eq. (26) guarantees the so-called F-invariance23 of the
The most significant physical quantities in the theory action S +S under the global rotation of the matrix
σ F
containing information on its low-energy dynamics are Q:
physicalobservablesσ′ ,z′,andz′ associatedwiththe
xx s,v
mean-fieldparametersσ ,z,andz oftheaction(16).
xx s,v
Theobservableσ′ istheDCconductivityasonecanob- Q(r)→eiχˆQ(r)e−iχˆ, χˆ= χαIα. (27)
xx n n
tainfromthe linearresponseto anelectromagneticfield.
αn
The observablez′ isrelatedwiththe specific heat.21 The X
observables z′ and z′ determine the static spin (χ′) and
valley (χ′) sussceptibvilities of the 2D electron systesm5,25 Here, χαn is the unit matrix in the spin-valley space. In
as χ′ =v2z′ /π. Extremely important to remind that virtueofEq.(26),itisconvenienttointroducethetriplet
the os,bvservabsl,evparameters σ′ , z′ and z′ are precisely interactionparameterγ =Γ2/zsuchthatΓ=(1+γ)z/4.
xx s,v Wenoticethatthetripletinteractionparameterisrelated
the same as those determined by the background field
procedure.26 with F0σ as γ =−F0σ/(1+F0σ).
The conductivity σ′ is obtained from
xx
σ
σ′ (iω )=− xx tr[Iα,Q][Iα ,Q] (23)
xx n 16n n −n
σ2 (cid:10) (cid:11) B. Perturbative expansions
+ xx dr′hhtrIαQ(r)∇Q(r)trIα Q(r′)∇Q(r′)ii
64Dn n −n
Z
after the analytic continuation to the real frequencies, To define the theory for the perturbative expansions
iω → ω +i0+ at ω → 0. Here, D = 2 stands for the we use the “square-root” parameterization
n
space dimension, and the expectations are defined with
respect to the theory (16).
0 w
A natural definition of z′ is obtained23 through the Q=W +Λ 1−W2, W = . (28)
w† 0
derivativeofthethermodynamicpotentialΩpertheunit (cid:18) (cid:19)
p
volume with respect to T,
The action (16) can be written as the infinite series in
1 ∂ Ω
z′ = . (24) the independent fields wα1α2,σ1σ2 and w†α1α2,σ1σ2. We
2πtrηΛ∂T T n1n2;τ1τ2 n4n3;τ1τ2
usethe conventionthatthe Matsubarafrequencyindices
with odd subscripts n ,n ,... run over non-negative in-
The observables z′ are given as 1 3
s,v tegers whereas those with even subscripts n ,n ,... run
2 4
over negative integers. The propagators can be written
π ∂2Ω
z′ = . (25) in the following form
s,v 2N ∂∆2
r s,v
5
16 32πTzγ
hwα1α2;σ1,σ2(p)w†α4α3;σ4σ3(−p)i= δα1α3δα2α4δ δσ1σ3δσ2σ4δ δ δ D (ω )− δα1α2
n1n2;τ1τ2 n4n3;τ4τ3 σ n12,n34 τ1τ3 τ2τ4 n1,n3 p 12 σ
xx ( xx
h
8πTz(1+γ)
×D (ω )Dt(ω ) + δα1α2δσ1σ3δσ2σ4δ δ Ds(ω )Dt(ω ) , (29)
p 12 p 12 σ τ1τ3 τ2τ4 p 12 p 12
xx )
i
where ω =ω −ω and behavior of the observable parameters with changing of
12 n1 n2
the length scale L
16zω
D−1(ω )=p2+ n, [Ds(ω )]−1 =p2, (30)
p n σxx p n dσxx = β =−2 [1+15f(γ)], (36)
σ
16(z+Γ )ω dξ π
[Dt(ω )]−1 =p2+ 2 n.
p n σ dγ (1+γ)2
xx
= β = , (37)
γ
dξ πσ
xx
dlnz 15γ−1
C. Relation of zs,v with z and γ = γz = . (38)
dξ πσ
xx
The dynamical spin susceptibility χ (ω,p) can be ob-
s Here, f(γ) = 1 − (1 + γ−1)ln(1 + γ), ξ = lnL/l and
tained from5
we omit prime signs for a brevity. Physically, the mi-
χ (iω ,p)=χ0−Tz2htrIασ Q(p)trIα σ Q(−p)i croscopic length l is the mean-free path length. It is the
s n s s n z −n z length at which the bare parameters of the action (16)
(31)
are defined. Renormalization group Eqs. (36)-(38) are
by the analytic continuation to the real frequencies,
valid at short length scales L≪L , L .
