aa r X i v : . [ c ond - m a t . o t h e r] J un dc conductivity as a geometric phase Bal´azs Het´enyi
Department of PhysicsBilkent UniversityTR-06800 Bilkent, Ankara, Turkey (Dated: October 8, 2018)The zero frequency conductivity ( D c ), the criterion to distinguish between conductors and insu-lators is expressed in terms of a geometric phase. D c is also expressed using the formalism of themodern theory of polarization. The tenet of Kohn [ Phys. Rev.
A171 (1964)], namely, thatinsulation is due to localization in the many-body space, is refined as follows. Wavefunctions whichare eigenfunctions of the total current operator give rise to a finite D c and are therefore metallic.These states are also delocalized. Based on the value of D c it is also possible to distinguish purelymetallic states from states in which the metallic and insulating phases coexist. Several exampleswhich corroborate the results are presented, as well as a numerical implementation. The formalismis also applied to the Hall conductance, and the quantization condition for zero Hall conductance isderived to be e Φ B Nhc = QM , with Q and M integers. I. INTRODUCTION
What makes conductors conducting and insulators in-sulating? In classical physics this question is answered byconsidering the localization of individual charge carriers.Localized, bound charges do not contribute to conduc-tion. Quantum mechanics has rendered the answering ofthis question more difficult. In band theory, conductioncan be attributed to the density of electron states at theFermi level: if ρ ( ǫ F ) = 0, the system is conducting, if ρ ( ǫ F ) = 0 it is insulating. However, simple band theoryis not able to explain insulation of strongly correlatedsystems. In 1964 Kohn suggested [1] that the criterionthat distinguishes metals from insulators is localizationof the total position of all charge carriers. Kohn also de-rived [1] the quantum criterion of dc conductivity, theDrude weight ( D c ).For several decades, testing Kohn’s hypothesis was dif-ficult, due to the fact that in crystalline systems (systemswith periodic boundary conditions) the total position op-erator is ill-defined. This limitation was overcome by themodern theory of polarization [2–5], in which the expec-tation value of the total position is cast in terms of ageometric phase [6–8]. The geometric phase arises uponvarying the crystal momentum across the Brillouin zone.In numerical applications the polarization is easiest tocalculate in terms of the ground state expectation valueof the total momentum shift operator [9, 10]. These de-velopments have simplified the calculation of the polar-ization considerably, and are now in widespread use inelectronic structure calculations.Moulopoulos and Ashcroft [11] have also suggested aconnection between conduction and a Berry phase re-lated to the center of mass. Recently, the author hasshown [12] that the total current can be expressed as aphase associated with moving the total position acrossthe periodic cell, and that it can be written as a groundstate expectation value of the total position shift opera-tor. We note that topological invariants can also charac-terize metals [13] as well as insulators. II. PURPOSE
We demonstrate that D c can also be expressed in termsof a geometric phase. The formal expression for D c derived here consists of an expectation value of single-body operators and a geometric phase arising from thevariation of the total momentum and the total position.Its form is similar to that of the Hall conductance [14].The second term is also expressed in terms of the totalmomentum and total position shift operators, in otherwords, based on a formalism similar to that of the “mod-ern” theory of polarization. The resulting formula es-tablishes the precise connection between localization andconductivity as suggested by Kohn [1]. If the groundstate wavefunction of a system is an eigenstate of thetotal current operator, D c is finite. Such wavefunctionsare also delocalized according to the criterion defined byResta [9, 10]. The calculation of the Drude weight isalso straightforward: for metals the D c = παL . (Eq. (6),where L denotes the size of the system), for insulatorsit is zero. For wavefunctions corresponding to coexis-tence between metallic and insulating phases it holds that0 < D c < παL . One calculates the spread in total current,and if this spread is zero, then D c = παL . These resultsare indepedent of dimensionality. The formalism is alsoused to derive the Hall conductance [14], and a quanti-zation condition for that quantity being zero is derived.The condition coincides with the well-known experimen-tal results for the fractional quantum Hall effect [15]. III. DEFINITIONS
