Berry phase of the composite Fermi-liquid
BBerry phase of the composite Fermi-liquid
Guangyue Ji ( 棘 广 跃 ) and Junren Shi ( 施 均 仁 )
1, 2, ∗ International Center for Quantum Materials, Peking University, Beijing 100871, China Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
We derive the definition of the Berry phase for the adiabatic transport of a composite fermion(CF) in a half-filled composite Fermi-liquid (CFL). It is found to be different from that adopted inprevious investigations by Geraedts et al. For the standard CFL wave function, we analytically showthat the Berry curvature is uniformly distributed in the momentum space. For the Jain-Kamillawave function, we numerically show that its Berry curvature has a continuous distribution insidethe Fermi sea and vanishes outside. We conclude that the CF with respect to both the microscopicwave-functions is not a massless Dirac particle.
I. INTRODUCTION
The ubiquitous presence of the Berry phase is notablein recent theoretical investigations of condensed matterphysics. For non-interacting systems, it becomes a uni-fying concept for characterizing the orbital effects of thespin or other internal degrees of freedom [1], and playscentral roles in systems such as topological insulators [2],Dirac/Weyl semimetals [3] and valleytronic materials [4].Recently, it becomes clear that the Berry phase also playsa role in the theory of composite Fermions (CFs) [5].CFs can be regarded as weakly interacting particles re-siding in a hidden Hilbert space [6]. A wave function ofnon-interacting CFs in the hidden Hilbert space can bemapped into a wave function appropriate for describingthe physical state of a strongly correlated, fractionallyfilled Landau level. The theory of CFs achieves tremen-dous successes in understanding the fractional quantumHall effect and related phenomena. [5].However, although the wave functions prescribed bythe CF theory are shown to be very accurate and widelyaccepted [7], the effective theory of CFs interpreting thewave functions is still open to debate. The conven-tional interpretation, as explicated in Halperin-Lee-Read(HLR) theory of the composite Fermi-liquid (CFL) [8–10], treats the CF as an ordinary Newtonian particle. Inits pristine form, it suffers from an apparent difficulty:it can not correctly predict the CF Hall conductance ofa half-filled Landau level [11]. The difficulty motivatesSon to propose that the CF should be a massless Diracparticle [12]. An alternative interpretation, i.e., the CFis neither a Newtonian particle nor a Dirac particle, buta particle subject to a uniformly distributed Berry cur-vature in the momentum space and the Sundaram-Niudynamics [13], is also put forward [14–16]. It is alsoshown that the picture is equivalent to the dipole pic-ture of CFs [17]. The three pictures imply three differentdistributions of the Berry curvature, i.e., zero, singularlydistributed and uniformly distributed, respectively. Theclarification of the issue then hinges on the determinationof the Berry curvature for CFs.A “first principles” approach for determining the Berry curvature of CFs should be based on the microscopic CFwave-functions prescribed by the CF theory. To this end,several attempts have been made. In Ref. [16], the dy-namics of the CF Wigner crystal is derived. It showsthat the CF is subject to a uniformly distributed Berrycurvature in the momentum space. For the half-filledCFL phase, the Berry curvature distribution is found tobe uniform by determining the dynamics of a test (dis-tinguishable) CF added to the CF Fermi sea [15]. Aheuristic argument based on the dipole picture of CFsalso suggests the same [14, 15]. These works may drawcriticism for neglecting the particle exchange symmetryin their treatments. It is in this context that the recentworks by Geraedts et al. stand out [18–20]. Their calcu-lations are based on a microscopic CFL wave function inits full antisymmetric form. However, a close scrutinyto the works reveals a number of difficulties. Firstly, thedefinition of the Berry phase is a prescribed one and isnot fully justified. Secondly, the evaluation of the Berryphase