iω → ω +i0+. Similar expression is valid for the val- s v
n As is well-known,9 solution of the RG Eqs. (36)-(37)
ley susceptibility. Evaluating Eq. (31) in the tree level
yields the dependence of the resistivity ρ=1/πσ on ξ
approximation with the help of Eqs. (29), we obtain xx
whichhasthemaximumpointandγ(ξ)dependence that
2z 16z ω monotonically increases with ξ.
χ (iω ,p)= s 1− s nDt(ω ) . (32)
s n π σ p n
(cid:18) xx (cid:19)
In the case ∆s = ∆v = 0 the total spin conserves, i.e., IV. SU(2)×SU(2) SYMMETRY CASE
χ(ω,p = 0) = 0. In order to be consistent with this
physical requirement, the relation
A. Effective action
z =z+Γ ≡z(1+γ) (33)
s 2
Inthisandnextsectionsweassumethatthespinsplit-
should hold. Similarly, the total valley isospin conserva- ting is much larger than the valley splitting, ∆s ≫ ∆v.
tion guarantees that Then, at intermediate length scales Ls ≪ L ≪ Lv the
symmetry breaking term S becomes important. In the
sb
z =z+Γ ≡z(1+γ). (34) quadratic approximation it reads
v 2
Being related with the conservation laws, Eqs. (33) and iz ∆ αj,σj
(34) are valid also for the observables: Ssb = s2 s dr (σ2−σ1)wnα11nα22;;τσ11τσ22w¯nα22nα11;;τσ22τσ11
Z nXj,τj
z′ =z′ =z′(1+γ′). (35) (39)
s v
Hence, the modes in Qαβ;σ1σ2 with σ 6= σ acquire a
nm;τ1τ2 1 2
Therefore,threephysicalobservablesσ′ ,z′ andγ′ com- finite mass of the order of z ∆ and, therefore, are neg-
xx s s
pletelydeterminestherenormalizationofthetheory(16) ligible at length scales L ≫ L . As a result, Q becomes
s
at short length scales L≪L ,L . a diagonal matrix in the spin space. Then, the spin sus-
s v
ceptibility has no renormalizationonthese lengthscales,
i.e,
D. One loop renormalization group equations
dz
s
=0, L ≪L≪L . (40)
s v
As is shownin Ref. [9], the standardone-loopanalysis dξ
for the action S +S yields the following renormaliza-
σ F
tiongroupfunctionsthatdeterminethezero-temperature Let us denote Qαβ;±1±1 = [Qαβ ] . Then, the ac-
nm;τ1τ2 nm;τ1τ2 ±
6
tion (16) becomes S =S +S +S where B. Perturbative expansions
σ F vb
σ
S =− xx d2rtr(∇Q )2 (41) In order to resolve the constraint Q2 = 1 we use the
σ σ ±
32
σ=±Z “square-root”parameterization:
X
and Q =W +Λ 1−W2. (48)
± ± ±
q
S = 4πTz d2rtrη(Q −Λ) (42) Then, the action (41) and (42) determines the propaga-
F σ
tors as follows
σ Z
X
32
+ πT d2r Γσ1σ2trInαQσ1trI−αnQσ2 h[wnα11nα22;τ1,τ2(q)]σ[wn†α4n4α33;τ4τ3(−q)]σ′i= σ Dˆσσ′, (49)
xx
Z Xαn σ1,Xσ2=±
where
− πTΓ d2r (trIαQ )⊗(trIα Q ).