Let | Ψ i denote the ground state wavefunction of an N particle system. In coordinate space one can writeΨ( x + X, ..., x N + X ) where X denotes a shift of allcoordinates, or equivalently one can write in momentumspace Ψ( k + K, ..., k N + K ). A wavefunction can belabeled by X or K ( | Ψ( X ) i , | Ψ( K ) i ). One can definethe shift operators in position or momentum space as e − i ∆ K ˆ X | Ψ( K ) i = | Ψ( K + ∆ K ) i (1) e − i ∆ X ˆ K | Ψ( X ) i = | Ψ( X + ∆ X ) i , where ˆ X = P Ni =1 ˆ x i , and ˆ K = P Ni =1 ˆ k i . In lattice modelsthe current operator in momentum space takes the formˆ K = P Ni =1 sin(ˆ k i ). The explicit construction of the shiftoperators is given in Refs. [12, 16]. IV. MAIN RESULTSA. Conductivity as a geometric phase
The Drude weight [1] is defined as D c = πL ∂ E (0) ∂ Φ , (2)where Φ denotes a perturbing field, and the derivativeis the adiabatic derivative. The second derivative withrespect to Φ can be expressed as ∂ E (0) ∂ Φ = α + γ, (3)where α = i N X j h Ψ | [ ˆ ∂ k j , ˆ ∂ x j ] | Ψ i , (4) and where γ = − i π Z π/L − π/L Z L d K d X (5)( h ∂ K Ψ( K, X ) | ∂ X Ψ( K, X ) i − h ∂ X Ψ( K, X ) | ∂ K Ψ( K, X ) i ) . This expression is derived in Appendix B. γ has the formof an integrated Berry curvature over a surface in thetwo-dimensional space K − X , and can be converted intoa geometric phase by application of the Stokes theorem.Note that the Drude weight is the sum of two terms, oneproportional to the sum of the commutators of each mo-mentum and position, and a “commutator” of the vari-ables related to the total position and total momentumof the system. B. Analog of D c based on the modern theory ofpolarization D c , in particular the term γ , can also be expressedusing total momentum and total position shift operators.For charge carriers with mass one, the one-body term is α = (cid:26) N for continuyous models, − h Ψ | ˆ T (0) | Ψ i for lattice systems . (6)the geometric phase term can be written as γ = − lim ∆ X, ∆ K → X ∆ K " Im ln h Ψ | e i ∆ K ˆ X e i ∆ X ˆ K | Ψ ih Ψ | e i ∆ X ˆ K | Ψ i + Im ln h Ψ | e i ∆ X ˆ K e − i ∆ K ˆ X | Ψ ih Ψ | e i ∆ X ˆ K | Ψ i . (7)This expression is derived in Appendix C. V. INTERPRETATION
The first term of D c , proportional to α , is an extensivequantity, a sum over single-body operators. For any non- trivial system it is expected to be finite. For an insulatorthe many-body term (proportional to γ ) must cancel thesingle-body term.We consider a general wavefunction of the formΨ( x , ..., x N ) corresponding to an unperturbed groundstate. Acting on this function with the shift operatorsaccording to the first and second terms of γ (Eq. (7)),respectively, results in e i ∆ K ˆ X e i ∆ X ˆ K Ψ( x , ..., x N ) = e iN ∆ K ∆ X e i ∆ K P Ni =1 x i Ψ( x + ∆ X, ..., x N + ∆ X ) , (8) e i ∆ X ˆ K e − i ∆ K ˆ X Ψ( x , ..., x N ) = e − i ∆ K P Ni =1 x i Ψ( x + ∆ X, ..., x N + ∆ X ) . Evaluating the scalar products, one can then show thatapart from the term e iN ∆ K ∆ X in Eq. (8) the two terms inEq. (7) are complex conjugates of each other. The term e iN ∆ K ∆ X gives a contribution of − N to the conductivitycancelling the single-body term. When this derivation isvalid the system is insulating. This derivation, of course,has limits of validity, for example, if discontinuities arepresent in the momentum distribution [17].If the function | Ψ i is an eigenfunction of the currentoperator, then γ is zero, hence the system is metallic.To show this, one considers that the eigenvalue of