Spin-valley interplay in two-dimensional disordered electron liquid
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n 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 andDepartment of Theoretical Physics, Moscow Institute of Physics and Technology, 141700 Moscow, Russia (Dated: December 1, 2018)We report the detailed study of the influence of the spin and valley splittings on such physicalobservables of the two-dimensional disordered electron liquid as resistivity, spin and valley suscepti-bilities. We explain qualitatively the nonmonotonic dependence of the resistivity with temperaturein the presence of a parallel magnetic field. In the presence of either the spin splitting or the valleysplitting we predict novel , with two maximum points, temperature dependence of the resistivity.
PACS numbers: 72.10.-d 71.30.+h, 73.43.Qt 11.10.Hi
I. INTRODUCTION
Disordered two-dimensional (2D) electron systemshave been in the focus of experimental and theoreticalresearch for several decades. Recently, the interest to 2Delectron systems has been renewed because of the exper-imental discovery of metal-insulator transition (MIT) ina high mobility silicon metal-oxide-semiconductor field-effect transistor (Si-MOSFET).
Although, during lastdecade the behavior of resistivity similar to that ofRef. [2,3] has been found experimentally in a wide va-riety of two-dimensional electron systems, the MIT in2D calls still for the theoretical explanation.Very likely, the most promising framework is providedby the microscopic theory, initially developed by Finkel-stein, that combines disorder and strong electron-electroninteraction. Punnoose and Finkelstein have shown pos-sibility for the MIT existence in the special model of 2Delectron system with the infinite number of the spin andvalley degrees of freedom. The current theoretical re-sults do not support the MIT existence for electronswithout the spin and valley degrees of freedom. There-fore, it is natural to assume that the spin and valley de-grees of freedom play a crucial role for the MIT in the2D disordered electron systems.Usually, in the MIT vicinity, from the metallic side,i.e., for an electron density higher than the critical one,and at low temperatures T ≪ τ − the initial increaseof the resistivity ( ρ ) with lowering temperature is re-placed by the decrease of ρ as T becomes lower thansome sample specific temperature. Here, τ tr denotes theelastic scattering time. This nonmonotonic behavior ofthe resistivity has been predicted from the renormaliza-tion group (RG) analysis of the interplay between dis-order and electron-electron interaction in the 2D disor-dered electron systems. As a weak magnetic field B is applied parallel to the 2D plane, decrease of the re-sistivity is stopped at some temperature and ρ increasesagain. Further increase of B leads to the monotonicgrowth of the resistivity as temperature is lowered, i.e.,to an insulating behavior, in the whole T -range. Theseexperimental results suggest the significance of the elec-tron spin for the existence of the metallic phase in the2D disordered electron systems. As is well known, in both Si-MOSFET and n-AlAsquantum well
2D electrons can populate two valleys.Therefore, these systems offer the unique opportunityfor an experimental investigation of an interplay betweenthe spin and valley degrees of freedom. Recently, using asymmetry breaking strain to tune the valley occupationof the 2D electron system in the n-AlAs quantum well, aswell as a parallel magnetic field to adjust the spin polar-ization, the spin - valley interplay has been experimen-tally studied.
However, the electron concentrationsin the experiment were at least three times larger thanthe critical one. Therefore, the spin - valley interplayhas been studied in the region of a good metal very farfrom the metal-insulator transition.In the present paper we report the detailed theoreti-cal results on the T -behavior of the 2D electron systemwith two valleys in the MIT vicinity. In particular, westudy the effect of a parallel magnetic field and/or a val-ley splitting (∆ v ) on the transport, and the spin andvalley susceptibilities. We find that in the presence ofeither the magnetic field or the valley splitting the metal-lic behavior of the resistivity survives down to the zerotemperature. For example, this result implies that at B = 0 the metallic ρ ( T ) dependence can be observed ex-perimentally at temperatures T ≪ ∆ v . Only if both themagnetic field and the valley splitting are present, thenthe metallic behavior of the resistivity crosses over to theinsulating one. Next, we predict novel, with two maxi-mum points , T -behavior of the resistivity in the presenceof the magnetic field and/or the valley splitting. Finally,we find that as T vanishes the ratio of the valley sus-ceptibility ( χ v ) to the spin one ( χ s ) becomes sensitiveto the ratio of the valley splitting to the spin one. Athigh temperatures the ratio χ v /χ s is temperature inde-pendent and can be chosen equal unity. If the spin split-ting is larger (smaller) than the valley splitting, then atlow temperatures the ratio χ v /χ s < ( > )1. If the spinand valley splittings are equal each other, then the ratio χ v /χ s = 1 as temperature vanishes.The presence of the parallel magnetic field and thesymmetry-breaking strain introduces new energy scales∆ s = g L µ B B and ∆ v in the problem. Here, g L and µ B stand for the g -factor and the Bohr magneton, re-spectively. Let us assume that the following conditionshold: ∆ v ≪ ∆ s ≪ /τ tr . In addition, a magnetic field B ⊥ & T / ( De ) is applied perpendicular to the 2D elec-tron system in order to suppress the Cooper channel.Here, e and D denote the electron charge and diffusioncoefficient, respectively. Due