2 n σ −n σ Dˆ =δα1α3δα2α4δ δ δτ1τ3δτ2τ4D (ω ,τ ,τ )
Z αn σ=± n12,n34 n1,n3 q 12 1 2
X X
32πT h
Now, the symbol tr stands for the trace over replica, the − Γ δα1α2δτ1τ3δτ2τ4D (ω ,τ ,τ )Dt(ω ,τ ,τ )
σ 2 q 12 1 2 q 12 1 2
Matsubara frequencies, and the valley indices whereas xx
Tr = d2rtr. The action (42) corresponds to the fol- +32πTΓˆδα1α2δτ1τ2δτ3τ4Dˆs(ω )Dt(ω ) (50)
lowing low energy part of the Hamiltonian describing σxx q 12 q 12
R
electron-electron interactions:
with
1 16
Hint = 2 dr ρσ1Γsσ1σ2ρσ2 +maΓtma , (43) [Dˆqs(ωn)]−1 =q2+ σxx(z+Γ2−2Γˆ)ωn (51)
Z hσX1,σ2 i 8z ∆
ρσ = ψ¯τσψτσ, ma = ψ¯τσ(ta)σττσ′ψτσ′ Dq−1(ωn,τ1,τ2)=Dq−1(ωn)+i σvxxv(τ1−τ2), (52)
Xτ σXττ′ [Dt(ω ,τ ,τ )]−1 =[Dt(ω )]−1+i8zv∆v(τ −τ ).
ItisworthwhilementioningthatEq.(43)isinagreement q n 1 2 q n σxx 1 2
with the ideas of Ref. [27,28]. (53)
The symmetry breaking part reads
In the same way as in Sec. IIIC, the conservation of the
total valley isospin guarantees the relation z = z+Γ .
v 2
S =iz ∆ d2rtrτ Q . (44) The conservation of the z-component of the total spin,
sb v v z σ
σX=±Z ρ+−ρ−, implies that zs = 4Γ+− (see Eq. (31)). There-
fore,
AtlengthscalesL∼L ,thecouplingsΓ areallequal
s σ1σ2 dlnΓ
toeachother,Γσ1σ2(L∼Ls)=Γ=(z+Γ2)/4. However, +− =0 (54)
the symmetryallowsthe followingmatrixstructureofΓˆ: dξ
for the length scales L ≪L≪L .
s v
Γ Γ
Γˆ = ++ +− . (45)
Γ Γ
+− ++
(cid:18) (cid:19)
C. One-loop approximation
As we shall see below, this matrix structure is consis-
tent with the renormalization group. Physically, Γ
++ Evaluation of the conductivity according to Eq. (23)
andΓ describeinteractionsbetweenelectronswiththe
+− in the one-loop approximation yields
same and opposite spins, respectively.
The action (41) and (42) is invariant under the global σ′ (iω )=σ + 28π p2T min ωm,1 Dt(ω )
irnottahteiovnasll[eQyαnsmβp;aτc1τe2]fσo(rru)σ→∈SuUστ1(τ23[)Q. αnImnβ;;oτ3rτd4e]σr(tro)[pur−e1s]eστr4vτ2e xx n xx Dσxx Zp ωXm>0 (cid:26)ωn (cid:27) p m
the invariance under the global rotations ×D (ω +ω ) ΓˆDˆs(ω ) −4Γ D (ω ) .
p m n p m 2 p m
σσ
Q (r)→eiχˆQ (r)e−iχˆ, χˆ= χαIα, (46) hσX=±(cid:16) (cid:17) i(55)
± ± n n
αn
X Hence, we find
where χαn is the unit matrix in the valley space, the fol- 27π ω
lowing relation has to be fulfilled σ′ (iω )=σ + p2T min{ m,1}D (ω )
xx n xx Dσ ω p m
xx Zp ωXm>0 n
z+Γ −2Γ =2Γ . (47)
2 ++ +− ×D (ω +ω ) zDs(ω )−6Γ Dt(ω )
p m n p m 2 p m
Physically,thisequationcorrespondstotheparticlenum- h
−(z+2Γ −4Γ )D˜t(ω ) (56)
berconservationandiscompletelyanalogoustoEq.(26). 2 ++ p m
i
7
where Γv
3
64
[D˜t(n)]−1 =q2+ ω Γ . (57)
q σ n +−
xx
Performingtheanalyticcontinuationtotherealfrequen- 2
cies, iω → ω + i0+ in Eq. (56), one obtains the DC
n
conductivity in the one-loop approximation:
1
a
28π ∞
σ′ =σ − p2 dωD2(ω) zDs(ω) b
xx xx Dσ p p
−6Γ Dt(ω)−xx(Zzp+2ΓZ0−4Γ )D˜th(ω) (58) -1 c 1 2 3 Γs
2 p 2 ++ p
In order to compute z′ and z′ we have to evailuate the -1
v
thermodynamic potential Ω in the presence of the finite
FIG. 1: The projection of the RG flow in the three dimen-
valley splitting ∆ . In the one-loop approximation we
v
find sionalparameterspace(σxx,γv,γs)onto(γv,γs)planeforthe
SU(2)×SU(2) symmetrycase (Eqs.(62)-(64)). Dotsdenote
T2∂Ω/T =8N T ω z+ 4 2Γ D˜t(ω ) the line at which 1+6f(γv)+f(γs) = 0. The dashed line
∂T r ωXn>0 nh σxx Zph +− p n indicates the line γv =γs (see text).
