Localized - delocalized electron quantum phase transitions
aa r X i v : . [ c ond - m a t . d i s - nn ] M a r LOCALIZED (cid:22) DELOCALIZED ELECTRON QUANTUM PHASE TRANSITIONSV.F.Gantmakher, V.T.DolgopolovInstitute of Solid State Physi s RAS,Chernogolovka 142432, Russiatel. +7-496-5225425, +7-496-5222946,fax +7-496-5249701E-mail: gantmissp.a .ru, dolgopissp.a .ruMetal(cid:21)insulator transitions and transitions between di(cid:27)erent quantum Hall liquids are used todes ribe the physi al ideas forming the basis of quantum phase transitions and the methods ofappli ation of theoreti al results in pro essing experimental data. The following two theoreti als hemes are dis ussed and ompared: the general theory of quantum phase transitions, whi h hasbeen developed a ording to the theory of thermodynami phase transitions and relies on the on eptof a partition fun tion, and a theory whi h is based on a s aling hypothesis and the renormalization-group on ept borrowed from quantum ele trodynami s, with the results formulated in terms of (cid:29)owdiagrams.CONTENT1. Introdu tion1.1. Thermal and quantum (cid:29)u tuations.2. Quantum phase transitions2.1. De(cid:28)nition of a quantum phase transition . 2.2. Parallels and di(cid:27)eren es between lassi al and quantum phasetransitions 2.3. Criti al region of a quantum phase transition.3. Flow diagrams for metal(cid:21)insulator transitions4. Three-dimensional ele tron gas5. Two-dimensional ele tron gas5.1. Gas of nonintera ting ele trons 5.2. Spin(cid:21)orbit intera tion 5.3. Gas of intera ting ele trons.6. Quantum transitions between the di(cid:27)erent states of a Hall liquid7. Con lusion 1. INTRODUCTIONQuantum phase transitions that o ur in an ele tron gas and result in lo alization represent the mainstream inthe study of ele trons in solids; it is dire ted toward low temperatures and intera tions. We think that a dangerousgap between theory and experiment has appeared in this (cid:28)eld of s ien e. The related theories are so omplex thatexperimentalists, as rule, use ready re ipes and formulas of these theories without knowing (and, hen e, without ontrolling) all the initial assumptions and limitations.In this review, we will try to lose this gap. Our review does not ontain a sequential des ription of the mathemati alte hniques involved in formulating hypotheses and theories. Instead, we outline the physi al ideas, on epts, andassumptions that are often omitted in theoreti al works and reviews, espe ially at the stage of a developed theory.This spe i(cid:28) feature is thought to make our review interesting and useful for experimentalists.On the other hand, we did not tend to a umulate, lassify, and estimate the huge experimental data obtained inthis (cid:28)eld. We address experiment to demonstrate the use of a theory for its interpretation and the related di(cid:30) ultiesand problems. This spe i(cid:28) feature is thought to make our review interesting and useful for theorists.We (cid:28)rst qualitatively dis uss the ides that onstitute the basis of the theory of quantum phase transitions. Wethen onsider how the general theoreti al s heme an des ribe the well-known data on metal(cid:21)insulator transitions andtransitions between di(cid:27)erent quantum Hall liquids.1.1. Thermal and quantum (cid:29)u tuationsBefore dis ussing quantum phase transitions in essen e, we brie(cid:29)y re all some fa ts from statisti al physi s. We onsider a ma ros opi system onsisting of a huge number of parti les. This system is almost isolated and behavesmainly as a losed system; however, it an ex hange parti les and energy with a larger system, whi h serves as areservoir. In other words, our system is a subsystem of this reservoir.We (cid:28)rst analyze the lassi al subsystem at a (cid:28)nite temperature T. A ording to lassi al statisti s formulas, theprobability p i for the subsystem to o upy the state with energy ε i is p i = e − ε i /T Z , (1)where the oe(cid:30) ient /Z is determined by the normalization ondition P p i = 1 Z = X i exp( − ε i /T ) (2)(hereafter, temperature is given in energy units). Fun tion Z is alled the partition fun tion. It plays a key role inthe des ription of the thermodynami properties of lassi al obje ts. That the subsystem energy obeys distribution(1) rather than being (cid:28)xed means the presen e of lassi al thermal (cid:29)u tuations.When quantum me hani s appeared, the basis of lassi al thermodynami s and its prin ipal formulas were revised.The lassi al expressions were found to have a limited (cid:28)eld of appli ation. Quantum statisti s requires a developedquantum-me hani s math-based environment. Before addressing this environment, we assume that the subsystemis ompletely losed and does not intera t with the large system. As a result, we an use the standard quantumme hani s te hnique and use the S hr(cid:4)odinger steady-state equation to des ribe the subsystem. The resulting set ofsteady-state energies ε i and the orresponding wavefun tions ϕ i ( q ) of the subsystem are onsidered to be subsystemattributes in the zeroth approximation. These energies, ε i , enter Eqn (2). The set of ϕ i ( q ) fun tions is onvenientbe ause it is related to the subsystem under study and is a omplete set; therefore, it an be used for expansion intoa series.The presen e of a huge number of parti les in the subsystem implies that it has a very dense energy distribution ofquantum levels. Due to a weak intera tion with the reservoir, the subsystem is in the so- alled mixed state, in whi hno measurement or a set of measurements an lead to unambiguously predi ted results. Upon determining the mixedstate, the repeated measurements of any physi al quantity give a near-average result that di(cid:27)ers from the previousone. The s atter of the measured physi al quantities is interpreted as a result of (cid:29)u tuations.Be ause the subsystem intera ts with the environment, it annot be des ribed by a wave fun tion; therefore, thewave fun tion is substituted by a density matrix, ρ ( q ′ , q ) = Z ψ ∗ ( q ′ , X ) ψ ( q, X ) dX, (3)where q orresponds to the set of subsystem oordinates, X are the remaining oordinates of the system, and ψ ( q, X ) is the wave fun tion of the losed system.The mean values of any physi al quantity h s i are now al ulated using the density matrix h s i = 1 A Z nb s [ ρ ( q ′ , q )] o q ′ = q dq, A = Z ρ ( q, q ) dq. (4)rather than a wave fun tion. In Eqn (4), we apply the operator b s , whi h a ts on fun tions of the variable q , to thedensity matrix ρ ( q ′ , q ) ; we then set q ′ = q and integrate. The operator that formally enters the expression for thenormalization fa tor A is identi al with unity; therefore, the ondition q ′ = q is naturally taken into a ount in theintegrand.A ording to general rule(4), the mean of oordinate h q i , for example, is given by h q i = 1 A Z Z ψ ∗ ( q, X ) qψ ( q, X ) dqdX = R qρ ( q, q ) dq R ρ ( q, q ) dq . (5)The physi al meaning of the density matrix an be lari(cid:28)ed if we write it in an expli it form for the subsystem instatisti al equilibrium at a (cid:28)nite temperature T : ρ ( q ′ , q ) = X i ϕ ∗ i ( q ′ ) ϕ i ( q ) e − ε i /T . (6)For the subsystem in statisti al equilibrium, it follows from Eqns (5) and (6) h q i = 1 A X i Z ϕ ∗ i ( q ) ϕ i ( q ) e − ε i /T qdq = 1 A X i h q i i e − ε i /T , h q i i = Z ϕ ∗ i ( q ) ϕ i ( q ) qdq. (7)The averaging with the density matrix performed in Eqn (7) a ount for both the probabilisti des ription in the formof h q i i in quantum me hani s and in omplete information about the system (statisti al averaging).We use the ϕ i ( q ) fun tions and write the fun tion ρ ( q ′ , q ) in the matrix form ρ ( q ′ , q ) = k ρ ij k = (cid:13)(cid:13)(cid:13)(cid:13)Z ϕ ∗ j ( q ) ρ ( q ′ , q ) ϕ i ( q ) dq (cid:13)(cid:13)(cid:13)(cid:13) . (8)In Eqn (4), we then have h s i = P ij s ij ρ ji P i ρ ii = X ij s ij w ji , w ij = ρ ij P i ρ ii . (9)The w i j matrix onstru ted from the set of ϕ i ( q ) fun tions is alled the statisti al matrix and, in essen e, is thenormalized density matrix.We repla e b s with the energy operator b H in Eqn (4) and obtain h ε i = X i w ii ε i . (10)This means that the probability p i of dete ting the energy ε i in the subsystem is equal to the diagonal element of thematrix w ii , p i = w ii . (11)Equation (11) is the quantum analog of Eqn (1).The statisti al matrix has a number of universal properties. By de(cid:28)nition, this matrix is normalized: X i w ii = 1 . (12)As follows from Eqn (6), the statisti al matrix is diagonal in statisti al equilibrium. Its diagonal elements are fun tionsof only the energy of the orresponding subsystem state ε i . Until at least one w ii with i = 0 has a nonzero value, thesubsystem state remains mixed, and any measured quantity undergoes (cid:29)u tuations.The ma ros opi system is in a mixed state mainly due to a (cid:28)nite temperature. A omparison of Eqns (1) and (11)demonstrates that in the high-temperature limit, the statisti al matrix obeys the Gibbs distribution w ii = exp( − ε i /T ) /Z. (13)with a good a ura y.Be ause the integral of the squared wave fun tion ϕ i ( q ) over the entire oordinate spa e is Z ϕ ∗ i ( q ) ϕ i ( q ) dq ≡ , the partition fun tion Z an be represented as the sum of the diagonal matrix elements of the operator exp ( − b H/T ) over the omplete set of eigenfun tions ϕ i : Z = X i Z ϕ ∗ i ( q ) ϕ i ( q ) e − ε i /T dq = X i h ϕ i | exp( − b H/T ) | ϕ i i . (14)Here, we use that ϕ i are eigenfun tions of the Hamiltonian b H to obtain exp( − b H/T ) ϕ i = e − ε i /T ϕ i . A omparison of Eqns (4)(cid:22)(7) with Eqn (13) gives another form of the partition fun tion, Z = Z ρ ( q, q ) dq ≡ Sp [ ρ ( q ′ , q )] . (15)If Z is regarded as a normalization fa tor, Eqns (4) and (9) demonstrate that representation (15) is valid in both the lassi al limit and general ase.As the subsystem is open, it intera ts with the environment. Even if this intera tion is very weak, the states of thema ros opi system be ome mixed in a ertain small energy range due to the extremely high density of the systemenergy levels. Therefore, even at zero temperature, the un losed subsystem is in a mixed state. This means that as thetemperature de reases, ea h w ii ( i = 0) element approa hes a temperature-independent onstant, whi h depends on ε i and the sizes and properties of the subsystem, rather than tending to zero exponentially. Thermal (cid:29)u tuations arerepla ed by quantum (cid:29)u tuations. This statement is illustrated by Fig. 1a, whi h qualitatively shows the temperaturedependen e of the probability p i of dete ting the energy ε i in the subsystem. The dashed lines separate the regions ofpredominantly quantum and predominantly thermal (cid:29)u tuations. The limiting value p i (0) is spe i(cid:28)ed by the energy ε i , sizes, and properties of the subsystem. TZ ( /T) - exp -e i p i p i ThermalfluctuationsQuantumfluctuations e i T e w i i =h h t i a b FIG. 1: (a) Low-temperature behavior of the diagonal elements w ii of the statisti al matrix orresponding to low energies. (b)Simultaneous ex itation of thermal and quantum (cid:29)u tuationsIn this subsystem, p i (0) obviously de reases with an in rease in ε i . We assume that this dependen e is a power-lawfun tion, p i (0) ∝ /ε αi , and obtain the lower estimate of the exponent α . Energy for thermal (cid:29)u tuations is takenfrom the reservoir. A (cid:28)nite, not exponentially small, probability of dete ting the subsystem in a state ε i ≫ T meansa violation of the energy onservation