thecurrent operator for an unperturbed ground state is zero,which means that the total position shift operator willhave no effect at all. In this case the two terms of γ are complex conjugates of each other, and their sum willhave no imaginary part.If a wavefunction is an eigenstate of the total currentoperator, it also follows that the system is delocalized.Indeed the localization criterion defined by Resta [9, 10]is σ X = − K Re ln h Ψ | e − i ∆ K ˆ X | Ψ i . (9)The function resulting from the total momentum shiftoperator acting on an eigenfunction of the total currentwill be orthogonal to the original function, resulting in adivergent σ X .To decide whether a particular ground state eigenfunc-tion is an eigenfunction of the current one can calculatethe spread in current [12], defined as σ K = − X Re ln h Ψ | e − i ∆ X ˆ K | Ψ i . (10)If σ K is zero then the wavefunction is indeed a currenteigenstate, the system is metallic, moreover γ = 0 and the D c = παL . Otherwise the wavefunction corresponds to aninsulating state. To show this one can use the fact thatfor an eigenfunction of the current with eigenvalue zerothe expectation value h Ψ | e − i ∆ X ˆ K | Ψ i = 1, must give one,but for any other case h Ψ | e − i ∆ X ˆ K | Ψ i <
1. In calculatingconductivity, one can also use Eq. (9), but this quantityis expected to diverge when the system becomes metallic,hence calculations based on σ K can be expected to bemore stable.A wavefunction can also be a linear combination of aneigenstate of the current operator and a localized statecorresponding to the coexistence of the insulating andmetallic states. In this case the single body term willbe partially cancelled by the many-body term and a fi-nite Drude weight will result, but in that case D c willbe smaller than the contribution due to single-particleoperators (for continuous models D c < N ). VI. EXAMPLESA. Fermi sea, BCS
For both the Fermi sea and BCS wavefunctions D c = παL . The Fermi sea is diagonal in the momentum rep-resentation and corresponds to an eigenstate of ˆ K witheigenvalue zero. A BCS wavefunction consists of a lin-ear combination of wavefunctions with different numberof particles, but all have eigenvalue of ˆ K = 0, and theargument for the Fermi sea extends. B. Gutzwiller metal
The Gutzwiller variational wavefunction was proposedto understand the Hubbard model [18–20], and is of theform | Ψ G (˜ γ ) i = e − ˜ γ P i ˆ n i ↑ ˆ n i ↓ | F S i . (11)The state | F S i denotes the Fermi sea, out of which dou-bly occupied sites are projected out via the projector e − ˜ γ P i ˆ n i ↑ ˆ n i ↓ . This wavefunction has been shown [21, 22]to be metallic for finite values of the variational param-eter ˜ γ , ( D c = απL ).Indeed, the geometric phase term γ vanishes. To seethis, consider that the shift operator e i ∆ X ˆ K commuteswith the projector e − ˜ γ P i ˆ n i ↑ ˆ n i ↓ , since shifting the posi-tion of every particle will not affect the number of doublyoccupied sites [12]. Thus e i ∆ X ˆ K will operate on the Fermisea, which has eigenvalue ˆ K | F S i = 0, and then the samereasoning applies as in the case of the Fermi sea. C. Baeriswyl insulating wavefunction for a spinlesssystem
An insulating variational solution for spinless fermionson a lattice with nearest neighbor interaction ( t - V model)in one dimension, is the Baeriswyl wavefunction [23],which in this case has the form | Ψ B (˜ α ) i = Y RBZ[ e − ˜ αǫ k c † k + e ˜ αǫ k c † k + π ] | i , (12)where the product is over the reduced Brillouin zone.This wavefunction is easily shown to be insulating [23],hence we expect that it gives D c = 0.This can be shown readily by considering again the ac-tion of the shift operators on | Ψ B (˜ α ) i . The scalar prod-ucts in γ evaluate to h Ψ B (˜ α ) | e i ∆ K ˆ X e i ∆ X ˆ K | Ψ B (˜ α ) i = Y RBZ[ e i ∆ X sin( k +∆ K ) e − ˜ α ( ǫ k + ǫ k +∆ K ) + e − i ∆ X sin( k +∆ K ) e ˜ α ( ǫ k + ǫ k +∆ K ) ] , (13) h Ψ B (˜ α ) | e i ∆ X ˆ K e − i ∆ K ˆ X | Ψ B (˜ α ) i = Y RBZ[ e i ∆ X sin( k ) e − ˜ α ( ǫ k + ǫ k − ∆ K ) + e − i ∆ X sin( k ) e ˜ α ( ǫ k + ǫ k − ∆ K ) ] . U σ K D c × L/π − h T i σ X .