−(z+Γ )Dt(ω )+(z+Γ ) Dt(ω ,τ ,τ )
2 p n 2 p n 1 2 Ρ
τX1,τ2
−z D (ω ,τ ,τ ) . (59)
p n 1 2
τX1,τ2 ii
Following definitions (24) and (25) of the physical ob-
servables, we obtain from Eq. (59)
8
z′ =z+ (2Γ −Γ ) D (0) (60)
2 ++ p
σ
xx Zp c
and b
16 3 a
z′ =z 1+4π (z+Γ )T ω zD3(ω ) Ξ
v v σ 2 n p n
" (cid:18) xx(cid:19) ωXn>0 Zph FIG.2: Schematicdependenceoftheresistanceρ=1/(πσxx)
−(z+Γ )Dt3(ω ) . (61) on ξ. Curves a, b, and c corresponds to the flow lines a, b,
2 p n and c in Fig. 1 (see text).
#
i
Wementionthattheresults(58),(60),(61)canbeob-
tained with the helpof the backgroundfield procedure29
Eqs. (62)-(65) constitute one of the main results of the
applied to the action (41)-(42).
present paper and describe the system at the intermedi-
ate length scales L ≪ L ≪ L . We mention that the
s v
length scale l involved in ξ = lnL/l is now of the order
D. One loop RG equations
of L .
s
InFigure1wepresenttheprojectionoftheRGflowin
Using the standard method,30 we derive from
the three dimensional parameter space (σ ,γ ,γ ) onto
xx v s
Eqs.(58),(60)and(61)one-loopresultsfortheRGequa-
(γ ,γ )plane. Thereistheunstablefixedpointatγ =0
v s v
tionswhichdeterminetheT =0behaviorofthephysical
and γ =1. However,for the physicalsystemconsidered
s
observableswithchangingthelengthscaleL. Itisconve-
the fixedpointisinaccessiblesince aninitialpointofthe
nient to define γ =Γ /z andγ =−1+4Γ /z. Then,
v 2 s +− RG flow is always situated near the line γ = γ . As
for D=2 we obtain v s
showninFig.2,there arepossiblethree distincttypesof
dσ 2
xx the ρ(ξ) behavior for such initial points. Along the RG
=− [1+6f(γ )+f(γ )] (62)
v s
dξ π flow line a (Fig. 1) that crosses the curve d described by
dγ 1+γ theequation1+6f(γ )+f(γ )=0theresistancedemon-
v v v s
= (1+2γ −γ ) (63)
dξ πσ v s strates the metallic behavior: ρ decreases as ξ grows. If
xx
we move along the RG flow line b which intersects the
dγ 1+γ
s s
= (1−6γv−γs) (64) curved twice,then the resistancedevelopsthe minimum
dξ πσ
xx andthemaximum. Atlast,theresistanceontheRGflow
dlnz 1
=− [1−6γ −γ ]. (65) line c which has single crossing with the curve d has the
v s
dξ πσxx maximum. Remarkably,inallthreecases,thebehaviorof
8
theresistanceisofthemetallictypeforrelativelylargeL. group. It is worthwhile to mention that if the matrix
The reason of this metallic behavior can be understood Γˆ is diagonal then the theory (69) and (71) would in-
from the following arguments. At large ξ, the coupling clude four copies of the singlet U(1) theory studied in
γ flowstolargepositivevalueswhereasγ →−1. Then, Refs. [7,8]. The action (71) corresponds to the follow-
v s
Γ /Γ ∼1/γ ≪1 and the RG Eqs. (62)-(65) trans- ing low energy part of the electron-electron interaction