law. This violation an exist in the framework determined by the un ertaintyrelation ε i τ i ∼ ~ , (16)where τ i is the lifetime of the state ε i . Therefore, even if transitions to all ε i states were equiprobable, p i (0) wouldde rease a ording to the law p i (0) ∝ ε − i , as follows from Eqn (16) . As a result, we have α > . (17)It is essential that the energy range in whi h quantum (cid:29)u tuations o ur is temperature independent, whi h followsfrom un ertainty relation (16).The relative role of thermal and quantum (cid:29)u tuations is illustrated in Fig. 1b, whi h shows the ex itation spe trumof the subsystem in one of the phases, i.e., far from phase transition point x = x c . The bra e in the bottom leftpart of Fig. 1b indi ates the temperature-related s ale. Only modes with frequen ies ω i . T / ~ are lassi ally ex itedmodes. Although the range of thermal modes is bounded above by temperature, they have large o upation numbers.Modes with ω i > T / ~ are mainly ex ited due to quantum pro esses and have small o upation numbers. However,as the temperature de reases, the role of quantum ex itations in reases.2. QUANTUM PHASE TRANSITIONS2.1 De(cid:28)nition of a quantum phase transition.Quantum phase transitions are phase transitions that an o ur at the absolute zero T = 0 and onsist of a hangein the ground state of a system in the ase where a ertain ontrol parameter x takes a riti al value x c . (The groundstate is taken to be the lowest possible mixed state.) The ontrol parameter an be, for instan e, a magneti (cid:28)eld oran ele tron on entration. Quantum phase transitions belong to the lass of ontinuous transitions at whi h none ofthe physi al fun tions of state has a dis ontinuity at the transition point.A quantum (cid:29)u tuation is the only ause that an hange the ground state of the system at zero temperature;the phase transitions under study are therefore alled quantum phase transitions. As in the ase of thermodynami transitions, the on ept of a orrelation length ξ , whi h has the meaning of the average quantum-(cid:29)u tuation size,is introdu ed into the theory of quantum phase transitions. At the absolute zero temperature, ξ is only determinedby the deviation δx of the ontrol parameter from the riti al value. The dependen e of ξ on δx is assumed to be apower-law fun tion, ξ ∝ | x − x c | − ν . (18)Real experiments are always performed at (cid:28)nite temperatures, where thermal (cid:29)u tuations exist in addition to quantum(cid:29)u tuations. The goal of the theory is to predi t the manifestation of the phase transition that o urs only at zerotemperature in the properties of the system at a (cid:28)nite temperature.The theory of quantum phase transitions (see, e.g., book [1℄ or reviews [2, 3℄) is analogous to the theory ofthermodynami phase transitions. In the ( x, T )) plane (i.e., in a plane where temperature is plotted versus a ontrolparameter), the quantum transition point an be a (cid:28)nite point on the line of thermodynami transitions, x c ( T ) → x c (0) as T → . For example, this behavior is hara teristi of magneti transitions with the magneti (cid:28)eld used as a ontrol parameter. Quantum phase transitions of another type also exist; they are imaged by isolated point x c on theabs issa axis of the ( x, T ) plane. The metal(cid:21)insulator transition is an example of su h a transition. In this review,we restri t ourselves to the ase of isolated-point transitions.2.2 Parallels and di(cid:27)eren es between lassi al and quantum phase transitionsAs noted above, quantum phase transitions belong to the lass of ontinuous transitions; there are no two oexisting ompetitive phases and, hen e, no stationary boundary between them. This lass also in ludes thermodynami se ond-order phase transitions. At a thermodynami phase transition point, the system transforms into another phase as awhole as a result of thermal (cid:29)u tuations. Flu tuations exist on either side of the transition, and their hara teristi size ξ is alled the orrelation length. As the transition is approa hed from either side, ξ diverges [5, 6℄.It is natural to expe t a similar situation to o ur in the vi inity of a quantum transition with the parti ipationof quantum (cid:29)u tuations. An analogy between lassi al and quantum phase transitions does exist, and it is ratherunexpe ted. The behavior of a quantum system in the vi inity of a transition point at a (cid:28)nite temperature in a d-dimensional spa e is analogous to the behavior of a lassi al system in a spa e with dimension D > d . This statementrequires extensive explanations, whi h should in lude an algorithm for the introdu tion of su h an imaginary systemand the determination of its dimension.First, we have to learly distinguish between the dimension of the geometri spa e in whi h the system is lo atedand the dimension of the generalized- oordinate spa e, whi h depends on the number of parti les N in the system. If ̺ is the density of parti les in the spa e, we have N = R R R ̺d d X , where the dimension d determines the multipli ityof the integral. Usually, a d -dimensional spa e is assumed to be in(cid:28)nite in all dire tions, but the range of one orseveral X i oordinates an be limited.Se ond, we note a similarity between the operator exp ( − b H/T ) , whi h was used to write Eqn (14), and the operator b S that des ribes the evolution of a losed quantum system with time a ording to the S hr(cid:4)odinger equation i ~ ∂ψ∂t = b Hψ, b S = exp( − i ~ b Ht ) . (19)This similarity be omes obvious if we hange the variables as i t/ ~ = 1 /T. (20)With substitution (20), we an interpret the matrix elements in Eqn (14) di(cid:27)erently. The element (cid:10) i (cid:12)(cid:12) exp ( − b H/T ) (cid:12)(cid:12) i (cid:11) an be regarded as the amplitude of the probability that, starting from state h i | , the subsys-tem evolves under the a tion of b S and returns to the initial state | i i within imaginary time ˜ t , whi h is equal to − i ~ /T .The imaginary time ˜ t is often alled the Matsubara time. For de(cid:28)niteness, we assume that the number of steps is(cid:28)xed and equal to N + 1 , N ≫ and that, in time i ~ /T , the subsystem has passed through N virtual states ando upied ea h state for a time ( δ ˜ t ) j ; as a result, we have N X j =0 ( δ ˜ t ) j = i ~ /T. (21)The amplitude of `the probability of returning to the initial state' means the sum of the amplitudes of the proba-bilities of returning for all possible traje tories in the spa e of states. We onsider a set of traje tories onsisting of N steps. The operator entering ea h matrix element in Eqn (14) is represented as exp( − b H/kT ) = exp i ~ b H N X ( δ ˜ t ) j ! . (22)We take a omplete set of wavefun tions | m j i of an arbitrary operator that does not ommute with the Hamiltonianand assume that the traje tories are realized in these states. Then, instead of Eqn (14), we obtain Z = X i X m ,m ,...m N < i | exp( − i b H ( δ ˜ t ) / ~ ) | m >< m | exp( − i b H ( δ ˜ t ) / ~ ) | m > ... < m N | exp( − i b H ( δ ˜ t ) N / ~ ) | i > . (23)The produ t of the matrix elements in the summand orresponds to a hain of onse utive virtual transitions. Thesummation over all { m j } ombinations means that all possible losed hains of N links are taken into a ount. Themodi(cid:28) ation of the expression for Z (cid:22) the repla ement of Eqn (14) with Eqn (23) (cid:22) means that we take the quantumproperties of the system into a ount by adding virtual transitions to real ones. e i hT t dt j dt j - e i | ñ m j+1 d X | ñ m j | ñ m j 1 - FIG. 2: Set of quantum statisti al d -dimensional systems lo ated in a one-dimensional band of width ~ /T and virtual transitionbetween themThe introdu tion of virtual transitions is illustrated in Fig. 2, where the abs issa axis stands for the initial d -dimensional spa e { d X } and all possible lassi al states of the system are lo ated along this axis in the order ofin reasing energy. All other horizontal lines are repli as of this spa e. These repli as form a set of N elements in thesegment [0 , ~ /T ] . The bla k dot on the abs issa axis represents the initial state of the system with energy ε i . These ond bla k dot on the upper horizontal line orresponds to the `(cid:28)nal' state of the system with the same energy ε i at the maximum value of the imaginary time | ˜ t | = ~ /T . The arrows indi ate the hain of virtual transitions throughthe set of virtual states | m j i designated by white dots. These are mixed states without a de(cid:28)nite energy; therefore,they are not eigenstates of the operator b H . The virtual transitions re(cid:29)e ting the quantum properties of the statisti alsystem under study are shown by arrows. The hain of virtual transitions shown in Fig. 2 orresponds to one term inthe sum in (23). The summation over all { m j } ombinations means that we a ount for all possible hains betweenthe (cid:28)xed initial and (cid:28)nal ε i points.Now, the problem is to onstru t a lassi al system whose partition fun tion is represented by Eqn (23). In theoriginal Eqn (14), the summation over i meant the summation over all the states realized in a d -dimensional system.The number of terms in the sum is now in reased, and our real system does not have su h a large number of di(cid:27)erentstates. Nevertheless, we an onstru t an imaginary lassi al system with a higher dimension using the s heme shownin Fig. 2.We add an additional axis for an imaginary time to the d axes of the original spa e. In the graphi al terms of Fig.2, this means that an ordinate axis is added to the abs issa axis. It is seen from Eqn (21) that the oordinate alongthis new axis hanges in the range | e t | ~ /kT. (24)Ea h point in the strip (24) in Fig. 2 orresponds to some virtual state of the quantum system; it an be named theimage point. The width (24) of strip in whi h an image point is lo ated in reases with de reasing the temperature.At T = 0 , the strip transforms into a half-plane; an in rease in the temperature, in ontrast, narrows the band andde reases its ontribution to the statisti al properties of the quantum system.We repeat the d -dimensional lassi al system N times by pla ing repli as along the imaginary time ˜ t axis at adistan e ( δ ˜ t ) j from ea h other. The states of the repli as in this ensemble an be di(cid:27)erent. Let these states be m j .We (cid:28)x a ertain `initial' state of the `lower' subsystem ε i . The energy of the lassi al D -dimensional system, we are onstru ting, is equal to the sum of the energies of all layers, with allowan e for the intera tion between them. Ea h ε i, { m } orresponds to a ertain energy ε i of the initial d -dimensional lassi al system and a ertain hain of statesentering Eqn (23). Therefore, the number of terms in the sum P { m } s i, m in Eqn (23) is equal to the number ofdi(cid:27)erent ε i, { m } energies. We note that ea h s i, m term in sum has a form typi al for partition sums, sin e the produ tof the matrix elements in this term eventually redu es to the produ t of exp (cid:0) − i ε j ( δ ˜ t ) j / ~ (cid:1) . The only di(cid:27)eren e isthat for the lassi al system, ea h term in Z is a real positive number and, in Eqn (23), a real positive number isrepresented by the sum s i = P { m } s i, m over { m } of all s i, m produ ts. When negle ting this di(cid:27)eren e, we an onsider Eqn (23) the partition fun tion Z of the imaginary D -dimensional lassi al system.Of ourse, in the general form, the resulting mathemati al onstru tion is absolutely unpra ti al. However, as is ustomary in deriving s aling relations, it is important to reveal some signi(cid:28) ant properties of sum (23) rather thanto al ulate it. As is shown in Se tion 2.3, one of these properties is anisotropy, i.e., non equivalen e of the axes ofthe D -dimensional spa e. De e e -e