based on the definition seems to be not numeri-cally robust, sensitive to the choices of paths and proneto statistical errors. Moreover, there exist extraneous ± π/ phases preventing direct interpretations of numer-ical results. Finally, the microscopic CFL wave functionadopted for the calculation is of the Jain-Kamilla (JK)type [21], which is numerically efficient in implementingthe projection to the Landau level (LLL). However, it isunclear whether or not it yields the same result as thatfrom the standard CFL wave function prescribed by thetheory of CFs [5].In this paper, we solve these issues and determine thedistribution of the Berry curvature for CFs. First, wederive the definition of the Berry phase directly fromthe original definition of the Berry phase. It is foundto be different from the prescribed one adopted by Ger-aedts et al. [18, 19]. Then, we analytically show that theBerry curvature distribution is uniform in the momentumspace for the standard CFL wave function. On the otherhand, to compare with the results in Refs. [18, 19], wealso evaluate the Berry curvature of the JK wave func-tion. With our definition, the numerical evaluation ofthe Berry phase becomes robust and free of the extra- a r X i v : . [ c ond - m a t . s t r- e l ] J u l neous phases observed in Refs. [18, 19]. It enables us tonumerically determine the distribution of the Berry cur-vature in the whole momentum space. We find that theBerry curvature has a continuous distribution inside theFermi sea and vanishes outside, and is different from theuniform distribution of the standard CFL wave function.We analytically show that the difference originates fromthe different quasi-periodicities of the two wave functionsin the reciprocal space.The reminder is organized as follows. In Sec. II, wederive the definition of the Berry phase of the CFL. InSec. III, we determine the Berry phase and Berry cur-vature of the standard CFL wave function analytically.In Sec. IV, we evaluate the Berry curvature of the JKwave function numerically. In Sec. V, we analyze thequasi-periodicities of the two wave functions in the re-ciprocal space, and determine the uniform backgroundof the Berry curvature. In Sec. VI, we summarize anddiscuss our results. II. DEFINITION OF THE BERRY PHASE OFCFL
In this section, we derive the definition of the Berryphase for CFL systems. We first introduce the genericdefinition of the Berry phase. Next, we derive the defini-tion of the Berry phase for CFL systems from the genericdefinition. Then, we discuss different representations ofthe Berry phases. Finally, we discuss and interpret Ger-aedts et al.’s definition and results.
A. Berry phase
A quantum system acquires a geometric phase, i.e., theBerry phase, when it is adiabatically transported alonga path C by varying parameters α in its Hamiltonian ˆ H ( α ) [22]. The Berry phase is determined by a lineintegral in the parameter space γ ( C ) = ˆ C i (cid:104) Ψ α | ∇ α Ψ α (cid:105) · d α , (1)where | Ψ α (cid:105) is the eigenstate of ˆ H ( α ) , and the integrandis called the Berry connection. The phase is independentof how the path is traversed as long as it is slow enoughfor the adiabaticity to hold. For a closed path, the phaseis gauge invariant, i.e., independent of the choice of thephase factor of the wave function.The Berry phase formula Eq. (1) can be recast into analternative form as a time integral: γ ( C ) = i ˆ t t (cid:28) Ψ α ( t ) (cid:12)(cid:12)(cid:12)(cid:12) dΨ α ( t ) d t (cid:29) d t, (2)where the wave function evolves with time via the time-dependence of its parameters, and α ( t ) is an arbitrary time-dependent function that traverses the path C with t ( t ) being the beginning (ending) time of the evolu-tion. The integrand is actually a part of the SchrödingerLagrangian [13, 23]: L = (cid:28) Ψ( t ) (cid:12)(cid:12)(cid:12)(cid:12) i (cid:126) dd t − ˆ H (cid:12)(cid:12)(cid:12)(cid:12) Ψ( t ) (cid:29) , (3)which governs the time evolution of a quantum system.This is why one sees the ubiquitous presence of the Berryphase in various contexts such as effective dynamics [13,23] and path-integral formalisms. B. Definition of the Berry phase of CFL