+− ++ v
formsintoequationsforthesinglevalleysystemwiththe Hamiltonian:
conductance equal σ /2. The metallic behavior of this
xx
system is well-known.5 Hint = 21 dr ρστ (Γs)σττσ′′ +Γt(ta)σττσ(ta)τσ′′τσ′′ ρστ
Z σσX′;ττ′ h i
ρσ =ψ¯σψσ (73)
V. COMPLETELY SYMMETRY BROKEN CASE τ τ τ
In order to have the invariance under the global rota-
A. Effective action
tions
AtthelonglengthscalesL≫Lv thesymmetrybreak- Q →eiχˆQ e−iχˆ, χˆ= χαIα, (74)
j j n n
ing term S becomes important. In the quadratic ap-
vb αn
proximation it reads X
the following relation has to be fulfilled
iz ∆ αj,σ
Svb = v2 v d2r (τ2−τ1)[wnα11nα22;τ1τ2]σ[w¯nα22nα11;τ2τ1]σ. z+Γ2−Γ++−Γ˜++ =Γ+−+Γ˜+−. (75)
Z nXj,τj
(66)
Hence, the modes in [Qαβ ] with τ 6= τ acquire
nm;τ1τ2 σ 1 2 B. Perturbative expansions
a finite mass of the order of z ∆ . Therefore, they are
v v
negligibleatlonglengthscalesL≫L . Astheresult,the
v
matrix Q becomes diagonal matrix in the valley isospin As above, in order to resolve the constraints Q2 = 1,
j
space. The valley susceptibility remains constant under we shalluse the “square-root”parameterizationfor each
the action of the renormalization group on these length Q : Q = W +Λ 1−W2. Then, the propagators are
j j j j
scales:
defined by the theoqry (69) and (71) as
dz
v
=0, L≫L . (67)
v 32
dξ h[wα1α2(q)] [w†α4α3;τ4τ3(−q)] i= Dˆ , (76)
n1n2 j n4n3 k σ jk
xx
Let us define
32πT
Dˆ =δα1α3δα2α4δ δ D (ω )+ Γˆδα1α2
Qαjβ ={[Qα11β]+,[Qα−β1−1]+,[Qα11β]−,[Qα−β1−1]−}. (68) n12,n34h n1,n3 q 12 σxx
×D (ω )Dˆc(ω ) , (77)
Then the action S =S +S reads q 12 q 12
σ F
i
S =−σxx d2rtr(∇Q )2 (69) where
σ j
32
j Z 16
X [Dˆc(ω )]−1 =q2+ (z−Γˆ)ω . (78)
q n σ n
and xx
S =πT d2r trIαQ Γˆ trIα Q (70) The conservation of the z-components of the total
F n j jk −n k spin, σρσ, and the total valley isospin, τρσ,
Z Xj,k Xαn implies (σsτee Eτq. (31)) that z = 2Γ + 2Γ˜ στ anτd
P s +− P+−
+4πTz d2rtrηQ , (71) z = 2Γ˜ + 2Γ˜ . Since, for L ≫ L both z and
j v ++ +− v s
Xj Z zv are not renormalized, we obtain
where dΓ˜ dΓ dΓ˜
+− +− ++
= = =0. (79)
Γ −Γ Γ˜ Γ˜ Γ dξ dξ dξ
++ 2 ++ +− +−
Γ˜ Γ −Γ Γ Γ˜
Γˆ = Γ˜++−+ +Γ++− 2 Γ+++−−Γ2 Γ˜++−+ . Since, both Γ˜+− and Γ+− coincides at the length scales
 Γ Γ˜ Γ˜ Γ −Γ  L ∼ Lv and they are not renormalized we shall not dis-
 +− +− ++ ++ 2 tinguish Γ˜ and Γ from here onwards. If we intro-
 (72) +− +−
duce γ and γ such that Γ =Γ˜ =z(1+γ )/4 and
s v +− +− s
Initially, at the length scale of the order of L , the cou- Γ˜ = z(1+2γ −γ )/4 then both γ and γ coincide
v ++ v s s v
pling Γ˜ =Γ and Γ˜ =Γ . However, more gen- withthecorrespondingcouplingsoftheprevioussections
++ ++ +− +−
eralstructure(72)is consistentwiththe renormalization at the length scales L∼L .