02 01 e xx c T Phase Phase d x FIG. 3: The di(cid:27)eren e in the lowest energies of two phases ∆ ε = ε − ε versus a ontrol parameter (dashed line). At a ontrol parameter x = x c − δx , phase 1 is subje ted to (cid:29)u tuation ex itation inside equilibrium phase 2. Be ause | ∆ ε | < T ,thermal (cid:29)u tuation play a key role in this ex itationAt a thermodynami phase transition point, the partition fun tion Z has spe ial features. The sensitivity of fun tion(14) to the presen e of a phase transition is aused by the fa t that, when the transition is approa hed, the energyrange T determining signi(cid:28) ant terms in sum (14) ontains not only ϕ i levels from the set orresponding to theequilibrium phase but also ϕ ′ i levels from the nonequilibrium phase (Fig. 3). This allows (cid:29)u tuation transitionsbetween | i i and | i ′ i levels from di(cid:27)erent sets. Additional possibilities appear after Eqn (14) is repla ed with Eqn(23).If the temperature is low ( | ∆ ε | > T ) , the (cid:29)u tuation-indu ed appearan e of another phase is also possible, but dueto quantum (cid:29)u tuations.2.3. Criti al region of a quantum phase transitionA region in whi h all physi al quantities depend only on the orrelation length ξ always exists in the phase planenear a lassi al phase transition. It is alled the riti al or s aling region. Near a quantum phase transition at T = 0 , there also exists a ontrol parameter δx range in whi h physi al quantities are expressed through the length ξ determined by Eqn (18). In Se tion 4, we show this behavior by the example of a metal(cid:21)insulator transition ina three-dimensional system of nonintera ting ele trons, and determine the boundaries of this region. At a (cid:28)nitetemperature T = 0 , however, the s aling region of a quantum phase transition is more omplex.The spa e { d X, ˜ t } has dimension d + 1 be ause of the additional imaginary time axis. We introdu e orrelationlengths in the spa e { d X, ˜ t } . We retain the traditional notation ξ for the orrelation length in an ordinary d -dimensional subspa e and let ξ ϕ denote the orrelation length along the additional axis. The subs ript ϕ is a reminderthat ξ ϕ is related to the quantum aspe t of the problem and to the spe i(cid:28) features of wave fun tions. The dimensionof ξ ϕ is ~ /T rather than length; that is, it is measured in se onds rather than entimeters.As x → x c and T → , both orrelation lengths diverge at a transition. A ording to the theory of ontinuousthermodynami transitions, the divergen e is des ribed by power fun tions; but the exponents of the two orrelationlengths an be di(cid:27)erent in general. This is usually written as ξ ∝ δx − ν , δx = | x − x c | , (25) ξ ϕ ∝ ξ z . (26)The exponents ν and z are alled riti al indi es; z is alled the dynami riti al index. Both names and designationsoriginate from the theory of thermodynami pro esses. In parti ular, the dynami riti al index in the theory ofthermodynami pro esses enters the relation between the lifetime and size of thermal (cid:29)u tuations, τ ∝ ξ z . Inthe quantum problem at T = 0 , the situation is similar: the length ξ hara terizes spatial orrelations, i.e., the hara teristi size of quantum (cid:29)u tuations, and the time ξ ϕ hara terizes time orrelations.Formula (26) immediately (cid:28)xes the dimension of the spa e in whi h the imaginary lassi al system should be pla ed.A real length must be the oordinate along all axes of this spa e; that is, length axes should be made from the ˜ t axis.As follows from dimensional onsiderations and Eqn (26), the length equivalent to the ` orrelation pseudolength' ξ ϕ is proportional to ξ /zϕ , ζ ∝ ξ /zϕ . (27)Therefore, the volume element in the spa e is ( dξ ) d dξ ϕ ∝ ( dξ ) d ( dζ ) z (28)and a z -dimensional subspa e with the onventional `spatial' oordinates appears instead of the one-dimensionalimaginary time axis. Hen e, we have D = d + z. (29) xx c T a a
Criticalregion d x = bx x FIG. 4: Shape of the riti al region of a quantum phase transition depi ted by an isolated point in the ( x, T ) planeIt is onvenient to dis uss the onsequen es of Eqns (25) and (26) using the ( x, T ) plane (Fig. 4). At T = 0 , the ontrol parameter x on the abs issa axis of the ( x, T ) plane only a(cid:27)e ts one independent orrelation length, ξ , and ξ ϕ (and the length ζ ) an be formally obtained from ξ using Eqn (26). All the physi al quantities an be expressedonly in terms of x c in a ertain segment [ x , x ] that ontains the point x c .Now, let the temperature T = 0 . We move in the ( x, T ) plane along a horizontal line T = 0 (Fig. 4, line aa ) in the x → x c dire tion. At a ertain value of δx , the length ξ ϕ , whi h hanges a ording to Eqns (25) and (26), rea hes itsmaximum value ~ /T [whi h is determined by inequality (24)℄. At lower values of δx , Eqn (26) does not hold on line aa , be ause ξ in reases a ording to Eqn (25) and ξ ϕ remains equal to its maximum value ~ /T . The parameters ξ and ξ ϕ be ome mutually independent. The region of this independen e is alled the riti al region. If we now movetoward the transition inside the riti al region (e.g., along line bx c ) , both parameters still diverge at the transition.However, the divergen e of one of them is ontrolled by δx , ξ ∝ δx − ν , and the divergen e of the other is ontrolledby temperature, ξ ϕ = ~ /T .The name of the region derives from the fa t that, for a quantum transition, the riti al region is onsidered to bethe region where quantum (cid:29)u tuations play an essential role in mixing the two-phase states. As an be seen fromFig. 3, this o urs only under the ondition | ε − ε | > T. (30)As the temperature de reases, the range of the ontrol parameter δx where ondition (30) is satis(cid:28)ed narrows. As aresult, the shape of the riti al region is unusual: it widens when moving from the transition.Inside the riti al region, the quantity ξ ϕ = ~ /T an be asso iated with a real length. The quantity L ϕ ∝ ( ~ /T ) /z , (31)has the required properties: it is equivalent to the parameter ξ ϕ = ~ /T and, a ording to Eqn (26), has the dimensionof length. The physi al meaning of L ϕ an be understood from the following onsiderations. Be ause of a (cid:28)nite tem-perature, the quantum problem a quires a hara teristi energy T that separates lassi al and quantum (cid:29)u tuations(see Fig. 1b). Quantum (cid:29)u tuations are (cid:29)u tuations with energies ~ ω ϕ > T . The spatial size of these (cid:29)u tuations l ϕ ∝ /ω ϕ is bounded above be ause frequen y ω ϕ > T / ~ is bounded below. The length L ϕ is the upper boundary ofthe size of quantum (cid:29)u tuations. Therefore, L ϕ is often alled the dephasing length, i.e., the length beginning fromwhi h oheren e in a system of ele trons is broken.The appearan e of two independent parameters in the riti al region of a quantum phase transition is aused byinequality (24). Therefore, a s aling des ription in the riti al region of a quantum phase transition is therefore alleda (cid:28)nite-size s aling. An imaginary thermodynami system is thought to exist in a hyperstrip in a D -dimensional spa ewith d variables ranging from 0 to ∞ and z variables ranging from 0 to ( ~ /T ) /z .Inside the riti al region, L ϕ < ξ , and, along its boundary, L ϕ = ξ. (32)The lengths L ϕ and ξ are determined by Eqns (25) and (26) up to a multipli ative onstant; however, Eqns (25) and(26) rigidly (cid:28)x a power relation between these variables. Therefore, the equation for the riti al-region boundarieshas the form T = C ( δx ) νz , (33)where the onstant C an be di(cid:27)erent on either side of the transition in general.Be ause the riti al region has two independent parameters, the s aling expressions for physi al quantities be omemore omplex. We only present and dis uss expressions for the ele tri ondu tivity and resistivity, σ = ξ − d F ( L ϕ /ξ ) , ρ = ξ d − F ( L ϕ /ξ ) , F ( u ) = 1 /F ( u ) , (34)where F ( u ) is an arbitrary fun tion. The exponent of the (cid:28)rst fa tor in Eqn (34) is spe i(cid:28)ed by how the length entersthe expressions for ondu tivity at various dimensions d . As an be seen from Fig. 4, this fa tor is determined bythe x - omponent of the distan e from the transition. The L ϕ /ξ ratio is dimensionless; therefore, s aling rules an beused to regard L ϕ /ξ as an argument of an arbitrary fun tion. This relation depends on both T (i.e., the y - omponentof the distan e from an image point to the transition in the phase plane) and the distan e from this point to the riti al-region boundary along the x -axis.As an argument, we an also take any power of the L ϕ /ξ ratio; as a result, we write the argument in various forms,su h as L ϕ ξ , ~ /Tξ ϕ , ~ /Tξ z , ~ ( δx ) zν T or δx ( T / ~ ) /zν . (35)These various forms elu idate various aspe ts of the physi al meaning of this ratio. In the last two forms, theargument is dimensional and the dimensionality of the ontrol parameter is not spe i(cid:28)ed a priori. These forms showhow temperature enters the argument of the s aling fun tion.In on lusion of this se tion, as an example, we present an impli ation of Eqn (34) for the resistivity of a two-dimensional system. If a quantum phase transition indu ed by ele tron lo alization o urs in a two-dimensionalsystem and manifests itself in the transport properties of this system, we an write d = 2 : ρ ( x = x c ) ≡ /F (0) , i.e. ρ ( x c , T ) = onst = ρ c . (36)0 T r x c x x > c x x < c r ln uMetalInsulator (a) (b) r c r c FIG. 5: (ï¨DZ) Temperature dependen e of the resistivity ρ ( T ) at various values of x with the horizontal separatrix x = x c . (b)Redu tion of the all the ρ ( x, T ) urves to two urves (34) of the dependen e of the resistivity ρ s aling variable (35)This impli ation (a horizontal separatrix in the set of temperature dependen es at various values of x ) is s hemati allyshown in Fig. 5a. We note that it is valid only under assumption (25), whi h indi ates that the orrelation length ξ is temperature independent. In general, ρ ( x c , T ) may have a (cid:28)nite slope at the point T = 0 ( f. below, se tion 5.3and Fig. 13).Usually, with δx repla ed by the modulus | δx | , the ρ ( u ) urve is represented in the form of two bran hes as afun tion of ln u (Fig. 5b). The values of zν are then hosen su h that these urves (cid:28)t all the experimental pointsobtained at various temperatures.3. FLOW DIAGRAMS FOR METAL(cid:21)INSULATOR TRANSITIONSSo far, we have not spe i(cid:28)ed the type of a quantum phase transition. Hereafter, we speak about transitions relatedto a hange in the ele tron lo alization, whi h, in turn, is aused by the degree of disorder in the system. The generaltheory of quantum phase transitions initially assumes that a ontrol parameter a(cid:27)e ts the interparti le intera tion andthat a disorder is not more then a perturbative fa tor in the initial s heme of quantum phase transitions. Therefore,the appli ability of this s heme to metal(cid:21)insulator transitions, in whi h the degree of disorder is the main fa tor andthe intera tion is a se ondary fa tor, is not obvious a priori.The fundamental di(cid:27)eren e between an insulator and a metal is that the ele troni states at the Fermi level arelo alized in an insulator and delo alized in a metal. If an