to the symmetry breaking,the spin and valley splittings set the cut-off for a polein the diffusion modes (“diffusons”) with opposite spinand valley isospin projections. In the temperature range∆ s ≪ T ≪ τ − this cut-off is irrelevant and the 2D elec-tron system behaves as if no symmetry breaking termsare applied. The temperature behavior of the resistiv-ity is governed by one singlet and 15 triplet diffusivemodes. At low temperatures ∆ v ≪ T ≪ ∆ s , eight dif-fusive modes with opposite spin projections do not con-tribute. Then, the ρ ( T ) dependence is determined bythe remaining one singlet and seven triplet modes. Aswe shall demonstrate below the behavior of the resistiv-ity can be either metallic or insulating. Surprisingly, wefound that the seven triplet diffusive modes are not equiv-alent. They have to split into two groups of six and onemodes for the spin susceptibility be T -independent. Fortemperatures T ≪ ∆ v , next four diffusive modes withopposite isospin projections become ineffective. In thiscase, the temperature dependence of the resistivity is de-termined by one siglet and three triplet diffusive modes.Although, the number of the remaining diffusive modescorresponds formally to single-valley electrons with spin,the ρ ( T ) behavior is insulating.The paper is organized as follows. In Section II weintroduce the nonlinear sigma model that describes thedisordered interacting electron system. Then, we con-sider the short length scales at which the system has SU (4) symmetry in the combined spin and valley space(Sec. III). The behavior of the system at the interme-diate and long length scales is studied in Sec. IV andSec. V, respectively. We end the paper with discussionsof our results and with conclusions (Sec. VI). II. FORMALISMA. Microscopic Hamiltonian
To start out, we consider 2D interacting electronswith two valleys in the presence of a quenched disor-der and a parallel magnetic field at low temperatures T ≪ τ − . We assume that the perpendicular magneticfield B ⊥ & T / ( De ) is applied in order to suppress theCooper channel. Using one electron orbital functions, wewrite an electron annihilation operator as ψ σ ( R ) = X τ = ± ψ στ ( r ) ϕ ( z ) e iτzQ/ , (1)where z denotes the coordinate perpendicular to the 2Dplane, r the in-plane coordinate vector, and R = r + z e z .The subscript τ enumerates two valleys and ψ στ is theannihilation operator of an electron with the spin andisospin projections equal σ/ τ /
2, respectively. Let us assume that the wave functions ϕ ( z ) exp( ± iQz/ R dz ϕ ( z ) exp( iQz ). The vector Q = (0 , , Q ) corre-sponds to the shortest distance between the valley min-ima in the reciprocal space: Q ∼ a − , with a lat being thelattice constant. In the path-integral formulation 2D interacting elec-trons in the presence of the random potential V ( r ) aredescribed by the following grand partition function Z = Z D [ ¯ ψ, ψ ] e S [ ¯ ψ,ψ ] , (2)with the imaginary time action S = Z /T dt h − ¯ ψ στ ( r , t ) ∂ t ψ στ ( r , t ) −H −H dis −H int i . (3)The one-particle Hamiltonian H = Z d r ¯ ψ στ ( r ) h − ∇ m e − µ + ∆ s σ + ∆ v τ i ψ στ ( r ) (4)describes a 2D quasiparticle with mass m e in the pres-ence of the parallel magnetic field and the valley splitting.Here, µ denotes the chemical potential. Next, H dis = Z d r ¯ ψ στ ( r ) V τ τ ( r ) ψ στ ( r ) (5)involves matrix elements of the random potential: V τ τ ( r ) = Z dz V ( R ) ϕ ( z ) e i ( τ − τ ) Qz/ . (6)In general, the matrix elements V τ τ ( r ) induce both theintravalley and intervalley scattering. We suppose that V ( R ) has the Gaussian distribution, and h V ( R ) i = 0 , h V ( R ) V ( R ) i = W ( | r − r | , | z − z | ) , (7)where the function W decays at a typical distance d . If d is larger than the effective width of the 2D electronsystem, i.e., d ≫ [ R dz ϕ ( z )] − , then one can neglect the z -dependence of V ( R ) under the integral sign in Eq. (6).In this case, the intravalley scattering survives only: h V τ τ ( r ) V τ τ ( r ) i = W ( | r − r | , δ τ τ δ τ τ . (8)In the opposite case, d ≪ [ R dz ϕ ( z )] − , one finds h V τ τ ( r ) V τ τ ( r ) i = h δ τ τ δ τ τ ˜ W ( | r − r | ,
0) (9)+ δ τ τ δ τ ,τ δ τ , − τ ˜ W ( | r − r | , Qτ ) i Z dz ϕ ( z ) , where ˜ W ( r, Q ) = R dz W ( r, z ) exp( iQz ). The other cor-relation functions, e.g. with τ = − τ and τ = τ , vanishdue to integration over ( z + z ) / /τ v . Assuming Q − ≪ d , one can ne-glect the intervalley scattering rate in comparison withthe intravalley scattering rate 1 /τ i ∼ ˜ W ( r, n e in 2D elec-tron systems, we consider the case when the followinginequality holds, n e d ≪
1. Then, both Eqs. (8) and (9)read h V τ τ ( r ) V τ τ ( r ) i = 12 πντ i δ τ τ δ τ τ δ ( r − r ) , (10)1 τ i = 2 πν Z d r dz dz W ( r, | z − z | ) ϕ ( z ) ϕ ( z ) ,Q − ≪ d, [ Z ϕ ( z ) dz ] − ≪ n − / e . (11)Here, ν is the thermodynamic density of states. Underconditions (11), the interaction part of the Hamiltonianis invariant under global SU (4) rotations of the electronoperator ψ στ in the combined spin-valley space: H int = e ǫ Z d r d r ¯ ψ σ τ ( r ) ψ σ τ ( r ) ¯ ψ σ τ ( r ) ψ σ τ ( r ) | r − r | . (12)A dielectric constant of a substrate is denoted as ǫ . Thelow energy part of H int can be written as H int = 12 Z d r d r h ρ ( r )Γ s ( r − r ) ρ ( r ) (13)+ m a ( r )Γ t ( r − r ) m a ( r ) i (14)where ρ ( r ) = X στ ¯ ψ στ ( r ) ψ στ ( r ) , (15) m a ( r ) = X σ σ ; τ τ ¯ ψ σ τ ( r )( t a ) σ σ τ τ ψ σ τ ( r ) . Here, Γ s ( q ) = U ( q ) + F ρ / (4 ν ) involves the long-rangepart of the Coulomb interaction U ( q ) = 2 πe / ( qǫ ) andΓ t ( q ) = F σ / (4 ν ). Quantities F σ and F ρ are the stan-dard Fermi liquid interaction parameters in the singletand triplet channels, respectively. The matrices t a with a = 1 , . . .