v
9
C. One-loop approximation Γ
v
3
a
Evaluating the conductivity with the help of Eq. (23)
in the one-loop approximation, we find 2
28π ω
σx′x(iωn)=σxx+ Dσ p2T min ωm,1 1
xx Zp ωXm>0 (cid:26) n (cid:27)
Dp(ωm+ωn)Dp(ωm) ΓˆDˆpc(ωm) jj. (80) -1 1 2 3 Γs
Xj (cid:16) (cid:17)
Hence, -1
28πz ω FIG. 3: The projection of the RG flow in the three dimen-
σ′ (iω )=σ + p2T min m,1
xx n xx Dσ ω sionalparameterspace(σxx,γv,γs)onto(γv,γs)planeforthe
xx Zp ωXm>0 (cid:26) n (cid:27) completely symmetry broken case (Eqs. (86)-(88)). Dots de-
D (ω +ω )D (ω ) Ds(ω )−2γ D¯t(ω )−γ D˜t(ω ) notethe line at which 1+2f(γv)+f(γs)=0 (see text).
p m n p m p m v p m s p m
h (81) i
Ρ
where
32
[D¯t(ω )]−1 =q2+ (Γ +Γ˜ )ω . (82)
q n σ ++ ++ n
xx
Performingtheanalyticcontinuationtotherealfrequen-
cies in Eq. (81), we find
28πz ∞
σ′ =σ − p2 dωD2(ω) Ds(ω)
xx xx Dσ p p
xx Zp Z0 h Ξ
−2γ D¯t(ω)−γ D˜t(ω) . (83)
v p s p
i FIG.4: Schematicdependenceoftheresistanceρ=1/(πσxx)
As in the previous Section, in order to compute z′ we on ξ along theflow line a in Fig. 3 (see text).
evaluate the thermodynamic potential in the one-loop
approximation. The result is
(D=2):
∂Ω/T 8 (1+γ )
T2 =8TN z ω 1+ s D˜t(ω ) dσxx 2
∂T r n σ 2 p n =− [1+2f(γv)+f(γs)] (86)
ωXn>0 h xx Zph dξ π
dγ 1+γ
+(1+γv)D¯pt(ωn)−2Dp(ωn) . (84) dξv = πσ v(1−2γv−γs) (87)
xx
ii
dγ 1+γ
Hence, we obtain s = s(1−2γ −γ ) (88)
v s
dξ πσ
xx
16 dlnz 1
z′ =z+ (Γ −Γ ) D (0). (85) =− [1−2γ −γ ] (89)
2 ++ p v s
σ dξ πσ
xx Zp xx
Therenormalizationgroupequations(86)-(89)constitute
We mention29 that the results (79), (83), and (85)
oneofthemainresultsofthepresentpaper. Wemention
can be obtained with the help of the background field
that the length scale l involved in ξ = lnL/l is now of
procedure applied to the action (69)-(71).
the order of L and Eqs. (86)-(89) describe the system
v
at the long length scales L≫L .
v
The projection of the RG flow for Eqs. (86)-(88) on
D. One loop RG equations
the γ – γ plane is shown in Fig. 3. There exits the
v s
line of the fixed points that is described by the equation
Equations (79), (83) and (85) allow us to derive the 2γ +γ =1. If the initial point has large γ or γ then
v s v s
following one-loop results for the renormalization group the RG flow line crosses the curve that is determined by
functions which determine the T = 0 behavior of the thecondition1+2f(γ )+f(γ )=0. Therefore,theρ(ξ)
v s
physical observables with changing the length scale L dependencealongtheRGflowlinedevelopstheminimum
10
Ρ Ρ
c
b
b
a
a
T
Dv TmHILax Ds Dv TmHILax T
FIG. 5: The schematic ρ(T) dependence in the case of zero FIG. 6: The schematic ρ(T) dependence in the presence of
parallel magnetic field. See text both spin and valley splitting in the case ∆s < ∆v. For the
opposite case, thebehavior will be similar. See text
and will be of the insulating type as is shown in Fig. 4.