insulator is transformed into a metal due to a hange in a ertain parameter, the properties of the wave fun tions at the Fermi level hange. The main physi al property that isradi ally di(cid:27)erent in materials of these two types is the ondu tivity, i.e., the possibility of arrying an ele tri urrentat an arbitrarily weak ele tri (cid:28)eld. This gives a `yes(cid:21)no' type signature: the ondu tivity is either zero, σ = 0 , ornonzero, though it may be arbitrary small. However, at a (cid:28)nite temperature T = 0 , an insulator also arries a urrentowing to hopping ondu tivity. Therefore, the de(cid:28)nition of an insulator given above is only related to the temperature T = 0 , and the on ept of a metal(cid:21)insulator transition makes sense only at T = 0 .That the ondu tivity σ is not a fun tion of the state (be ause it is realized only under nonequilibrium onditions) isan additional argument against the appli ability of the general theory. However, Thouless [4℄ noted that the transportproperties an be used to hara terize the de rease in the wave fun tion of an ele tron pla ed at the enter of an L d ube in a d -dimensional spa e when it moves toward the edges of the ube. Therefore, in analyzing the behavior of ondu tan e or ondu tivity near the transition, we a tually follow the evolution of wave fun tions.Thus, we have every reason to apply the general theory of quantum phase transitions to metal(cid:21)insulator transitions.However, the (cid:28)rst su essful version of a theoreti al des ription of metal(cid:21)insulator quantum transitions [7℄ was basedon the renormalization group theory [8℄ borrowed from quantum (cid:28)eld theory. The essen e of its on lusions, relatedto systems of nonintera ting ele trons in a random potential, is illustrated in Fig. 6, whi h shows the logarithmi derivative of the ondu tan e of a sample with respe t to the size L , β = d ln yd ln L = Ly dydL at the temperature T=0 as a fun tion of the ondu tan e y , d ln yd ln L = β d (ln y ) . (37)1 -
101 ln yy C d 1 = d 2 = d 3 = b = d y ln d L ln y x L= x FIG. 6: Universal β (ln y ) fun tions for di(cid:27)erent dimentionalities [7℄.The β d (ln y ) urves des ribe the universal laws of a hange in the ondu tan e of the system when its sizes hange.The behavior of the system of nonintera ting ele trons in any material is des ribed by a part of the urve β d for the orresponding dimension d . The interpretation and method of using these urves are des ribed in detail in review [9℄and book [10℄.For metal(cid:21)insulator quantum phase transitions, the theory [7℄ plays the role of an existen e theorem. The shape ofthe β d (ln y ) urves and their position in the ( y, β ) plane indi ate that these transitions are absent in one- and two-dimensional systems of nonintera ting ele trons and that in three-dimensional systems, su h a transition an o ur,with the ondu tivity of the material hanging ontinuously during this transition. This means that su(cid:30) iently largeone- and two-dimensional samples should be always insulators at absolute zero whereas a three-dimensional material an be either an insulator or a ondu tor. Below, we dis uss this feature in detail. Here, we note that the β d (ln y ) urves are alled di(cid:27)erential (cid:29)ow diagrams or Gell-Mann(cid:21)Low urves.On the one hand, the theory in [1, 2, 3℄ is more spe ialized than that in [7℄ in some respe ts, be ause it onlydes ribes the vi inity of a quantum transition. On the other hand, the theory in [1, 2, 3℄ is more universal, be auseits appli ability is not limited by either intera tion or a strong magneti (cid:28)eld. Therefore, it is of interest to omparethe on lusions of these theories for the systems to whi h they both an be applied.4. THREE-DIMENSIONAL ELECTRON GASThe possibility of a phase transition in a three-dimensional material follows from the fa t that the β (ln y ) urveinterse ts the abs issa axis β = 0 . A ertain point on the β (ln y ) urve (an image point) orresponds to the transportproperties of a sample of size L made from a ertain material (Fig. 7a). If this point is lo ated in the lower half-plane β < the material is an insulator, and the image point shifts to the left as L in reases; as a result, the ondu tan e ofthe sample be omes exponentially small. If this point is lo ated in the upper half-plane β > the image point shiftsto the right; as a result, the material is a metal, and its ondu tan e y in reases with the size L .The urves in Figs 6 and 7a are s ar ely adapted for a dire t omparison with experimental data be ause of aspe ial hoi e of their oordinate axes. This disadvantage an be partly orre ted by integrating the equation d ln yβ (ln y ) = d ln L. (38)for three-dimensional (3D) systems. Sin e the left-hand side of di(cid:27)erential equation (38) be omes in(cid:28)nite at thepoint y = y c , the urves orresponding to di(cid:27)erent integration onstants de ompose into two families: one of them orresponds to an insulator region and the other to a metalli region. To demonstrate this by the simplest way, werestri t ourselves to the immediate vi inity of point y c , in whi h the β (ln y ) urve an be approximated by a straightline, d ln yd ln L = s ln yy c (39)2 y l b y x L= x d = ln yy c L= l (a) ln y/y c L s L= l / y/y c L/ x Insulator Metal
FIG. 7: Nonintera ting 3D ele tron gas: (a) di(cid:27)erential s aling diagram from [7℄ (see also Fig. 6); the interse tion of the urve β (ln y ) with the abs issa axis β = 0 means existen e of a quantum phase transition; (b) (cid:29)ow diagram for a nonintera ting 3Dele tron gas in the vi inity of the transition point; ( ) two universal urves of the dependen e of the ondu tan e of the 3Dsystem on its size L obtained by s alingwhere s is the slope of the line with respe t to the abs issa axis. Correspondingly, Eqn (38) an be repla ed by linearequation (39). The general solution of linear di(cid:27)erential equation (39) is given by ln yy c = (cid:18) Lλ (cid:19) s ln y λ y c , (40)where λ plays the role of the initial ondition that (cid:28)xes the initial point in the β (ln y ) urve; for example, it an bethe length k − determined by the ele tron on entration. On the metalli side, k − is equal to the minimum freepath length l min . The ondu tan e y λ orresponds to the point λ .Figure 7b shows parti ular solutions of Eqn (39). If y λ > y c , we have ln y λ /y c > , the solution is lo ated in theright quadrant, and the ondu tan e y in reases with L (metal). If y λ < y c , we have ln y λ /y c < , the solution islo ated in the left quadrant, and the ondu tan e y de reases as L in reases (insulator). The set of urves in Fig. 7bis alled a (cid:29)ow diagram. The arrows in this diagram show the dire tion of the image point motion when the systemsize in reases. The (cid:29)ow lines orresponding to the parti ular solutions of Eqn (39) (cid:28)ll both upper quadrants of the (ln y λ /y c , L s ) plane. The boundary between them, the ln y λ /y c = 0 axis, is alled the separatrix; in Fig. 7b, it isindi ated by a dashed line.We extend straight line (39) in Fig. 7a to its interse tion with the asymptote β = 1 ; as a result, we approximatethe β ( y ) urve in the upper half-plane β > by a broken line onsisting of segments of two straight lines. The size L whi h brings the image point to the interse tion point is alled the orrelation length ξ . From Eqn (40), we have ξ = λ (cid:18) s ln y λ y c (cid:19) − /s . (41)With Eqn(41), we an rewrite general solution (40) as ln yy c = (cid:18) Lξ (cid:19) s (42)normalize the size L by the length ξ (whi h is spe i(cid:28) for every material), and redu e ea h of the two families in Fig.7b to one s aling urve (Fig. 7 ).We make several remarks.First, Eqns (39)(cid:21)(42) involve a parameter s , whi h is unknown a priori. However, in the ase of a nonintera ting3D-gas, this parameter is either equal to unity or very lose to it [9℄. Be ause s enters all formulas as a fa tor orexponent, this parameter an be dropped.Se ond, the (cid:29)ow lines in Fig. 7b are straight only be ause we integrated the Gell-Mann(cid:21)Low equation only in thesmall vi inity of the transition point. If we integrated the β (ln y ) fun tion over a wide range of its argument ln y using some model representation, we would obviously obtain urved (cid:29)ow lines.The third note on erns the role of temperature. The parameter that spe i(cid:28)es system motion along the (cid:29)ow linesindi ated by arrows in Fig. 7b is the size. If the temperature is taken to be zero (as was assumed until now), this3parameter is the sample size L . However, we an initially suppose that the system is large, as this is done in thetheory of quantum phase transitions [1, 2, 3℄. In this ase, the limiting size is onsidered to be the dephasing length L ϕ , i.e., the size at whi h quantum oheren e in an ele tron system is realized. The image point then moves along(cid:29)ow lines in the dire tion indi ated by arrows as the temperature de reases. Be ause L ϕ depends on the temperatureand be omes in(cid:28)nite as T → , this pro edure introdu es the temperature into the set of physi al quantities relatedto the (cid:29)ow diagram. However, we do not study this relation; based on the `existen e theorem' of a quantum transitionin a 3D spa e, we turn to the ( x, T ) phase plane to onstru t the riti al region (see Fig. 8 and ompare it with Fig.4).Let small values of the ontrol parameter x orrespond to metalli states. Then, at small x on the abs issa axis,the Drude formula σ = σ = ne l/ ~ k F holds, and, at small x and (cid:28)nite T , the quantum orre tion σ = σ + e ~ L − ee , L ee = p ~ D/T . (43)is added to σ (here D is the di(cid:27)usion oe(cid:30) ient). The di(cid:27)usion length in Eqn (43) is taken to be the length L ee determined by the Aronov(cid:21)Altshuler e(cid:27)e t [11℄, i.e., by the ele tron(cid:21)ele tron intera tion. In other words, we assumethat dephasing is aused by internal pro esses that o ur in the ele tron gas in the time τ ee = ~ /T, (44)without external impa ts, su h as the ele tron(cid:21)phonon intera tion (see analogous dis ussion of the physi al meaningof L ϕ in review [2℄, p. 324).At any ontrol parameter x , another spe ial point x m (the Mott limit) exists on the right of the transition point x c . At this spe ial point, we have σ ( x m , T = 0) = σ Mott = e ~ k F = e ~ l . (45)In the range between x c and x m , the Drude formula is invalid: des ribing ondu tivity with this formula impliesintrodu ing a free path length l that is shorter than the de Broglie wavelength. In this range at T = 0 , σ is expressedthrough the orrelation length ξ ; this orresponds to the general s heme of the theory of quantum phase transitions.As a result, the ondu tivity along the abs issa axis is expressed as σ ( T = 0) = (cid:18) e ~ (cid:19) × x > x c , /ξ x c > x > x m , ( k F l ) /l x x m . (46)To mat h the last two expressions at x = x m , it su(cid:30) es to set ξ ( x m ) = l .Thus, to des ribe ondu tivity in the metalli region, we introdu ed two parameters, ξ and L ee , whi h have thedimension of length and the properties required for the riti al region (they are mutually independent and diverge atthe point x c ) . Following [12℄, we onstru t the following interpolation fun tion in the riti al region: σ = e ~ (cid:18) ξ + 1 L ee (cid:19) = e ~ ξ F (cid:18) L ee ξ (cid:19) F ( u ) = (1 + 1 /u ) . (47) T Critical region x z =3 x m Metal s - - - = + ( ) e h x L ee s - --- - = + ( ) e h ( ) k l F 2 l L ee Quantumcorrection k l = F x = LT hopp µ x - x = L ee T µ x - s µ ee = r/a - - x e r/ x c e he h x x l - ( ) T/T m Insulator a BB -- x i FIG. 8: Vi inity of a metal(cid:21)insulator transition in a nonintera ting 3D ele tron gas in the ( x, T ) phase plane4 n [10 cm ]