15 are the non-trivial generators of the SU (4)group. B. Nonlinear sigma model
At low temperatures,
T τ tr ≪
1, the effective quantumtheory of 2D disordered interacting electrons describedby the Hamiltonian (3) is given in terms of the non-linear σ -model. This theory involves unitary matrix field vari-ables Q α α ; σ σ mn ; τ τ ( r ) which obey the nonlinear constraint Q ( r ) = 1. The integers α j = 1 , , . . . , N r denote thereplica indices. The integers m, n correspond to the dis-crete set of the Matsubara frequencies ω n = πT (2 n + 1).The integers σ j = ± τ j = ± S = S σ + S F + S sb + S vb + S , (16) where S σ represents the free electron part S σ = − σ xx
32 Tr( ∇ Q ) . (17)Here, σ xx denotes the mean-field conductivity in units e /h . The symbol Tr stands for the trace over replica,the Matsubara frequencies, spin and valley indices as wellas integration over space coordinates.The term S F = 4 πT z Tr η ( Q − Λ) + πT Γ Z d r X αn tr I αn Q tr I α − n Q − πT Γ Z d r X αn (tr I αn Q ) ⊗ (tr I α − n Q ) . (18)involves the electron-electron interaction amplitudeswhich describe the scattering on small (Γ) and large(Γ ) angles and the quantity z originally introduced byFinkelstein which is responsible for the specific heatrenormalization. The interaction amplitudes are re-lated with the standard Fermi liquid parameters as Γ = − zF σ / (1 + F σ ), 4Γ = Γ + zF ρ / (1 + F ρ ), and z = πν ⋆ / ν ⋆ = m ⋆ / (2 π ) with m ⋆ being the ef-fective mass. The case of the Coulomb interaction corre-sponds to the so-called “unitary” limit, F ρ → ∞ .The symbol tr involves the same operations as in Trexcept the integration over space coordinates, and tr A ⊗ tr B = A αα ; σ σ nn ; τ τ B ββ ; σ σ mm ; τ τ . The matrices Λ, η and I γk aregiven as Λ αβ ; ζ ζ nm = sign ( ω n ) δ nm δ αβ δ ζ ζ ,η αβ ; ζ ζ nm = nδ nm δ αβ δ ζ ζ , (19)( I γk ) αβ ; ζ ζ nm = δ n − m,k δ αγ δ βγ δ ζ ζ . In the absence of ∆ s and ∆ v , the action S σ + S F is invariant under the global rotations Q αβ ; σ σ nm ; τ τ ( r ) → u τ τ σ σ Q αβ ; σ σ nm ; τ τ ( r )[ u − ] τ τ σ σ in the combined spin-valleyspace for u ∈ SU (4). The presence of the parallel mag-netic field and the valley splitting generates the symme-try breaking terms: S sb = iz s ∆ s Tr σ z Q, S vb = iz v ∆ v Tr τ z Q, (20)where σ z and τ z are Pauli matrices in the spin and val-ley spaces, respectively. The Q -independent part of theaction reads S = − πT z Tr η Λ + N r T Z d r (cid:0) χ s ∆ s + χ v ∆ v (cid:1) . (21)with χ s,v = 2 z s,v /π being a bare value of the spin (valley)susceptibility. C. F -algebra The action (16) involves the matrices which are for-mally defined in the infinite Matsubara frequency space.In order to operate with them we have to introduce acut-off for the Matsubara frequencies. Then, the set ofrules which is called F -algebra can be established. Atthe end of all calculations one should tend the cut-off toinfinity.The global rotations of Q with the matrix exp( i ˆ χ )where ˆ χ = P α,n χ αn I αn play the important role. Forexample, F -algebra allows us to establish the followingrelationssp I αn e i ˆ χ Qe − i ˆ χ = sp I αn Q + 2 inχ α − n , tr ηe i ˆ χ Qe − i ˆ χ = tr ηQ + X αn in ( χ αn ) σ σ τ τ sp I αn Q σ σ τ τ − X αn n ( χ αn ) σ σ τ τ ( χ α − n ) σ σ τ τ , (22)where sp stands for the trace over replica and the Mat-subara frequencies. D. Physical observables
The most significant physical quantities in the theorycontaining information on its low-energy dynamics arephysical observables σ ′ xx , z ′ , and z ′ s,v associated with themean-field parameters σ xx , z , and z s,v of the action (16).The observable σ ′ xx is the DC conductivity as one can ob-tain from the linear response to an electromagnetic field.The observable z ′ is related with the specific heat. Theobservables z ′ s and z ′ v determine the static spin ( χ ′ s ) andvalley ( χ ′ v ) susceptibilities of the 2D electron system as χ ′ s,v = 2 z ′ s,v /π . Extremely important to remind thatthe observable parameters σ ′ xx , z ′ s,v and z ′ are preciselythe same as those determined by the background fieldprocedure. The conductivity σ ′ xx is obtained from σ ′ xx ( iω n ) = − σ xx n (cid:10) tr[ I αn , Q ][ I α − n , Q ] (cid:11) (23)+ σ xx D n Z d r ′ hh tr I αn Q ( r ) ∇ Q ( r ) tr I α − n Q ( r ′ ) ∇ Q ( r ′ ) ii after the analytic continuation to the real frequencies, iω n → ω + i + at ω →
0. Here, D = 2 stands for thespace dimension, and the expectations are defined withrespect to the theory (16).A natural definition of z ′ is obtained through thederivative of the thermodynamic potential Ω per the unitvolume with respect to T , z ′ = 12 π tr η Λ ∂∂T Ω T . (24)The observables z ′ s,v are given as z ′ s,v = π N r ∂ Ω ∂ ∆ s,v . (25) III. SU (4) SYMMETRIC CASEA. F -invariance At short length scales L ≪ L s , L v where L s,v = p σ xx / (16 z s,v ∆ s,v ), the symmetry breaking terms S sb and S vb can be omitted and the effective theory becomes SU (4) invariant in the combined spin-valley space. Then,Eqs. (17) and (18) should be supplemented by the impor-tant constraint that the combination z + Γ −