95 327 .
95 —1 5 . . . . . . . . . . . . . . K = ∆ X = 0 . Substituting into the definiton of γ and taking the lim-its ∆ K, ∆ X → D c = 0 as expected for aninsulating state. The above derivation is also valid forthe mean-field spin or charge-density wave solutions ofstrongly correlated lattice models. D. Anderson localized system
We have evaluated the above formula for a model whichexhibits Anderson localization [24], with Hamiltonian ofthe form H = − t X i c † i c i +1 + H. c. + U X i ξ i n i , (14)where ξ i is a number drawn from a uniform Gaussiandistribution. By diagonalizing the Hamiltonian we havecalculated the localization parameter [9, 10] for differ-ent system sizes, and have found that the larger systemsizes are always more localized for finite U (results not shown). We have also calculated the Drude weight andthe quantity σ K . The results are shown in Table I.For the metallic state σ K gives zero as expected, andthe Drude weight is equal to minus one-half the kineticenergy. For all insulating cases the Drude weight is verynear zero, in particular if one compares its magnitude tothat of the kinetic energy. While one can calculate theDrude weight directly, this may be difficult in some ap-plications, since phases have to be evaluated. Howeverevaluating the kinetic energy and the spread in currentallows the determination of the Drude weight unambigu-ously. VII. HALL CONDUCTANCE
The Hall conductance can also be expressed in termsof a Berry phase [14], similar in form to the conductivityderived above (Eq. (5)). It is possible to express the Hallconductance as a ground state observable. [25, 26] Herewe express it via shift operators, and derive a quantiza-tion condition for zero Hall conductance in a quantumHall system. The momentum shift operators in this casetake forms which are different from those used in express-ing dc conductivity.Our starting point is the form derived by Thouless etal. [14], σ Hxy = ie πh Z d K x d K y [ h ∂ K x Ψ | ∂ K y Ψ i − H.c.] , (15)which, using the formalism above converts to σ Hxy = e h lim ∆ K x ∆ K y → K x ∆ K y (cid:20) Im ln h Ψ | U x (∆ K x ) U y (∆ K y ) | Ψ ih Ψ | U y (∆ K y ) | Ψ i + Im ln h Ψ | U y (∆ K y ) U x ( − ∆ K x ) | Ψ ih Ψ | U y (∆ K y ) | Ψ i (cid:21) , (16)where U x (∆ K x ) and U y (∆ K y ) are momentum shift op-erators in the x and y directions. Using the forms of thetotal momentum shift operators in Eqs. (1) (applicablewhen the wavefunctions can be written in the coordinateor momentum representations) we can show that in thelimit ∆ K x , ∆ K y → σ Hxy = ie h X i h Ψ | [ˆ x i , ˆ y i ] | Ψ i . (17) Using Eq. (16) applied to a Landau state one canalso derive a quantization condition for the values of themagnetic field at which σ Hxy must be zero. A Landau levelhas the form ψ ( x, y ) = e ik x x φ n ( y − y ) , (18)where y = k x ~ ceB . As far as the x direction is concernedthis function is neither in the momentum nor in the posi-tion representations. However, the momentum shift oper-ators can be constructed, considering that a momentumshift in the x -direction is also a position shift in the y direction. It is easy to check that in this case U x (∆ K x ) = e i ∆ K x x e i ∆ Y k y , (19)with ∆ Y = ∆ K x ~ ceB . The momentum shift in the y di- rection remains U y (∆ K y ) = e i ∆ K y y . (20)Applying the shift