17 3 - ¸ T (K )403020100 s W ( - c m ) - Ge:As B = 9.3 T9.49.59.69.79.91010.210.5 9.559.659.750 0.4 1.00.6 T (K ) 0.81.51.00.5 s W ( c m ) -- GaAs3.5 ×
10 cm
16 3 - FIG. 9: (a) Temperature dependen e of the ondu tivity of Ge:As samples with various level of doping in the region of ametal(cid:21)insulator transition (from [18℄); the riti al on entration is determined by the extrapolation of the experimental datato T = 0 . b) Temperature dependen e of the ondu tivity of a GaAs sample in various magneti (cid:28)elds in the region of ametal(cid:21)insulator transition (from [16℄); the riti al (cid:28)eld determined by extrapolation is 9.78 T.Expression (47) mat hes Eqn (43) on the straight line x = x m and gives the orre t values of ondu tivity in the x c < x x m segment at T = 0 . The se ond form of the fun tion demonstrates that fun tion (47) satis(cid:28)es generalrelation (34).We now move from right to left along the line T = const (Fig. 4, line aa ). As long as the quantum orre tion isrelatively small, ele tron di(cid:27)usion o urs via s attering by impurities; i.e., it is ontrolled by the (cid:28)rst term in Eqn (43).Therefore, the di(cid:27)usion oe(cid:30) ient D in L ee is temperature independent. However, when we enter the riti al region, σ transforms into ( e / ~ ) ξ − and begins to de rease rapidly. Under these onditions, D eases to be a onstant:di(cid:27)usion is likely to be aused by the ele tromagneti -(cid:28)eld (cid:29)u tuations that determine L ee ; as a result, this di(cid:27)usionbe omes temperature independent. We an then write a set of equations for the σ ( T ) and D ( T ) fun tions with theEinstein relation σ = e ~ ξ + r T ~ D ! ,σ = e g F D (48)used as a se ond equation. Here g F is the density of states at the Fermi level.We eliminate D from two equations (48), use that /ξ ≪ /L ee near the transition, and obtain the temperaturedependen e of the ondu tivity on the right-hand side of the riti al region [12, 13℄, σ ( T ) = e ~ (cid:18) ξ + ( T g F ) / (cid:19) ≡ α + βT / . (49)This means that inside the riti al region, L ee is given by L ee = ( T g F ) − / . (50)rather than being determined by Eqn (43). Exa tly at the transition, we have ξ = ∞ and α = 0 .Temperature dependen e (49) was repeatedly dete ted in experiments performed on a variety of materials [14, 15,16, 17℄. Figure 9 shows two examples in whi h the ontrol parameter is represented by the ele tron on entration andmagneti (cid:28)eld.Using ondition (32) and Eqn (49), we an write the relation T ∝ ( g F ξ ) − . (51)5for the right boundary of the riti al region. If the relation ξ ∝ ( δx ) − holds (i.e., if ν = 1 , as is assumed in Fig. 8),the boundary of the riti al region is represented by a ubi parabola T ∝ ( δx ) . In the general ase, we have ν = 1 and T ∝ ( δx ) ν /g F . (52)A omparison with Eqn (33) demonstrates that in the metal(cid:21)insulator transition in a 3D system of nonintera tingele trons, the dynami riti al index is z = 3 . (53)Below urve (52) in the x c > x > x m segment, the ondu tivity is also des ribed by Eqn (47); however, L ee entersEqn (47) in the initial form (43) with the di(cid:27)usion oe(cid:30) ient D = const . Therefore, the temperature dependen e ofthe ondu tivity in this region should have the form σ ( T ) = α + βT / .Up to this point, we have dis ussed the right-hand side of the phase diagram. The left-hand side of the phasediagram, i.e., the insulator region (where hopping ondu tivity takes pla e), also has two hara teristi lengths. First,there is the de ay length ξ of the lo alized-state wave fun tions, ψ ∝ exp ( − r/ξ ) . Far from the transition, ξ de reasesto the Bohr radius a B = κ ~ / ( m ∗ e ) (where κ is the diele tri onstant and m ∗ is the e(cid:27)e tive ele tron mass); atthe transition, it diverges be ause the ele trons be ome delo alized. Se ond, there is the average hopping distan e r .If the hopping ondu tivity is des ribed by the Mott law, the average hopping distan e is [10℄ r = ( ξ/g B T ) / . (54)The lengths r and ξ annot be used as two independent lengths in the riti al region be ause they are onne ted byrelation (54). However, using L ee determined by Eqn (50), we an rewrite Eqn (54) as r = ( ξL ee ) / , (55)We take into a ount that expression (50) for L ee does not ontain the kineti hara teristi s of the ele tron gas and onsider L ee the dephasing length over the entire riti al region, in luding its left-hand side, above the x i > x > x c segment on the abs issa axis. Equation (52) then determines both the right and left boundaries of the riti al region.Equations (54) and (55) demonstrate that r = ξ = L ee at the left boundary.The on lusion that L ee an be onsidered to be the dephasing length over the entire riti al region is supportedexperimentally: as an be seen from Fig. 9, the σ ( T ) temperature dependen e straighten in terms of the ( T / , σ ) axes, not only on the right-hand side of the riti al region but also at values of the ontrol parameter x < x c (e.g.,at the ele tron on entration n = 3 × m − in Fig. 9). However, the free term α in Eqn (49) be omes negative.This means that we should either suppose that orrelation length ξ is negative in the insulator region or (whi h isformally preferred but is essentially the same) should repla e interpolation formula (49) on the left-hand side of the riti al region by the formula σ ( T ) = e ~ (cid:18) − ξ + 1 L ee (cid:19) . (56)As follows from Eqns (55) and (56), we have σ = 0 along the left boundary of the riti al region, whi h means thatthe ondu tivity along this boundary is determined up to an exponentially small hopping ondu tivity.The hopping ondu tion me hanism is still operative below the lower boundary of the riti al region over the x i > x > x c segment, and the wavefun tion de ay length is ξ ≫ a B rather than a B [19℄. Therefore, a B does not enterthe expression for the hopping ondu tivity, and σ is expressed in terms of the orrelation length ξ (see Fig. 8): σ ∝ exp( − r/ξ ) . The ( x, T ) phase diagram in Fig. 8 a umulates the results of the long-term experimental studies of the low-temperature transport properties of ondu ting systems, namely, the quantum orre tions to metalli ondu tivity,the evolution of ele troni spe tra during metal(cid:21)insulator transitions, the temperature dependen e of ondu tivityin the vi inity of the transitions, and hopping ondu tivity. In essen e, this phase diagram was plotted irrespe tiveof the theory of quantum phase transitions [1, 2, 3℄ in order to reveal the ompatibility of all experimental data.Nevertheless, the onsiderations given above demonstrate that this diagram is absolutely adequate for the on eptsfollowing from this theory.65. TWO-DIMENSIONAL ELECTRON GAS.5.1 Gas of nonintera ting ele tronsWe now pass to 2D systems. A ording to the theory by Abrahams et al. [7℄, a 2D system ( d = 2) of nonintera tingele trons is always an insulator in the sense that lo alization should inevitably o ur in a su(cid:30) iently large sample, L > ξ , at a su(cid:30) iently low temperature,
T < T ξ , and an arbitrarily small disorder. This statement follows from thefa t that (cid:29)ow line d = 2 in Fig. 6 asymptoti ally approa hes the β = 0 axis and does not interse t it (this line is shownin Fig. 10a). However, in weakly disordered (cid:28)lms, the orrelation length ξ , whi h bounds the size L below, an beunrealisti ally large, and the temperature T ξ of the rossover from the region with a logarithmi quantum orre tionto the ondu tivity to the region that is hara terized by an exponential temperature dependen e is, in ontrast, toolow. These (cid:28)lms are alled metalli (cid:28)lms.Estimates of ξ and T ξ an be obtained from the assumption [20℄ that the logarithmi ally diverging quantum orre tion ∆ σ in the ondu tivity σ = σ + ∆ σ = ( e / ~ )( k F l ) − ( e / ~ ) ln( L ee /l ) (57)is of the order of the lassi al ondu tivity σ , and hen e σ ≈ and ln( L ee /l ) ≈ k F l. (58)The di(cid:27)usion length L ee over whi h ele tron dephasing o urs is determined by Eqn (43). The value of L ee determined from Eqn (58) is de(cid:28)ned as the orrelation length ξ , ξ = l exp( k F l ) , (59)and the temperature T ξ = D ~ /ξ , (60)determined from the relation L ee = ξ using Eqn (43) is alled the rossover temperature. Of ourse, the ondu tivitydoes not vanish at T = T ξ ; however, the theory of weak lo alization is obviously not valid at this temperature, andthe resistivity should begin to in rease exponentially with de reasing the temperature at T < T ξ in samples of size L > ξ .We now plot fun tion (60) and, for onvenien e, dire t the /ξ axis to the left (see Fig. 10). As a result, thisdiagram an be onveniently ompared with the diagram for a 3D system shown in Fig. 8. We add an axis and layo(cid:27) the ontrol parameter x as abs issa; this parameter hara terizes the degree of disorder, with x = 0 orrespondingto an ideal system in whi h a disorder is absent.It is easily seen that the T (1 /ξ ) urve in Fig. 10 represents the left-hand side of the phase diagram in the vi inityof a metal(cid:21)insulator transition in a 3D material ( f. Fig. 8) whose phase transition point is lo ated at the edge of b T / x Ds ln µ L T ee ( ) l I d ea l s y s t e m z d = ln y T x µ x - s µ e - ( ) T /T m s - = ln e he he h x Disorder x d = + e e L ee t t ee = FIG. 10: Nonintera ting 2D ele tron gas: (a) di(cid:27)erential (cid:29)ow lines for systems with dimension d = 2 taken from the s alingdiagram in [7℄ (see also Fig. 6) and d = 2 + ε (see text), and (b) rossover from the logarithmi temperature dependen e of ondu tivity to its exponential dependen e, whi h an be treated as the boundary of the riti al region of a virtual quantumtransition (see text)7the diagram, at the origin ( T = x = 0) . Curve (60) is then the left boundary of the riti al region, and the dynami riti al index is z = 2 . (61)Using Eqns (53) and (61), one an on lude that for metal(cid:21)insulator transitions in systems of nonintera tingele trons, the dynami riti al index is equal to the dimension, z = d .Thus, systems with dimension d = 2 turn out to be boundary systems: a quantum transition is still present inthe ( x, T ) phase plane but is shifted toward its orner, to the unrea hable point x = 0 . The boundary properties of2D systems an also be found from the (cid:29)ow diagram in Fig. 6. We imagine that the dimension d is a ontinuousparameter and an take not only integer values. Straight lines β = d − represent the asymptoti s of the (cid:29)ow lines athigh values of y . Therefore, the (cid:29)ow lines of a system with dimension d = 2 + ǫ > inevitably ross the abs issa axis β = 0 , and su h systems have a metal(cid:21)insulator transition (Fig. 10a). As ǫ → , the transition point shifts towardhigh ondu tan e and goes to in(cid:28)nity.This interpretation of the urve in Fig. 10 implies that the domain over the T ∝ (1 /ξ ) parabola is the riti alregion of the quantum transition. For 2D systems, only one s aling variable u is retained in Eqn (34) written for the ondu tivity in the riti al region; it is equal to the ratio of two hara teristi lengths, σ = F ( u ) ≡ F ( L ϕ /ξ ) . (62)In this region, however, the ondu tivity is typi ally expressed as the di(cid:27)eren e between the lassi al ondu tivityand the quantum orre tion, σ = σ − ∆ σ = e ~ (cid:18) k F l − ln L ee l (cid:19) . (63)To resolve