4Γ remainsconstant in the course of the RG flow. Physically, it cor-responds to the conservation of the particle number inthe system. In the special case of the Coulomb or otherlong-ranged interactions which are of the main interestfor us in the paper the relation z + Γ −
4Γ = 0 (26)holds. With the help of Eqs. (22), one can check thatEq. (26) guarantees the so-called F -invariance of theaction S σ + S F under the global rotation of the matrix Q : Q ( r ) → e i ˆ χ Q ( r ) e − i ˆ χ , ˆ χ = X αn χ αn I αn . (27)Here, χ αn is the unit matrix in the spin-valley space. Invirtue of Eq. (26), it is convenient to introduce the tripletinteraction parameter γ = Γ /z such that Γ = (1+ γ ) z/ F σ as γ = − F σ / (1 + F σ ). B. Perturbative expansions
To define the theory for the perturbative expansionswe use the “square-root” parameterization Q = W + Λ p − W , W = (cid:18) ww † (cid:19) . (28)The action (16) can be written as the infinite series inthe independent fields w α α ,σ σ n n ; τ τ and w † α α ,σ σ n n ; τ τ . Weuse the convention that the Matsubara frequency indiceswith odd subscripts n , n , . . . run over non-negative in-tegers whereas those with even subscripts n , n , . . . runover negative integers. The propagators can be writtenin the following form h w α α ; σ ,σ n n ; τ τ ( p ) w † α α ; σ σ n n ; τ τ ( − p ) i = 16 σ xx δ α α δ α α δ n ,n ( δ σ σ δ σ σ δ τ τ δ τ τ h δ n ,n D p ( ω ) − πT zγσ xx δ α α × D p ( ω ) D tp ( ω ) i + 8 πT z (1 + γ ) σ xx δ α α δ σ σ δ σ σ δ τ τ δ τ τ D sp ( ω ) D tp ( ω ) ) , (29)where ω = ω n − ω n and D − p ( ω n ) = p + 16 zω n σ xx , [ D sp ( ω n )] − = p , (30)[ D tp ( ω n )] − = p + 16( z + Γ ) ω n σ xx . C. Relation of z s,v with z and γ The dynamical spin susceptibility χ s ( ω, p ) can be ob-tained from χ s ( iω n , p ) = χ s − T z s h tr I αn σ z Q ( p ) tr I α − n σ z Q ( − p ) i (31)by the analytic continuation to the real frequencies, iω n → ω + i + . Similar expression is valid for the val-ley susceptibility. Evaluating Eq. (31) in the tree levelapproximation with the help of Eqs. (29), we obtain χ s ( iω n , p ) = 2 z s π (cid:18) − z s ω n σ xx D tp ( ω n ) (cid:19) . (32)In the case ∆ s = ∆ v = 0 the total spin conserves, i.e., χ ( ω, p = 0) = 0. In order to be consistent with thisphysical requirement, the relation z s = z + Γ ≡ z (1 + γ ) (33)should hold. Similarly, the total valley isospin conserva-tion guarantees that z v = z + Γ ≡ z (1 + γ ) . (34)Being related with the conservation laws, Eqs. (33) and(34) are valid also for the observables: z ′ s = z ′ v = z ′ (1 + γ ′ ) . (35)Therefore, three physical observables σ ′ xx , z ′ and γ ′ com-pletely determines the renormalization of the theory (16)at short length scales L ≪ L s , L v . D. One loop renormalization group equations
As is shown in Ref. [9], the standard one-loop analysisfor the action S σ + S F yields the following renormaliza-tion group functions that determine the zero-temperature behavior of the observable parameters with changing ofthe length scale Ldσ xx dξ = β σ = − π [1 + 15 f ( γ )] , (36) dγdξ = β γ = (1 + γ ) πσ xx , (37) d ln zdξ = γ z = 15 γ − πσ xx . (38)Here, f ( γ ) = 1 − (1 + γ − ) ln(1 + γ ), ξ = ln L/l andwe omit prime signs for a brevity. Physically, the mi-croscopic length l is the mean-free path length. It is thelength at which the bare parameters of the action (16)are defined. Renormalization group Eqs. (36)-(38) arevalid at short length scales L ≪ L s , L v .As is well-known, solution of the RG Eqs. (36)-(37)yields the dependence of the resistivity ρ = 1 /πσ xx on ξ which has the maximum point and γ ( ξ ) dependence thatmonotonically increases with ξ . IV. SU (2) × SU (2) SYMMETRY CASEA. Effective action
In this and next sections we assume that the spin split-ting is much larger than the valley splitting, ∆ s ≫ ∆ v .Then, at intermediate length scales L s ≪ L ≪ L v thesymmetry breaking term S sb becomes important. In thequadratic approximation it reads S sb = iz s ∆ s Z d r α j ,σ j X n j ,τ j ( σ − σ ) w α α ; σ σ n n ; τ τ ¯ w α α ; σ σ n n ; τ τ (39)Hence, the modes in Q αβ ; σ σ nm ; τ τ with σ = σ acquire afinite mass of the order of z s ∆ s and, therefore, are neg-ligible at length scales L ≫ L s . As a result, Q becomesa diagonal matrix in the spin space. Then, the spin sus-ceptibility has no renormalization on these length scales,i.e, dz s dξ = 0 , L s ≪ L ≪ L v . (40)Let us denote Q αβ ; ± ± nm ; τ τ = [ Q αβnm ; τ τ ] ± . Then, the ac-tion (16) becomes S = S σ + S F + S vb where S σ = − σ xx X σ = ± Z d r tr( ∇ Q σ ) (41)and S F = 4 πT z X σ Z d r tr η ( Q σ − Λ) (42)+ πT Z d r X αn X σ ,σ = ± Γ σ σ tr I αn Q σ tr I α − n Q σ − πT Γ Z d r X αn X σ = ± (tr I αn Q σ ) ⊗ (tr I α − n Q σ ) . Now, the symbol tr stands for the trace over replica, theMatsubara frequencies, and the valley indices whereasTr = R d r tr. The action (42) corresponds to the fol-lowing low energy part of the Hamiltonian describingelectron-electron interactions: H int = 12 Z d r h X σ ,σ ρ σ Γ σ σ s ρ σ + m a Γ t m a i , (43) ρ σ = X τ ¯ ψ στ ψ στ , m a = X σττ ′ ¯ ψ στ ( t a ) σσττ ′ ψ στ ′ It is worthwhile mentioning that Eq. (43) is in agreementwith the ideas of Ref. [27,28].The symmetry breaking part reads S sb = iz v ∆ v X σ = ± Z d r tr τ z Q σ . (44)At length scales L ∼ L s , the couplings Γ σ σ are all equalto each other, Γ σ σ ( L ∼ L s ) = Γ = ( z +Γ ) /