Inthepresenceofthesufficientlylowparallelmagnetic
field∆ <T(I) ,theρ(T)behaviorofthreedistincttypes
VI. DISCUSSIONS AND CONCLUSIONS s max
is possible as plotted in Fig. 6. In all three cases, the
ρ(T) dependence has the maximum point at tempera-
The renormalization group equations discussed above ture T = T(I) and is of the insulating type as T → 0.
max
describetheT =0behavioroftheobservableparameters
As follows from Fig. 2, in the intermediate temperature
with changing of the length scale L. At finite tempera-
range,whenT isbetween∆ and∆ ,themetallic(curve
tures T ≫ σ /(zL2 ) where L is the sam- s v
xx sample sample a),insulating(curveb)andnonmonotonic(curvec)types
ple size, the temperature behavior of the physical ob-
of the ρ(T) behavior emerge. As a result, there has to
servables can be found from the RG equations stopped
exist the ρ(T) dependence with two maximum points in
at the inelastic length L rather than at the sample
in the presence of B.
size. Formally, it means that one should substitute
(I)
ξoTbe=yin12glnthσexxfo/l(lzoTwlin2)gfeoqruξatiinont3h1e RG equations with ξT maFxoirmuhmighpominatgnaettiTc fi=elTdsm(Ia)sxucishatbhsaetnt∆, san>d tTwmoaxtypthees
of the ρ(T) behavior are possible as is shown in Fig. 7.
dξT 1dlnz If T(II) < ∆ , then the dependence of the resistivity
=1− . (90) max v
dξ 2 dξ is monotonic and insulating, see the curve a in Fig. 7.
(II)
Here, T denotes the temperature of the maximum
max
Having in mind Eq. (90), we find that the T-behavior
point that appears in the resistivity in accord with the
of the resistivity at B =0 is described by Eqs. (36) and
RGEqs.(62) and(63). In the opposite caseT(II) >∆ ,
(37) for T ≫ ∆ and Eqs. (62)-(64) with interchanged max v
v
a typical ρ(T) dependence is illustrated by the curve b
γ and γ for T ≪ ∆ . In what follows, we assume
v s v
in Fig. 7. Therefore, if the valley splitting is sufficiently
that ∆ < T(I) where T(I) denotes the temperature of
v max max large, i.e., ∆ > T(II), then the monotonic insulating
the maximum point that appears in ρ(T) according to v max
behavior of the resistivity appears in the parallel mag-
theRGEqs.(36)and(37). Ourassumptionisconsistent
withtheexperimentaldatainSi-MOSFETwhere,forex- netic field which corresponds to ∆s ∼ Tm(Ia)x. This is the
ample,32 the valley splitting is of the order of hundreds case for the experiments on the magnetotransport in Si-
of mK and T(I) is about several Kelvins33. Then, de- MOSFET.10,33 However, if the valley splitting is small,
pendingonthmeainxitialconditionsatT ∼1/τ twotypesof ∆v < Tm(IaIx), then the maximum point of the ρ(T) de-
the ρ(T)behaviorarepossibleasis showninFig.5. The pendence surviveseveninhighmagnetic fields but shifts
curve a represents the typical ρ(T) dependence that was down to lower temperatures.
observedintransportexperimentsontwo-valley2Delec- In addition, to interesting T-dependences of the resis-
tron systems in Si-MOS samples2 and n-AlAs quantum tivity, the theory predicts strong renormalization of the
well.34Surprisingly,theotherbehaviorwiththetwomax- electron-electron interaction with temperature. In order
imumpoints ispossible,asillustratedbycurvebinFig.5. to characterize this renormalization, we consider the ra-
So far, this interesting non-monotonic ρ(T) dependence tio χ /χ of valley and spin susceptibilities. In Figure 8,
v s
has been neither observed experimentally nor predicted wepresenttheschematicdependenceofχ /χ onT fora
v s
theoretically. At very low temperatures T ≪ ∆ , the fixed valley splitting but with varying spin splitting. At
v
metallic behavior of ρ(T) wins even in the presence of high temperatures, T ≫ ∆ ,∆ the ratio of the suscep-
v s
the valley splitting. tibilities equals unity, χ /χ = 1. At low temperatures,
v s