this apparent dis repan y, let us substitute the expression l = ξ exp ( − k F l ) from Eqn (59) in the argumentof the logarithm in Eqn (63), move the exponent from the logarithm, and obtain σ = e ~ ln ξL ee . (64)The lassi al ondu tivity σ an els and the remaining part depends only on the s aling variable, as it should be inthe riti al region. Thus, in this regard, our ( x, T ) diagram also satis(cid:28)es the requirements of the theory of quantumphase transitions.Expression (64) holds not in the entire `quasi- riti al' region T > T ξ ; moreover, it diverges on the x = 0 axis.However, near this axis, the elasti mean free path l → ∞ and the standard expression for L ee lose their meaningbe ause the ne essary ondition τ ee ≫ τ is violated. The boundary of the part of the region where Eqn (64) is invalid, τ ee = τ , is shown in Fig. 10 by a dashed straight line plotted under the assumption that τ ∝ x − .As is shown in Se tion 5.3, the introdu tion of intera tion an lead to the appearan e of a metal(cid:21)insulator transitionin a 2D system. In terms of the phase diagram shown in Fig. 10, this means that the phase transition point shiftsfrom the origin to a point x c = 0 on the abs issa axis due to a ertain ause, and a boundary additionally appearsbetween the riti al and metalli regions. Standard expression (63), whi h was written for the ondu tivity in themetalli region and was transformed into Eqn (64), was already used for the des ription of the ondu tivity in the riti al region. This means that the expression for the ondu tivity in the metalli region should be radi ally di(cid:27)erentand that we should expe t the appearan e of a marginal metal instead of a Fermi liquid. The nature of the ele troni states in this hypotheti metal should be rather pe uliar, sin e ele trons are assumed to be lo alized when intera tionis turned o(cid:27) [21℄.5.2. Spin(cid:21)orbit intera tionThe boundary position of 2D systems makes them sensitive to various types of intera tion, e.g., spin(cid:21)orbit intera tionor ele tron(cid:21)ele tron intera tion, whi h an ause a phase transition.We (cid:28)rst onsider the spin(cid:21)orbit intera tion. This ase is onvenient be ause a (cid:29)ow diagram an be plotted using the ( y, β ) axes of the initial diagram shown in Fig. 6. At large y , the initial (cid:29)ow line β ( y ) deviates down from its rightasymptote β = 0 be ause of weak lo alization, whi h results in a de rease in the ondu tivity (Fig. 10a). But thespin(cid:21)orbit intera tion hanges the sign of the quantum orre tion, i.e., hanges weak lo alization into antilo alization.Therefore, the sign of the derivative on the right-hand side of the β (ln y ) urve should hange: the urve deviatesupward from the asymptote β = 0 , goes to the upper half-plane β > , and, hen e, inevitably rosses the abs issaaxis β = 0 and rea hes the left asymptote.8Antilo alization was studied in detail both experimentally [22℄ and theoreti ally [23℄. In the 2D ase ( d = 2) , thequantum orre tion to the ondu tivity is given by ∆ σ ≈ − ( e / ~ ) τ ee Z τ dtt (cid:18) e − t/τ so − (cid:19) , (65)where τ is the time between elasti ollisions, τ so ≥ τ is the time between spin-(cid:29)ip ollisions, and τ ee = ~ /T is thedephasing time. The motion of the image point in the (cid:29)ow diagram now depends on two parameters, the ondu tan e y and time τ so . For various values of τ so , Fig. 11 shows the family of (cid:29)ow lines lo ated between two envelope urves.The lower urve is β (ln y ) from the diagrams in Figs 6 and 10. It an be obtained from integral (65) if we set τ so = ∞ in the integrand, and hen e the parenthesis in the integrand be omes equal to unity. The upper urve has the sameasymptoti s; however, its onstru tion implies that τ so = τ . Then, we have t ≫ τ so over the major portion of theintegration range, and the parenthesis is onsidered to be − / . b yy c FIG. 11: Flow diagram for a 2D ele tron gas with spin-orbit intera tion.We now take a sample of size L of a material with an intermediate value of τ so , τ ≪ τ so ≪ ∞ . The size L bounds the di(cid:27)usion time t of an ele tron in a 2D sample until the ollision with boundary by the quantity τ L , t < τ L ∼ τ ( L/l ) . Therefore, to des ribe weak lo alization in a sample of size L , we must repla e the upperintegration limit in Eqn (65) with τ L , τ ee → τ L ≡ τ ( L/l ) . Let L be (cid:28)rst very small ( Lgtrsim and the temperature T = 0 . Di(cid:27)usion pro esses then have no time to develop; ele tron interferen e isvirtually absent; the ondu tivity is equal to its lassi al value; and the image point is on the right on the axis β = 0 .As the size L in reases to L so = l p τ so /τ (66)spin-(cid:29)ip ollisions are insigni(cid:28) ant and orre tion (65) to the ondu tivity is negative. The image point moves tothe left along the lower envelope urve, as in Fig. 10 in the absen e of the spin(cid:21)orbit intera tion. When the size L be omes larger than L so , integral (65) and ∆ σ hange their sign and the image point moves to the upper envelope urve. If this passage o urs at y < y c , the (cid:28)lm be omes an insulator at L → ∞ in any ase. But if the spin(cid:21)orbitintera tion is strong, τ so is small and the image point rea hes the upper envelope urve at y > y c . Then, the imagepoint ontinues to move along the upper envelope urve toward large y , and the (cid:28)lm retains its metalli properties as L → ∞ . Thus, a metal(cid:21)insulator transition ould be observed in an experiment with the spin(cid:21)orbit intera tion used asa ontrol parameter. So far, only the transformation of weak lo alization into antilo alization has been demonstratedthis way [22℄.5.3. Gas of intera ting ele tronsWe have to re(cid:28)ne what is meant by the absen e or presen e of the ele tron(cid:21)ele tron intera tion. The intera tionmanifests itself di(cid:27)erently in the properties of ele tron systems; for example, it determines the probability of ele tron(cid:21)ele tron s attering. In the lassi al theory of metals, ele tron(cid:21)ele tron s attering is onsidered not to ontribute to9 ondu tivity, sin e the total momentum of the ele tron system and the drift velo ity remain the same. When quantume(cid:27)e ts are taken into a ount, this statement be omes invalid be ause the ondu tan e depends on the dephasinglength and ele tron(cid:21)ele tron s attering hanges this length. Nevertheless, if the ele tron(cid:21)ele tron intera tion a(cid:27)e tsthe ondu tan e only through s attering, the ele tron system is onsidered nonintera ting, be ause the intera tiona(cid:27)e ts the ondu tan e as an external a tion, e.g., s attering by phonons.Ele tron(cid:21)ele tron s attering is not the only hannel of the e(cid:27)e t of the intera tion on the ondu tan e. Theintera tion is thought to be not the bare Coulomb intera tion but the intera tion between `dressed' quasiparti les,i.e., the s reened intera tion that depends on the ele tron density, the di(cid:27)usion oe(cid:30) ient of ele trons, and (under ertain onditions) the sample size or the quantum oheren e length [24℄.This intera tion determines the stru ture of quantum levels and the ground-state energy, a(cid:27)e ting the ompetitionbetween phases in the vi inity of a phase transition. Ele trons are mainly s attered by a s reened impurity potential,whose properties depend on the e(cid:27)e tive ele tron(cid:21)ele tron intera tion. This intera tion, in turn, depends on thedi(cid:27)usion oe(cid:30) ient and dephasing length, whi h results in the omplex dependen e of the e(cid:27)e tive intera tion on allexternal parameters.The ase of spin(cid:21)orbit intera tion onsidered in Se tion 5.2 represents an example of intera tion renormalizationas the sample size L hanges: the spin(cid:21)orbit intera tion is a tually turned on only when L be omes larger than thevalue determined from Eqn (66).As another example, we analyze the pi ture of possible states in the model with a multivalley ele tron spe trumdeveloped by Punnoose and Finkel'stein [25℄. In this model, a hange in L auses hanges in both the ondu tan e yand intera tion Θ . Therefore, an equation for Θ is added to Eqn (37). The (cid:29)ow diagram is a result of the solutionof these two equations. Figure 12 shows part of this diagram al ulated by the authors of [25℄ in slightly di(cid:27)erent oordinate axes. The variables were hanged to fa ilitate a omparison of this diagram with those shown in Figsï¨DZ6and 10, although the quantitative information ontained in the initial diagram in [25℄ is lost. The abs issa of the (cid:29)owdiagram shown in Fig. 12 is the ondu tan e. Thus, the abs issa of all the diagrams in Figs 6 and 10 is the same.The ordinate Θ of the (cid:29)ow diagram in Fig. 12 re(cid:29)e ts the e(cid:27)e tive intera tion. A nonintera ting ele tron gas, whi hwas dis ussed in Se tion 5.1, orresponds to the straight line Θ = 0 . As in all previous diagrams (Figs 6(cid:21)10), the sizeis the parameter that determines system motion along the (cid:29)ow lines indi ated by arrows. This size is given by thesample size L if T = 0 or by the dephasing length L ϕ (or L ee ), i.e., the maximum size in whi h quantum oheren eis retained in an ele troni system.The on(cid:28)gurations of the (cid:29)ow lines in Fig. 12 learly display the spe i(cid:28) feature of the intera tion Θ that wasdis ussed at the beginning of this se tion: as the size L or L ϕ hanges, the intera tion in the system hanges (isrenormalized), and this hange is di(cid:27)erent in di(cid:27)erent (cid:29)ow lines. The diagram in Fig. 12 is a two-parameter diagram:two parameters are required to spe ify the (cid:29)ow line of the image point. The (cid:29)ow lines (traje tories) in this diagramo upy the entire half-plane.We assume that the position of a point in a (cid:29)ow line is determined by L ϕ , i.e., by the temperature. We assume thatwe are at point A in a (cid:29)ow line, that is metalli be ause it re edes to the region of high ondu tivity at T → . We(cid:28)x the intera tion Θ and vary the ondu tan e y ; that is, we vary the degree of disorder. This pro ess is shown by adashed straight line in the diagram. When moving along this line, we an ross the separatrix and rea h point B in a(cid:29)ow line that des ribes an insulator, be ause it tends to the point y = 0 as T → . To spe ify the position of a pointin line AB, we an use a single parameter, whi h is alled a ontrol parameter. At another ontrol parameter, the y Q Crossover y c B AQCP