4. However,the symmetry allows the following matrix structure of ˆΓ:ˆΓ = (cid:18) Γ ++ Γ + − Γ + − Γ ++ (cid:19) . (45)As we shall see below, this matrix structure is consis-tent with the renormalization group. Physically, Γ ++ and Γ + − describe interactions between electrons with thesame and opposite spins, respectively.The action (41) and (42) is invariant under the globalrotations [ Q αβnm ; τ τ ] σ ( r ) → u τ τ σ [ Q αβ ; nm ; τ τ ] σ ( r )[ u − ] τ τ σ in the valley space for u σ ∈ SU (2). In order to preservethe invariance under the global rotations Q ± ( r ) → e i ˆ χ Q ± ( r ) e − i ˆ χ , ˆ χ = X αn χ αn I αn , (46)where χ αn is the unit matrix in the valley space, the fol-lowing relation has to be fulfilled z + Γ − ++ = 2Γ + − . (47)Physically, this equation corresponds to the particle num-ber conservation and is completely analogous to Eq. (26). B. Perturbative expansions
In order to resolve the constraint Q ± = 1 we use the“square-root” parameterization: Q ± = W ± + Λ q − W ± . (48)Then, the action (41) and (42) determines the propaga-tors as follows h [ w α α ; τ ,τ n n ( q )] σ [ w † α α ; τ τ n n ( − q )] σ ′ i = 32 σ xx ˆ D σσ ′ , (49)whereˆ D = δ α α δ α α δ n ,n h δ n ,n δ τ τ δ τ τ D q ( ω , τ , τ ) − πTσ xx Γ δ α α δ τ τ δ τ τ D q ( ω , τ , τ ) D tq ( ω , τ , τ )+ 32 πTσ xx ˆΓ δ α α δ τ τ δ τ τ ˆ D sq ( ω ) D tq ( ω ) (50)with [ ˆ D sq ( ω n )] − = q + 16 σ xx ( z + Γ − ω n (51) D − q ( ω n , τ , τ ) = D − q ( ω n ) + i z v ∆ v σ xx ( τ − τ ) , (52)[ D tq ( ω n , τ , τ )] − = [ D tq ( ω n )] − + i z v ∆ v σ xx ( τ − τ ) . (53)In the same way as in Sec. III C, the conservation of thetotal valley isospin guarantees the relation z v = z + Γ .The conservation of the z -component of the total spin, ρ + − ρ − , implies that z s = 4Γ + − (see Eq. (31)). There-fore, d ln Γ + − dξ = 0 (54)for the length scales L s ≪ L ≪ L v . C. One-loop approximation
Evaluation of the conductivity according to Eq. (23)in the one-loop approximation yields σ ′ xx ( iω n ) = σ xx + 2 π D σ xx Z p p T X ω m > min (cid:26) ω m ω n , (cid:27) D tp ( ω m ) × D p ( ω m + ω n ) h X σ = ± (cid:16) ˆΓ ˆ D sp ( ω m ) (cid:17) σσ − D p ( ω m ) i . (55)Hence, we find σ ′ xx ( iω n ) = σ xx + 2 π D σ xx Z p p T X ω m > min { ω m ω n , } D p ( ω m ) × D p ( ω m + ω n ) h zD sp ( ω m ) − D tp ( ω m ) − ( z + 2Γ − ++ ) ˜ D tp ( ω m ) i (56)where [ ˜ D tq ( n )] − = q + 64 σ xx ω n Γ + − . (57)Performing the analytic continuation to the real frequen-cies, iω n → ω + i + in Eq. (56), one obtains the DCconductivity in the one-loop approximation: σ ′ xx = σ xx − π D σ xx Z p p Z ∞ dω D p ( ω ) h zD sp ( ω ) − D tp ( ω ) − ( z + 2Γ − ++ ) ˜ D tp ( ω ) i (58)In order to compute z ′ and z ′ v we have to evaluate thethermodynamic potential Ω in the presence of the finitevalley splitting ∆ v . In the one-loop approximation wefind T ∂ Ω /T∂T = 8 N r T X ω n > ω n h z + 4 σ xx Z p h + − ˜ D tp ( ω n ) − ( z + Γ ) D tp ( ω n ) + ( z + Γ ) X τ ,τ D tp ( ω n , τ , τ ) − z X τ ,τ D p ( ω n , τ , τ ) ii . (59)Following definitions (24) and (25) of the physical ob-servables, we obtain from Eq. (59) z ′ = z + 8 σ xx (2Γ − Γ ++ ) Z p D p (0) (60)and z ′ v = z v " π (cid:18) σ xx (cid:19) ( z + Γ ) T X ω n > ω n Z p h zD p ( ω n ) − ( z + Γ ) D t p ( ω n ) i . (61)We mention that the results (58), (60), (61) can be ob-tained with the help of the background field procedure applied to the action (41)-(42). D. One loop RG equations
Using the standard method, we derive fromEqs. (58), (60) and (61) one-loop results for the RG equa-tions which determine the T = 0 behavior of the physicalobservables with changing the length scale L . It is conve-nient to define γ v = Γ /z and γ s = − + − /z . Then,for D = 2 we obtain dσ xx dξ = − π [1 + 6 f ( γ v ) + f ( γ s )] (62) dγ v dξ = 1 + γ v πσ xx (1 + 2 γ v − γ s ) (63) dγ s dξ = 1 + γ s πσ xx (1 − γ v − γ s ) (64) d ln zdξ = − πσ xx [1 − γ v − γ s ] . (65) Γ s Γ v - - FIG. 1: The projection of the RG flow in the three dimen-sional parameter space ( σ xx , γ v , γ s ) onto ( γ v , γ s ) plane for the SU (2) × SU (2) symmetry case (Eqs. (62)-(64)). Dots denotethe line at which 1 + 6 f ( γ v ) + f ( γ s ) = 0. The dashed lineindicates the line γ v = γ s (see text). ΞΡ ab c FIG. 2: Schematic dependence of the resistance ρ = 1 / ( πσ xx )on ξ . Curves a , b , and c corresponds to the flow lines a , b ,and c in Fig. 1 (see text). Eqs. (62)-(65) constitute one of the main results of thepresent paper and describe the system at the intermedi-ate length scales L s ≪ L ≪ L v . We mention that thelength scale l involved in ξ = ln L/l is now of the orderof L s .In Figure 1 we present the projection of the RG flow inthe three dimensional parameter space ( σ xx , γ v , γ s ) onto( γ v , γ s ) plane. There is the unstable fixed point at γ v = 0and γ s = 1. However, for the physical system consideredthe fixed point is inaccessible