operators to the Landau state resultsin U x (∆ K x ) U y (∆ K y ) ψ ( x, y ) = e i ∆ K y ( y − y ) e i ∆ K x x ψ ( x, y + ∆ y ) , (21) U y (∆ K y ) U x ( − ∆ K x ) ψ ( x, y ) = e i ∆ K y ∆ y e i ∆ K y ( y − y ) e i ∆ K x x ψ ( x, y − ∆ y ) , where ∆ y = ∆ K x ~ ceB . If ∆ K x ∆ y = ∆ K x ∆ K y ~ ceB =2 πM , with M integer, then the phase in the secondof Eqs. (21) is one, and in this case taking the limits∆ K x , ∆ K y → K x = q x πL x and ∆ K y = q y πL x , with q x , q y integers, which correspondsto equivalent states for the adiabatic case [27, 28] it fol-lows that for a system with N particles the quantizationcondition is e Φ B N hc = QM , (22)where Φ B denotes the magnetic flux, and Q is an inte-ger. Indeed, the maxima in the Hall resistivity occur [15]precisely at values of the magnetic flux given by Eq. (22). VIII. CONCLUSION
In this work it was shown that the zero frequency con-ductivity can be expressed in terms of a Berry phase.Subsequently the conductivity was also expressed interms of shift operators (total momentum and total posi-tion) leading to expressions which provide clear physicalinsight, as well as a good starting point for numericalwork. It was argued that a metallic state is one which isthe eigenstate of the total current operator. Such stateswere also shown to be delocalized. In this case the dcconductivity takes its maximum possible value for a givensystem (proportional to the number of charge carriers forcontinuous models). These conclusions were supportedby analytic and numerical calculations on a number ofexamples, both metallic and insulating. If the wavefunc-tion is a linear combination of a total current eigenstateand an insulating wavefunction then a finite dc condutiv-ity results which is smaller than the allowed maximum.Hence, based on the value of the dc conductivity it is pos-sible to distinguish metallic and insulating states fromones in which conducting and insulating states coexist.Subsequently the formalism was used to express the Hallconductance, and to derive the quantization condition atwhich the Hall conductance is zero. The condition coin-cides with the well-known experimental results.
ACKNOWLEDGMENTS
The author acknowledges a grant from the Turk-ish agency for basic research (T ¨UBITAK, grant no.112T176).
APPENDICESAPPENDIX A: Perturbed Hamiltonian
The dc conductivity [1] is proportional to the secondderivative of the ground state energy with respect to thePeierls phase Φ at Φ = 0. For a continuous system, tak-ing the mass of charge carriers to be unity, the Hamilto-nian has the formˆ H (Φ) = X j (ˆ k j + Φ) V , (23)in the case of discrete models one can writeˆ H (Φ) = ˆ T + ˆ V , (24)with ˆ T (Φ) = − X j te i Φ c † j +1 c j + H. c. . (25)For a detailed discussion see Refs. [1] and [29]) For bothcontinuous and lattice Hamiltonians it holds that H ′ (0) = i [ ˆ H, ˆ X ] = ˆ K, (26)and H ′′ (0) = i [ ˆ K, ˆ X ] , (27)where ˆ X ( ˆ K ) are defined asˆ X = P j ˆ x j (28)ˆ K = P j ˆ k j , for continuous systems andˆ X = P j j ˆ n j (29)ˆ K = − it P j c † j +1 c j + H.c., for lattice models. One can also write H ′′ (0) as a sum ofone-body operators as H ′′ (0) = − X j [ˆ k j , ˆ ∂ k j ] = − X j [ ˆ ∂ x j , ˆ x j ] . (30)One can also show that H ′′ (0) = (cid:26) N for continuous models, − ˆ T (0) for lattice systems . (31)One can expand the Hamiltonian and the ground statewavefunction up to second order as H (Φ) ≈ H (0) + Φ H ′ (0) + Φ H ′′ (0) (32) | Ψ(Φ) i ≈|