FIG. 12: Part of a (cid:29)ow diagram plotted for a 2D ele tron gas with intera tion [25℄0angle of interse tion of the separatrix an be di(cid:27)erent. For example, we an initiate the rossover from a metalli toa nonmetalli (cid:29)ow traje tory by hanging the ele tron on entration and, thus, simultaneously hanging the e(cid:27)e tiveintera tion Θ and the ondu tan e.By hanging the ontrol parameter, we an pass from one (cid:29)ow traje tory to another, and, by hanging the temper-ature (size), we an move along a (cid:29)ow traje tory. But be ause intera tion Θ an hange when any of two parameters hanges, fa torization is absent; that is, we annot suppose that the length ξ depends only on δx and the length L ϕ depends only on temperature. The onsequen e of this `mixing' of variables is learly visible in the (cid:29)ow diagram inFig. 12. As we move along the separatrix toward the quantum riti al point (QCP), the ondu tan e de reases andtends to y c . Hen e, we an draw the following important on lusion, whi h is qualitatively shown in Fig. 13: theseparatrix in the set of temperature dependen e is not a horizontal line, as in Fig. 5. y Ty c x x = c x x < c x x > c FIG. 13: Qualitative s heme for the evolution of the ondu tan e y ( T ) urves with a ontrol parameter in the model proposedin [25℄An indi ation of the presen e of a metal(cid:21)insulator transition in a 2D gas was (cid:28)rst obtained in the inversion layer ofa (cid:28)eld-e(cid:27)e t transistor on a Si surfa e [26℄. The presen e of the transition was questioned for a long time, be ause iswas in on(cid:29)i t with the on epts formulated in [7℄ and be ause the transition was not reprodu ed in other materials.However, the uniqueness of sili on was found to be related to a high ele tron mobility, whi h allows performingexperiments at a very low ele tron density, where the ele tron(cid:21)ele tron intera tion is espe ially important. Withthis fa t, it was possible to interpret the experimental data using not the theory in [7℄, whi h was developed fornonintera ting ele trons, but in a ordan e with the model in [25℄. As a result, the presen e of a metal(cid:21)insulatortransition was supported and a (cid:28)nite slope of the separatrix, whi h was predi ted in [25℄, was obtained (see Fig. 14aborrowed from [27℄.As is seen from the (cid:29)ow diagram in Fig. 12, the (cid:28)nite slope of the separatrix in the set of the temperature dependen eof σ or R of a system of 2D ele trons is ontrolled by the angle at whi h the separatrix in the (cid:29)ow diagram approa hesthe QCP. If the tangent to the separatrix is normal to the abs issa axis at the QCP due to any spe i(cid:28) reason, theseparatrix in the set of temperature dependen e has a zero derivative at T = 0 . Thus, the horizontal position of theseparatrix in the set of the temperature dependen e of ondu tivity in the vi inity of the quantum phase transitionresults from the symmetry of the (cid:29)ow diagram of a ertain system and is not an inherent property of all 2D systems.Conversely, a (cid:28)nite slope of the separatrix is not an indispensable onsequen e of a two-parameter (cid:29)ow diagram.An in lined separatrix ompli ates s aling, i.e., the redu tion of measurements performed along di(cid:27)erent (cid:29)ow linesto one universal urve by hanging the s ales. Nevertheless, the s aling of the resistan e R ( T ) data is possible. Figure14b shows the s aling arried out in the metalli region of the transition displayed in Fig. 14a. The three lower urvesin Fig. 14a are replotted in the oordinates ρ/ρ max (instead of ρ ) and ρ max ln ( T /T max ) (instead of T ; here, ρ max isgiven in dimensionless units), and the values ρ max and T max orrespond to the position of the maximum in ea h ofthe experimental urves, whi h are seen to merge into one urve and to oin ide with the theoreti al urve. This last urve was plotted using the al ulations in [25℄ and the intera tion parameter Θ determined in the same sample fromthe magnetoresistan e data obtained in a (cid:28)eld parallel to the 2D plane.An analogous problem of an in lined separatrix is also often en ountered during the experimental pro essing ofthe R x ( T ) temperature dependen e ( x is a ontrol parameter) in the vi inity of super ondu tor(cid:21)insulator quantumphase transitions. In Ref. [28℄, a pro edure was proposed for the ` orre tion' of the urves via the introdu tion of theadditional linear term R x ( T ) → R x ( T ) − αT, α = ∂R x c ( T ) ∂T (cid:12)(cid:12)(cid:12)(cid:12) T =0 , (67)into ea h of them to make the separatrix horizontal and for performing subsequent standard s aling for 2D systemsusing s aling variable (35). The idea of this pro edure is to ompensate for the slope of the separatrix in a (cid:29)owdiagram. However, the orre tness of this pro edure has not been proved theoreti ally.1 T (K)00.11 1 2 3 4 rp ( / ) h e n (10 cm )
10 2 - rr / m a x ln( T/T max )0 0.5 - - b r max p h e / FIG. 14: a) Temperature dependen e of the resistivity of a 2D gas in a (cid:28)eld-e(cid:27)e t sili on transistor in a on entration range ontaining the metal(cid:21)insulator transition [27℄. b) S aling of the three lower urves, whi h orrespond to the on entrations9.87 ( (cid:3) ), 9.58 ( (cid:13) ), and 9.14 · m − ( + ). Theoreti al results are shown by a solid line, [27℄ y Q y c B AQCP A B FIG. 15: Criti al region in the (cid:29)ow diagram of a 2D ele tron gas with intera tion (indi ated by a dotted urve)Formally, the ( x, T ) phase plane also has a meaning for a two-parameter (cid:29)ow diagram. However, the riti alvi inity of the transition an also be dire tly plotted in the diagram (see Fig. 15). It should be noted that thetwo-parameter s aling be omes one-parameter in the region adja ent to the separatrix inside the riti al region, where(cid:29)ow traje tories are parallel. Indeed, as noted above, one parameter is su(cid:30) ient to spe ify the position of a point online AB, and this parameter does not hange when line AB shifts parallel to the separatrix, e.g., to position A B ,sin e the (cid:29)ow lines are parallel to ea h other. As we move along the separatrix toward the quantum riti al point,near-separatrix traje tories turn aside alternately, and the strip in whi h the (cid:29)ow traje tories are parallel, as well asthe riti al region, narrows.6. QUANTUM TRANSITIONS BETWEEN THE DIFFERENT STATES OF A HALL LIQUIDStates in the plateaus of the quantum Hall e(cid:27)e t are the spe i(cid:28) phase states of a 2D ele tron gas with spe ialtransport properties des ribed by the longitudinal σ xx and transverse σ xy ondu tivities σ xx → at T → , σ xy = i ( e / π ~ ) , i = 0 , , , ... . (68)Su h phase states of a 2D ele tron gas are quantum Hall liquids with di(cid:27)erent quantum Hall numbers i , whi h aredetermined by the values of the Hall ondu tivity σ xy , Eqn (68), in the plateaus, i = σ xy / ( e / π ~ ) , i = 1 , , ... . (69)The transitions from one plateau to another indu ed by a hange in the magneti (cid:28)eld or the ele tron on entrationare learly visible in experimental urves. As an be seen from Fig. 16, σ xy jumps are a ompanied by narrow σ xx B (T) R xx ( k ) W r W xy ( k ) n = n = n = n = T = n = ×
10 cm
11 2 - GaAs Al Ga As -
0. 3 0.7
FIG. 16: The magnetoresistan e R xx and the Hall resistivity ρ xy versus the magneti (cid:28)eld B in a GaAs(cid:21)Al x Ga − x As het-erostru ture at T = 8 mK [29℄. The ele tron density in the 2D layer is . · m − , and the mobility is µ = 4 . · m /V · s.Figure 16 only displays the plateaus of the integer quantum Hall e(cid:27)e t, whi h is onsidered below. The integerquantum Hall e(cid:27)e t an also be realized in a nonintera ting ele tron gas; therefore, it an be des ribed without regardfor intera tion.To onstru t the (cid:29)ow diagram of a 2D system of nonintera ting ele trons in a strong magneti (cid:28)eld, we need two ondu tan e omponents that are equivalent to ondu tivity omponents σ xx and σ xy . Correspondingly, Eqn (37)transforms into the set of two equations d ln σ xx d ln L = f ( σ xx , σ xy ) ,d ln σ xy d ln L = f ( σ xx , σ xy ) , (70)When eliminating the variable L from two equations (70), we (cid:28)nd a relation between σ xx and σ xy , whi h an bedisplayed as urves in the ( σ xy , σ xx ) plane [30℄. These urves make up a (cid:29)ow diagram for a 2D system of nonintera tingele trons in a strong magneti (cid:28)eld (Fig. 17). The (cid:29)ow lines in this diagram are separated by separatri es periodi allyrepeated along the σ xy axis. This diagram is again a two-parameter diagram, but due to a strong magneti (cid:28)eldrather than to intera tion. As in the previous (cid:29)ow diagrams, we an move along arrows in (cid:29)ow traje tories by eitherin reasing the sample size L or de reasing the temperature at large L , i.e., in reasing L ϕ .In the region below points C i , the separatrix is split su h that two equivalent points lo ated at the same height (B and B ) appear in it. The (cid:29)ow lines inside the semi ir les in Fig. 17 are omitted be ause they are thought toexist separately from the (cid:29)ow lines outside these semi ir les: as the ontrol parameter in the region below points C i hanges, the motion leads to a hop between points (B and B ) and to a sharp in rease in σ xy . Stri tly speaking,the transfer should o ur along a line lo ated outside the semi ir le and bypassing the point C i above. However, forsimpli ity, it is indi ated by a horizontal dashed line.The ontrol parameter in the quantum Hall e(cid:27)e t regime is usually given by the ele tron on entration or themagneti (cid:28)eld. Their e(cid:27)e t on the state of a real system depends on the random (cid:28)eld of impurities and other defe ts,whi h transforms dis rete Landau levels into minibands and spe i(cid:28)es the energy stru ture and the hara ter of wavefun tions in them. Be ause a magneti (cid:28)eld determines the magneti length r B = (cid:0) ~ c/ | e | B (cid:1) / as a hara teristi s ale,we an speak about two limiting types of random potential, a potential with large-s ale (cid:29)u tuations of hara teristi sizes ζ ≫ r B and a short-range potential with ζ ≪ r B . In the long-period potential model, an energy value ε c existsnear the enter of ea h Landau miniband su h that a delo alized ele tron wave fun tion orresponds to this value. Ifthe wave fun tion is stri tly delo alized only at ε F = ε c and the random (cid:28)eld lifts the degenera y of energy levels, a3 s xx s xy A A A i =1 i =2 i =012 B B C CC FIG. 17: Flow diagram for a 2D ele tron gas in a strong magneti (cid:28)eld [30℄. The oordinates are represented by the ondu tivitytensor omponents σ xy and σ xx in the dimensionless units e / π ~ . Separatri es are indi ated by dashed lines. A i are stationarysingular points, C i are unstable singular points, whi h are quantum transition points similar to QCP in Fig. 12. The dashed lineindi ates the riti al region near C . Horizontal dashed line B (cid:21) B indi ates the image point hopping as the ontrol parameter hanges. The right-hand side of the (cid:28)gure shows the hypotheti al (cid:29)ow diagram that orresponds to a split phase transitionand the appearan e of a metalli phasesmooth hange in the ele tron on entration produ es a jumplike transformation of the system from one phase stateinto another through an isolated energy state with a delo alized wave fun tion at the Fermi level. This behavior isimplied in the (cid:29)ow diagram in Fig. 17, where ea h isolated metalli state orresponds to a pe uliar separatrix.The a tual width δε of the energy range with delo alized wavefun tions depends on (cid:28)ner pro esses, su h as tunnelingbetween two semi lassi al traje tories that are lose to ea h other in the vi inity of the saddle point (magneti breakdown). In essen e, δε is the energy un ertainty of any delo alized state. Another sour e of in reasing the δε range is the (cid:28)niteness of the lengths L and L ϕ . If the δε range is (cid:28)nite, a separatrix is split into two parallel linesand the phase transition is split into two transitions: a metalli state with a partly (cid:28)lled layer of extended statesat the Fermi level should appear between two Hall-liquid states whose indi es i di(cid:27)er by unity. The orrespondinghypotheti al (cid:29)ow diagram is shown on the right-hand side of Fig. 17.At (cid:28)rst glan e, it seems that experiment an distinguish between these two hypotheti al possibilities. For de(cid:28)nite-ness, we assume that the ele tron on entration n hanges in experiment ( onsiderations for a hange in the magneti (cid:28)eld are similar). As the on entration hanges, the