since an initial point of theRG flow is always situated near the line γ v = γ s . Asshown in Fig. 2, there are possible three distinct types ofthe ρ ( ξ ) behavior for such initial points. Along the RGflow line a (Fig. 1) that crosses the curve d described bythe equation 1+6 f ( γ v )+ f ( γ s ) = 0 the resistance demon-strates the metallic behavior: ρ decreases as ξ grows. Ifwe move along the RG flow line b which intersects thecurve d twice, then the resistance develops the minimumand the maximum. At last, the resistance on the RG flowline c which has single crossing with the curve d has themaximum. Remarkably, in all three cases, the behavior ofthe resistance is of the metallic type for relatively large L .The reason of this metallic behavior can be understoodfrom the following arguments. At large ξ , the coupling γ v flows to large positive values whereas γ s → −
1. Then,Γ + − / Γ ++ ∼ /γ v ≪ σ xx /
2. The metallic behavior of thissystem is well-known. V. COMPLETELY SYMMETRY BROKEN CASEA. Effective action
At the long length scales L ≫ L v the symmetry break-ing term S vb becomes important. In the quadratic ap-proximation it reads S vb = iz v ∆ v Z d r α j ,σ X n j ,τ j ( τ − τ ) [ w α α n n ; τ τ ] σ [ ¯ w α α n n ; τ τ ] σ . (66)Hence, the modes in [ Q αβnm ; τ τ ] σ with τ = τ acquirea finite mass of the order of z v ∆ v . Therefore, they arenegligible at long length scales L ≫ L v . As the result, thematrix Q becomes diagonal matrix in the valley isospinspace. The valley susceptibility remains constant underthe action of the renormalization group on these lengthscales: dz v dξ = 0 , L ≫ L v . (67)Let us define Q αβj = { [ Q αβ ] + , [ Q αβ − − ] + , [ Q αβ ] − , [ Q αβ − − ] − } . (68)Then the action S = S σ + S F reads S σ = − σ xx X j Z d r tr( ∇ Q j ) (69)and S F = πT Z d r X j,k X αn tr I αn Q j ˆΓ jk tr I α − n Q k (70)+4 πT z X j Z d r tr ηQ j , (71)whereˆΓ = Γ ++ − Γ ˜Γ ++ ˜Γ + − Γ + − ˜Γ ++ Γ ++ − Γ Γ + − ˜Γ + − ˜Γ + − Γ + − Γ ++ − Γ ˜Γ ++ Γ + − ˜Γ + − ˜Γ ++ Γ ++ − Γ . (72)Initially, at the length scale of the order of L v , the cou-pling ˜Γ ++ = Γ ++ and ˜Γ + − = Γ + − . However, more gen-eral structure (72) is consistent with the renormalization group. It is worthwhile to mention that if the matrixˆΓ is diagonal then the theory (69) and (71) would in-clude four copies of the singlet U (1) theory studied inRefs. [7,8]. The action (71) corresponds to the follow-ing low energy part of the electron-electron interactionHamiltonian: H int = 12 Z d r X σσ ′ ; ττ ′ ρ στ h (Γ s ) σσ ′ ττ ′ + Γ t ( t a ) σσττ ( t a ) σ ′ σ ′ τ ′ τ ′ i ρ στ ρ στ = ¯ ψ στ ψ στ (73)In order to have the invariance under the global rota-tions Q j → e i ˆ χ Q j e − i ˆ χ , ˆ χ = X αn χ αn I αn , (74)the following relation has to be fulfilled z + Γ − Γ ++ − ˜Γ ++ = Γ + − + ˜Γ + − . (75) B. Perturbative expansions
As above, in order to resolve the constraints Q j = 1,we shall use the “square-root” parameterization for each Q j : Q j = W j + Λ q − W j . Then, the propagators aredefined by the theory (69) and (71) as h [ w α α n n ( q )] j [ w † α α ; τ τ n n ( − q )] k i = 32 σ xx ˆ D jk , (76)ˆ D = δ α α δ α α δ n ,n h δ n ,n D q ( ω ) + 32 πTσ xx ˆΓ δ α α × D q ( ω ) ˆ D cq ( ω ) i , (77)where [ ˆ D cq ( ω n )] − = q + 16 σ xx ( z − ˆΓ) ω n . (78)The conservation of the z -components of the totalspin, P στ σρ στ , and the total valley isospin, P στ τ ρ στ ,implies (see Eq. (31)) that z s = 2Γ + − + 2˜Γ + − and z v = 2˜Γ ++ + 2˜Γ + − . Since, for L ≫ L v both z s and z v are not renormalized, we obtain d ˜Γ + − dξ = d Γ + − dξ = d ˜Γ ++ dξ = 0 . (79)Since, both ˜Γ + − and Γ + − coincides at the length scales L ∼ L v and they are not renormalized we shall not dis-tinguish ˜Γ + − and Γ + − from here onwards. If we intro-duce γ s and γ v such that Γ + − = ˜Γ + − = z (1 + γ s ) / ++ = z (1 + 2 γ v − γ s ) / γ s and γ v coincidewith the corresponding couplings of the previous sectionsat the length scales L ∼ L v . C. One-loop approximation
Evaluating the conductivity with the help of Eq. (23)in the one-loop approximation, we find σ ′ xx ( iω n ) = σ xx + 2 π D σ xx Z p p T X ω m > min (cid:26) ω m ω n , (cid:27) D p ( ω m + ω n ) D p ( ω m ) X j (cid:16) ˆΓ ˆ D cp ( ω m ) (cid:17) jj . (80)Hence, σ ′ xx ( iω n ) = σ xx + 2 πz D σ xx Z p p T X ω m > min (cid:26) ω m ω n , (cid:27) D p ( ω m + ω n ) D p ( ω m ) h D sp ( ω m ) − γ v ¯ D tp ( ω m ) − γ s ˜ D tp ( ω m ) i (81)where [ ¯ D tq ( ω n )] − = q + 32 σ xx (Γ ++ + ˜Γ ++ ) ω n . (82)Performing the analytic continuation to the real frequen-cies in Eq. (81), we find σ ′ xx = σ xx − πz D σ xx Z p p Z ∞ dωD p ( ω ) h D sp ( ω ) − γ v ¯ D tp ( ω ) − γ s ˜ D tp ( ω ) i . (83)As in the previous Section, in order to compute z ′ weevaluate the thermodynamic potential in the one-loopapproximation. The result is T ∂ Ω /T∂T = 8 T N r z X ω n > ω n h σ xx Z p h (1 + γ s )2 ˜ D tp ( ω n )+(1 + γ v ) ¯ D tp ( ω n ) − D p ( ω n ) ii . (84)Hence, we obtain z ′ = z + 16 σ xx (Γ − Γ ++ ) Z p D p (0) . (85)We mention that the results (79), (83), and (85)can be obtained with the help of the background fieldprocedure applied to the action (69)-(71). D. One loop RG equations