Ψ(0) i + Φ | Ψ ′ (0) i + Φ | Ψ ′′ (0) i and express the second derivative of the ground stateenergy with respect to Φ at Φ = 0 as ∂ E (Φ) | Φ=0 = h Φ(0) | H ′′ (0) | Φ(0) i (33)+2 h Φ ′ (0) | H ′ (0) | Φ(0) i + 2 h Φ(0) | H ′ (0) | Φ ′ (0) i . APPENDIX B: dc conductivity as a geometric phase
In this appendix the dc conductivity is derived in termsof a geometric phase. As shown in Ref. [12] the firstderivative of the ground state energy with respect to Φfor a continuous Hamiltonian is given by ∂ Φ E (Φ) = α Φ − iL Z L h Ψ( X ; Φ) | ∂ X | Ψ( X ; Φ) i , (34)where α = (cid:26) N for continuous models, −h Ψ | ˆ T (0) | Ψ i for lattice systems . (35) Taking the derivative with respect to Φ and setting Φ tozero results in ∂ E (Φ) | Φ=0 = α − iL Z L d X (36)[ h ∂ Φ Ψ( X ) | ∂ X | Ψ( X ) i + h Ψ( X ) | ∂ X | ∂ Φ Ψ( X ) i ] . Since Φ corresponds to a shift in the crystal momentum K the derivative with respect to Φ can be replaced witha derivative with respect to K . Subsequently an averageover K can be taken, resulting in ∂ E (Φ) | Φ=0 = α + γ, (37)with γ = − i π Z L Z π/L − π/L d X d K (38)[ h ∂ K Ψ( X, K ) | ∂ X Ψ( X, K ) i − h ∂ X Ψ( X, K ) | ∂ K Ψ( X, K ) i ] . The quantity γ in Eq. (37) is a surface integral overa Berry curvature, which can be converted into a lineintegral around the included surface via Stokes theorem,as for the Hall conductivity [14].The quantity α can be written with the help of Eq.(30) as α = i X j h Ψ | [ ∂ x j , ∂ k j ] | Ψ i . (39)In other words the conductivity corresponds to the dif-ference between the sum of one body commutators of theposition and momenta and the commutator of the totalposition and total momentum. APPENDIX C: dc conductivity in terms of shiftoperators
Our starting point is the current [12] written in termsof shift operators [16], ∂ Φ E (Φ) = α Φ − X Im ln h Ψ(Φ) | e i ∆ X ˆ K | Ψ(Φ) i . (40)Taking the derivative with respect to Φ results in ∂ Φ E (Φ) = α + γ, (41)with γ = 1∆ X Im " h ∂ Φ Ψ(Φ) | e i ∆ X ˆ K | Ψ(Φ) ih Ψ(Φ) | e i ∆ X ˆ K | Ψ(Φ) i + h Ψ(Φ) | e i ∆ X ˆ K | ∂ Φ Ψ(Φ) ih Ψ(Φ) | e i ∆ X ˆ K | Ψ(Φ) i (42)We can set the derivative in Φ equal to the derivative inthe crystal momentum, and set Φ = 0. For now we will consider only the first term in Eq. (42) but the steps forthe second term are essentially identical. We can writethis term as1∆ X ∆ K Im " ∆ K h ∂ K Ψ(0) | e i ∆ X ˆ K | Ψ(0) ih Ψ(0) | e i ∆ X ˆ K | Ψ(0) i , (43)where we have divided and multiplied by ∆ K . For small∆ K we can replace this term with1∆ X ∆ K Im ln " K h ∂ K Ψ(0) | e i ∆ X ˆ K | Ψ(0) ih Ψ(0) | e i ∆ X ˆ K | Ψ(0) i , (44) which can be converted to1∆ X ∆ K Im ln " h Ψ(∆ K ) | e i ∆ X ˆ K | Ψ(0) ih Ψ(0) | e i ∆ X ˆ K | Ψ(0) i , (45)and using the total momentum shift operator results in1∆ X ∆ K Im ln " h Ψ | e i ∆ K ˆ X e i ∆ X ˆ K | Ψ ih Ψ | e i ∆ X ˆ K | Ψ i . (46)Applying exactly the same steps to the second term ofEq. (42) results in γ = 1∆ X ∆ K " Im ln h Ψ | e i ∆ K ˆ X e i ∆ X ˆ K | Ψ ih Ψ | e i ∆ X ˆ K | Ψ i ! + Im ln h Ψ | e i ∆ X ˆ K e − i ∆ K ˆ X | Ψ ih Ψ | e i ∆ X ˆ K | Ψ i ! , (47)which is the discretized form for the Drude weight. [1] W. Kohn, Phys. Rev.
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