Fermi level moves along the energy s ale. When states at theFermi level are delo alized, the σ xx ondu tivity is (cid:28)nite, and the σ xy ondu tivity is in the intermediate regionbetween two plateaus. Therefore, the temperature dependen e of the on entration range of the intermediate region(i.e., the σ xx peak width, the ∂σ xy /∂n derivative at the enter of the intermediate region, et .) extrapolated to T = 0 must determine the energy range δε of the delo alized states.However, the experimental results were found to be ambiguous. On the one hand, the omprehensive experiments in[31, 32℄ give a (cid:28)nite value of δε . For example, Fig.18 shows the transition width measured from the ρ xx resistivity peakwidth of a 2D gas in the GaAs/AlGaAs heterojun tion. The measured fun tion is seen to be reliably extrapolatedto ∆ B ≈ . as T → . However, as we see below, many experiments give the opposite result: as the temperaturede reases, the transition width tends to zero (see, e.g., Fig. 20). The fa tor that determines δε has not yet beenrevealed. In any ase, there is no unique relation between δε and mobility. The statisti al hara teristi s of therandom potential, whi h are poorly ontrolled, are likely to play a key role.We now move to experiments that do not exhibit a (cid:28)nite energy layer with delo alized states. Although the (cid:29)owdiagram of i → i + 1 transitions between di(cid:27)erent states of a Hall liquid has two parameters (see Fig. 17), it issymmetri with respe t to both the σ xy = ( i + 1 / e / π ~ ) and σ xy = i ( e / π ~ ) axes. Therefore, we may usethe version of the general theory of quantum phase transitions that is based on Eqns (25), (26) and assumes thats aling formulas of type (34) are valid. As far as we onsider only 2D ele troni systems, all resistan es have the samedimension [ Ω ℄ and must have the form R uv = F uv ( L ϕ /ξ ) = F uv (cid:18) δxT /zν (cid:19) , (71)in the vi inity of the transition. Here, the u and v subs ripts stand for the x and y oordinates, F uv is an unknownfun tion, and the argument of the arbitrary F fun tion is written using the last form in Eqn (35).4 T (K) D B ( T ) B (T)0 2 4 6 r x x FIG. 18: Temperature dependen e of the peak width of the longitudinal resistivity ρ xx of a 2D ele tron gas in the GaAs/AlGaAsheterojun tion during the → transition in the magneti (cid:28)eld about 4 ï¨DZ [32℄. The arrier mobility and on entration at T = 1 . K are µ = 34000 m /V s and n = 1 . · m − . The inset shows ρ xx ( B ) urve re orded at 150 mKIn ontrast to the ase of metal(cid:21)insulator transitions, we analyze experimental results instead of al ulating orpredi ting the values of ν and z . As an example, Fig. 19 shows the magneti -(cid:28)eld dependen e at various temperaturesof the longitudinal R xx and transverse R xy resistan es of a Hall bar in a GaAs-based heterostru ture [33℄. B (T) R xy ( k ) W R xx ( k ) W i = 4 i = 3 B c B (T) T = 31 mK114 mK 510 mK FIG. 19: Transverse R xy and longitudinal R xx resistan es of an Al x Ga − x As − Al . Ga . As heterostru ture, x = 0 . , atvarious temperatures. The riti al magneti (cid:28)eld of the 4-3 transition determined from the point of interse tion of the R xy ( T ) urves is B c = 1 . T, [33℄A ording to Eqn (71), the s aling variable u = δx/T /zν (72)is identi ally zero at all temperatures in the ase where the ontrol parameter takes a riti al value; orrespondingly,we have R uv ( x c , T ) = const . (73)Therefore, separatrix (73) must be horizontal, and all isotherms R uv ( x, T = const) must interse t at one point x = x c .This is the (cid:28)rst test of the appli ability of Eqn (71).We (cid:28)rst fo us on the R xy ( T ) urves. As is seen from Fig. 19, the R xy ( T ) isotherms obtained for an Al x Ga − x As(cid:21)Al . Ga . As heterostru ture with x = 0 . [33℄ do interse t at one point, B c = 1 . T. Near the interse tionpoint, all the R xy ( T ) urves an be expanded into a series and repla ed by straight lines ( ∂R xy /∂B ) B c ( B − B c ) . Asthe slopes of these lines are hanged from ( ∂R xy /∂B ) B c to ( ∂R xy /∂B ) B c /T κ , where κ = 1 /zν , all the straight linesmust merge into one line. The hoi e of the value of κ at whi h the relation ( ∂R xy /∂B ) B c ( T ) /T κ = const , (74)5 T (K) d R / d B xy ( k / T ) W D B ( T ) dR /dB ~T xy - D B ~T
FIG. 20: Determination of the riti al index κ = 1 /zν for transition 4-3 using the R xx ( T ) peak width (a) and mat hing theslopes of the interse ting R xx ( T ) urves (b). The data were obtained from the sample used for Fig. 19 in a dilution refrigerator(triangles) and a liquid He ryostat ( ir les) [33℄holds is the se ond step in the appli ation of the s aling pro edure, and the possibility of this hoi e is the se ond ondition for the appli ability of the theory. To hoose κ , we plot ( ∂R xy /∂B ) versus T on a log − log s ale (Fig. 20).Formally, Eqn (71) an be applied to both longitudinal ( R xx ) and transverse ( R xy ) resistan es. However, the peakheight depends on the temperature; that is, the point of the maximum does not satisfy ondition (73). Leaving asidethe reasons for this fa t, we an use the longitudinal resistan e data for s aling analysis by a ounting for the integralproperty of the R xx ( B ) fun tions in the vi inity of the transition, namely, the peak half-width ∆ B determined by a ertain algorithm. In Fig. 20, ∆ B is determined as the distan e between the two maxima of the ( ∂R xx /∂B ) derivative.As an be seen from Fig. 20, an analysis of both families of the fun tions gives the same value of the riti al index κ = 0 . , whi h is an additional argument for our s aling pro edure.This value of the riti al index κ = 0 . was repeatedly obtained in heterostru tures made from various materials.However, ontrary to expe tations, this value is not universal: other values in the range from 0.2 to 0.8 were alsodete ted in a number of experiments on various heterostru tures. This s atter alls for explanations, sin e s alingrelations and riti al indi es are usually universal.At T = 0 , transition o urs when the ondition ε F = ε c is satis(cid:28)ed and therefore the di(cid:27)eren e δε = | ε F − ε c | isthe only `internal' ontrol parameter of the system, with the orrelation length ξ depending on it in a ordan e witha power law: ξ ∝ ( δε ) ν . (75)From this standpoint, the magneti (cid:28)eld B or the 2D ele tron on entration n , whi h depends on the gate voltage V g , are `external' ontrol parameters x , δx ≡ | B c − B | ∝ ( δε ) ν or δx ≡ | n c − n | ∝ | V gc − V g | ∝ ( δε ) ν . (76)In both ases, the exponent ν is the same. This is supported by the fa t that the R uv ( B ) and R uv ( V g ) experimental urves re orded using the same sample under equivalent onditions di(cid:27)er only in the s ales on the abs issa axis.Eventually, we an write Eqn (25) as ξ ∝ ( δx ) ν ( ν = ν ν ) , (77)where the relation between the ontrol parameter δx and the orrelation length ξ onsists of the following two links: ξ depends on the position of the Fermi level ε F with respe t to the delo alized level ε c , and the di(cid:27)eren e ε F − ε c ,in turn, depends on δx . Correspondingly, a ording to Eqn (77), the index ν turns out to be the produ t of ν and ν . The relation between ξ and δε and the related index ν are likely to be universal and the same for all transitionsbetween di(cid:27)erent quantum Hall liquids. But the index ν is determined by the density of states g ( ε ) in the vi inityof the energy ε c and, hen e, depends on the spe i(cid:28) features of the random potential; this potential an be long- orshort-range, statisti ally symmetri or asymmetri with respe t to the mean value, and so on. The authors of [33℄,whose urves are used in this se tion, just studied the e(cid:27)e t of spe i(cid:28) features of the random potential on κ .The ∆ B ( T ) urve in Fig. 20 di(cid:27)ers radi ally from the urve in Fig.18; in the latter ase, a free term ∆ B wasintrodu ed for the experimental data to be approximated by a power fun tion. Nevertheless, we an also performs aling analysis of the experimental data in this ase using the se ond hypotheti al version of the (cid:29)ow diagram (see6Fig. 17). For this interpretation, the presen e of the ∆ B term means that in the B range of the ontrol parameter B , the image point in the (cid:29)ow diagram moves a ross the metalli -phase orridor and falls on a separatrix not at B orresponding to a maximum of σ xx ( B ) and the derivative ∂σ xy /∂B but at B + ∆ B . This problem is dis ussed indetail in review [34℄. Here, we only note that Fig. 18 a tually ontains this s aling analysis. By extrapolating the ∆ B ( T ) dependen e to T = 0 , we an determine the delo alized-state layer width in units of magneti (cid:28)eld B and(cid:28)nd that ∆ B ( T ) − ∆ B depends linearly on T . This means that κ = 1 in this experiment. The same value of κ wasobtained earlier in [31℄. 7. CONCLUSIONAll the onsidered ases of metal(cid:21)insulator transitions were found to be adequately des ribed by (cid:29)ow diagrams.The list of theoreti al works that have su essfully used this te hnique begins with work [7℄, where a theoreti almodel for nonintera ting ele trons in a zero magneti (cid:28)eld was developed. The last a hievement in this (cid:28)eld is the onstru tion of a (cid:29)ow diagram for a 2D model system of intera ting ele trons and the demonstration of the possibilityof a metal(cid:21)insulator transition in this system .As regards the general theory of quantum phase transitions, a phase transition in a 3D system of nonintera tingele trons demonstrates that, in prin iple, this theory an be used to des ribe lo alized(cid:21)delo alized ele tron transitions.Neither ondu tan e, whi h is used as a physi al quantity spe ifying the state of the system, nor disorder, whi h is themain ontrol parameter, are substantial obsta les for this theory. However, as usual, various parti ular ases requiretheoreti al versions of various degrees of omplexity. For example, the version des ribed in this review annot beapplied to the model proposed in [25℄.The relative role and possibilities of both theoreti al approa hes are demonstrated when the integer quantum Halle(cid:27)e t is des ribed. The (cid:29)ow diagram in Fig. 17 is very onvenient for the dis ussion of the types of transitions thatare possible in a system and for the formulation of questions to be experimentally he ked. Many of these questionsare still open. For example, it is un lear whi h of the versions of the (cid:29)ow diagram in Fig. 17 is realized in reality andwhether the transition between the states with quantum indi es i and i ± , i ⇄ i ± , (78)is split [ i is determined by Eqn (69)). There are also problems related to the topology of the (cid:29)ow diagram. A ordingto Fig. 17, transitions where quantum number i hanges by more than unity are impossible [35, 36℄. However, whentheorists interpret many experimental data, they state that su h transitions o ur (see, e.g., review [34℄ and thereferen es therein).The general theory of quantum phase transitions does not onsider the problem of the relative position of varioustransitions in the phase plane. This theory des ribes the riti al vi inity of one ertain transition. Pro eeding fromthe assumptions that (a) a transition exists and (b) the fa torization ξ = ξ ( δx ) , L ϕ = L ϕ ( T ) o urs in its riti alregion, the resistivity an be des ribed by Eqns (34)(cid:21)(36) (see Se tions 2.3 and 5.3). Then, the transition point x = x c and riti al indi es an be determined by pro essing the R ( x, T ))