Equations (79), (83) and (85) allow us to derive thefollowing one-loop results for the renormalization groupfunctions which determine the T = 0 behavior of thephysical observables with changing the length scale L Γ s Γ v a - - FIG. 3: The projection of the RG flow in the three dimen-sional parameter space ( σ xx , γ v , γ s ) onto ( γ v , γ s ) plane for thecompletely symmetry broken case (Eqs. (86)-(88)). Dots de-note the line at which 1 + 2 f ( γ v ) + f ( γ s ) = 0 (see text). ΞΡ FIG. 4: Schematic dependence of the resistance ρ = 1 / ( πσ xx )on ξ along the flow line a in Fig. 3 (see text). ( D = 2): dσ xx dξ = − π [1 + 2 f ( γ v ) + f ( γ s )] (86) dγ v dξ = 1 + γ v πσ xx (1 − γ v − γ s ) (87) dγ s dξ = 1 + γ s πσ xx (1 − γ v − γ s ) (88) d ln zdξ = − πσ xx [1 − γ v − γ s ] (89)The renormalization group equations (86)-(89) constituteone of the main results of the present paper. We mentionthat the length scale l involved in ξ = ln L/l is now ofthe order of L v and Eqs. (86)-(89) describe the systemat the long length scales L ≫ L v .The projection of the RG flow for Eqs. (86)-(88) onthe γ v – γ s plane is shown in Fig. 3. There exits theline of the fixed points that is described by the equation2 γ v + γ s = 1. If the initial point has large γ v or γ s thenthe RG flow line crosses the curve that is determined bythe condition 1 + 2 f ( γ v ) + f ( γ s ) = 0. Therefore, the ρ ( ξ )dependence along the RG flow line develops the minimum0 T Ρ T max H I L ab D v FIG. 5: The schematic ρ ( T ) dependence in the case of zeroparallel magnetic field. See text and will be of the insulating type as is shown in Fig. 4. VI. DISCUSSIONS AND CONCLUSIONS
The renormalization group equations discussed abovedescribe the T = 0 behavior of the observable parameterswith changing of the length scale L . At finite tempera-tures T ≫ σ xx / ( zL sample ) where L sample is the sam-ple size, the temperature behavior of the physical ob-servables can be found from the RG equations stoppedat the inelastic length L in rather than at the samplesize. Formally, it means that one should substitute ξ T = ln σ xx / ( zT l ) for ξ in the RG equations with ξ T obeying the following equation dξ T dξ = 1 − d ln zdξ . (90)Having in mind Eq. (90), we find that the T -behaviorof the resistivity at B = 0 is described by Eqs. (36) and(37) for T ≫ ∆ v and Eqs. (62)-(64) with interchanged γ v and γ s for T ≪ ∆ v . In what follows, we assumethat ∆ v < T ( I )max where T ( I )max denotes the temperature ofthe maximum point that appears in ρ ( T ) according tothe RG Eqs. (36) and (37). Our assumption is consistentwith the experimental data in Si-MOSFET where, for ex-ample, the valley splitting is of the order of hundredsof mK and T ( I )max is about several Kelvins . Then, de-pending on the initial conditions at T ∼ /τ two types ofthe ρ ( T ) behavior are possible as is shown in Fig. 5. Thecurve a represents the typical ρ ( T ) dependence that wasobserved in transport experiments on two-valley 2D elec-tron systems in Si-MOS samples and n-AlAs quantumwell. Surprisingly, the other behavior with the two max-imum points is possible, as illustrated by curve b in Fig. 5.So far, this interesting non-monotonic ρ ( T ) dependencehas been neither observed experimentally nor predictedtheoretically. At very low temperatures T ≪ ∆ v , themetallic behavior of ρ ( T ) wins even in the presence ofthe valley splitting. T ΡD v D s abc T max H I L FIG. 6: The schematic ρ ( T ) dependence in the presence ofboth spin and valley splitting in the case ∆ s < ∆ v . For theopposite case, the behavior will be similar. See text In the presence of the sufficiently low parallel magneticfield ∆ s < T ( I )max , the ρ ( T ) behavior of three distinct typesis possible as plotted in Fig. 6. In all three cases, the ρ ( T ) dependence has the maximum point at tempera-ture T = T ( I )max and is of the insulating type as T → T is between ∆ s and ∆ v , the metallic (curve a ), insulating (curve b ) and nonmonotonic (curve c ) typesof the ρ ( T ) behavior emerge. As a result, there has toexist the ρ ( T ) dependence with two maximum points inthe presence of B .For high magnetic fields such that ∆ s > T ( I )max themaximum point at T = T ( I )max is absent, and two typesof the ρ ( T ) behavior are possible as is shown in Fig. 7.If T ( II )max < ∆ v , then the dependence of the resistivityis monotonic and insulating, see the curve a in Fig. 7.Here, T ( II )max denotes the temperature of the maximumpoint that appears in the resistivity in accord with theRG Eqs. (62) and (63). In the opposite case T ( II )max > ∆ v ,a typical ρ ( T ) dependence is illustrated by the curve b in Fig. 7. Therefore, if the valley splitting is sufficientlylarge, i.e., ∆ v > T ( II )max , then the monotonic insulatingbehavior of the resistivity appears in the parallel mag-netic field which corresponds to ∆ s ∼ T ( I )max . This is thecase for the experiments on the magnetotransport in Si-MOSFET. However, if the valley splitting is small,∆ v < T ( II )max , then the maximum point of the ρ ( T ) de-pendence survives even in high magnetic fields but shiftsdown to lower temperatures.In addition, to interesting T -dependences of the resis-tivity, the theory predicts strong renormalization of theelectron-electron interaction with temperature. In orderto characterize this renormalization, we consider the ra-tio χ v /χ s of valley and spin susceptibilities. In Figure 8,we present the schematic dependence of χ v /χ s on T for afixed valley splitting but with varying spin splitting. Athigh temperatures, T ≫ ∆ v , ∆ s the ratio of the suscep-tibilities equals unity, χ v /χ s = 1. At low temperatures,1 T ΡD v D s ab FIG. 7: The schematic ρ ( T ) dependence in the presence ofstrong parallel magnetic field: ∆ v , T ( I )max < ∆ s . See text T1 Χ v (cid:144) Χ s D v T a T c abc FIG. 8: The schematic dependence of the ration χ v /χ s ontemperature: a) for ∆ s < ∆ v ; b) for ∆ s = ∆ v ; c) for ∆ s > ∆ v . The temperature scales T a,c ≡ ∆ s . T ≪ ∆ v , ∆ s , we find χ v χ s = < , ∆ s < ∆ v , ∆ s = ∆ v > , ∆ s > ∆ v . (91)Therefore, the ratio χ v /χ s at T → v / ∆ s . This can be used for the experimental de-termination of the valley splitting in the 2D electron sys-tem.Finally, we remind that we do not consider abovethe contribution to the one-loop RG equations from theparticle-particle (Cooper) channel. It can be shown (seeAppendix) that neither the spin splitting nor the val-ley splitting does not change the “cooperon” contribu-tion to the RG equations in the one-loop approximation.Therefore, the Cooper-channel contribution to the RGequations discussed above can be taken into account bythe substitution of 1 + 2 for 1 in the square brackets ofEqs. (36), (62) and (86). The Cooper-channel contribu-tion does not change qualitative behavior of the resis-tivity, and the valley and spin susceptibilities discussedabove.To summarize, we have obtained the novel results on the temperature behavior of such physical observables asthe resistivity, spin and valley susceptibilities in 2D elec-tron liquid with two valleys in the MIT vicinity and inthe presence of both the parallel magnetic field and thevalley splitting. First, we found that the metallic be-havior of the resistivity at low temperatures survives inthe presence of only the parallel magnetic field or thevalley splitting. If both the spin splitting and the val-ley splitting exist then the metallic ρ -dependence crossesover to insulating one at low temperatures. Second, wehave predicted the existence of the novel, nonmonotonicdependence of resistivity at zero and finite magnetic fieldin which the ρ ( T ) has two maximum points. It would bean experimental challenge to identify this novel regime. Acknowledgments
The authors are grateful to D.A. Knyazev,A.A. Kuntzevich, O.E. Omelyanovsky, and V.M. Pudalovfor the detailed discussions of their experimental data.The research was funded in part by CRDF, the RussianMinistry of Education and Science, Council for Grants ofthe President of Russian Federation, RFBR 07-02-00998-a and 06-02-16708-a, Dynasty Foundation, Programs ofRAS, and Russian Science Support Foundation.
APPENDIX A: “COOPERON” CONTRIBUTIONTO THE CONDUCTANCE
We start from the standard equation for the“cooperon” C σ σ ; τ τ σ σ ; τ τ ( q ) = δ σ ,σ δ τ ,τ δ σ ,σ δ τ ,τ πντ i + I σ σ ; τ τ σ σ ; τ τ ( q ) C σ σ ; τ τ σ σ ; τ τ ( q ) (A1)where the “impurity ladder” is given as I σ ,σ ; τ τ σ σ ; τ τ ( q ) = 12 πντ i Z p G Rσ ,σ ; τ τ ( p + ) G Aσ σ ; τ τ ( p − )(A2)Here, p ± = p ± q/ G R ( A ) σ ,σ ; τ τ ( p )] − = δ σ ,σ δ τ τ h p m e − µ + ∆ s sgn σ +∆ v sgn τ ± i/ τ i i . (A3)Performing integration, we find for q → I σ ,σ ; τ τ σ σ ; τ τ ( q ) = δ σ ,σ δ σ σ δ τ τ δ τ τ h − Dq τ i − i ∆ s τ i (sgn σ − sgn σ ) − i ∆ v τ i (sgn τ − sgn τ ) i . (A4)2Next, solving Eq. (A1), we obtain C σ ,σ ; τ τ σ σ ; τ τ ( q ) = δ σ σ δ σ σ δ τ τ δ τ τ πντ i h Dq (A5)+ i ∆ s (sgn σ − sgn σ ) + i ∆ v (sgn τ − sgn τ ) i − The interference correction to the conductance is givenas δσ xx = − Dτ i Z p G R ( − p ) σ σ ; τ τ G Aσ σ ; τ τ ( − p ) (A6) × G Rσ σ ; τ τ ( p ) G Aσ σ ; τ τ ( p ) Z d q (2 π ) C σ σ ; τ τ σ σ ; τ τ ( q ) . Using the result: Z p G R ( − p ) σ σ ; τ τ G Aσ σ ; τ τ ( − p ) G Rσ σ ; τ τ ( p ) G Aσ σ ; τ τ ( p )= 4 πντ i δ σ σ δ σ σ δ τ τ δ τ τ s τ i (sgn σ − sgn σ ) + ∆ v τ i (sgn τ − sgn τ ) , (A7) we find δσ xx = − τ τ X σ σ δ σ σ δ τ τ h s τ i (sgn σ − sgn σ ) +∆ v τ i (sgn τ − sgn τ ) i − D Z d q (2 π ) h Dq + i ∆ s (sgn σ − sgn σ ) + i ∆ v (sgn τ − sgn τ ) i − = − π Z dqq . (A8)Therefore, neither the spin splitting nor the valley split-ting affect the Cooper channel (interference) contributionto the conductance. T. Ando, A.B. Fowler, and F. Stern, Rev. Mod. Phys. ,437 (1982). S.V. Kravchenko, G.V. Kravchenko, J.E. Furneaux,V.M. Pudalov, M. D’Iorio, Phys. Rev. B , 8039 (1994). S.V. Kravchenko, W.E. Mason, G.E. Bowker, J.E.Furneaux, V.M. Pudalov, and M.D’Iorio, Phys. Rev. B E. Abrahams, S.V. Kravchenko, and M.P. Sarachik, Rev.Mod. Phys. , 251 (2001); S.V. Kravchenko, andM.P. Sarachik, Rep. Prog. Phys, , 1 (2004). A.M. Finkelstein,
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