From the generic definition of the Berry phase, wecan infer a definition of the Berry phase appropriate forCFL systems. For the purpose, it is more convenient tofirst consider a much simpler system, i.e., a set of non-interacting electrons residing in a Bloch band. The defi-nition of the Berry phase for such a system is well knownin the single-particle form [13]. Here, we will treat thesystem as a many-particle system and find a many-bodygeneralization of the Berry phase definition. The gener-alization turns out to be general enough for applying toCFL systems.The many-body wave function of a set of non-interacting Bloch electrons is a Slater determinant ofBloch states: Ψ k ( z ) = det (cid:2) ψ k j ( z i ) (cid:3) , (4)where k ≡ { k , k , . . . } denotes the list of the quasi-wave-vectors of the Bloch states occupied by electrons,and ψ k j ( z i ) = exp(i k j · z i ) u k j ( z i ) is the Bloch wave func-tion with u k j ( z i ) being its periodic part.The wave function has a number of general proper-ties which are actually shared by the much more compli-cated CFL wave functions: (a) it is parameterized by aset of wave-vectors k ; (b) it is an eigenstate of the (mag-netic) center-of-mass translation operator ˆ T ( a ) such that ˆ T ( a )Ψ k ( z ) = exp(i (cid:80) i k i · a )Ψ k ( z ) , where a is one of thevectors of the Bravais lattice with respect to the period-icity of the system. As a result, two states with differenttotal wave-vectors are orthogonal to each other. Withproper normalizations of wave functions, we have: (cid:104) Ψ k | Ψ k (cid:48) (cid:105) = δ (cid:32)(cid:88) i k i − (cid:88) i k (cid:48) i (cid:33) f ( k , k (cid:48) ) , (5)where f ( k , k (cid:48) ) is a function with the property f ( k , k ) =1 , and the wave-vectors in the Dirac Delta function areregarded equal if they are only different by a reciprocallattice vector; (c) the wave function has the Fermionicexchange symmetry and can be obtained from an un-symmetrized wave function ϕ k by applying the anti-symmetrization operator ˆ P : ˆ P = 1 N ! (cid:88) P ( − P ˆ P , (6) Ψ k ( z ) = ˆ P ϕ k ( z ) ≡ N ! (cid:88) P ( − P ϕ k ( ˆ P z ) , (7)where ˆ P z denotes a permutation of electron coordi-nates, N is the total number of electrons, and ϕ k ( z ) = √ N ! (cid:81) i ψ k i ( z i ) for the Bloch system. In the unsym-metrized form, an electron is associated with a partic-ular wave-vector. The association is lost in the antisym-metrized form.With the wave function in hand, one may be temptedto directly apply Eq. (1) to determine the Berryphase. However, a difficulty immediately arises. Tosee that, we treat k as the parameters α , substi-tute Eq. (4) into Eq. (1), apply the identity ˆ P = ˆ P ,and obtain A k ≡ i (cid:104) Ψ k | ∂ k Ψ k (cid:105) = − (cid:104) Ψ k | r | ϕ k (cid:105) +i (cid:104) Ψ k | e i k · r | ∂ k u k ( r ) (cid:81) i ≥ ψ k i ( r i ) (cid:105) . Unfortunately,the resulting Berry connection A k is not a legitimateone because (cid:104) Ψ k | r | ϕ k (cid:105) does not have a deterministicvalue since the Bloch states have definite momenta andtherefore infinite position uncertainty. This is the diffi-culty we have to address before the generic definition canbe applied to wave functions like Eq. (4).The most straightforward approach to address theissue is to introduce a unitary transformation to theHamiltonian: ˆ H ( k ) = e − i k · r ˆ He i k · r . The result-ing Hamiltonian acquires dependence on the parame-ters k , and the corresponding eigenstate wave func-tion becomes e − i k · r ϕ k ( r ) = u k ( r ) (cid:81) i ≥ ψ k i ( r i ) . Onecan then apply Eq. (1) to obtain the well-known result A k = i (cid:104) u k | ∂ k u k (cid:105) . Such an approach is adoptedand generalized in Ref. [15] to show that the standardCFL wave function yields a uniform Berry curvature Ω( k ) = 1 /qB , where q is the unit charge of carriers and B is the perpendicular component of the external mag-netic field B . However, the approach is not compatiblewith the exchange symmetry because ˆ H ( k ) obviouslybreaks the symmetry of exchanging the first particle (theparticle being transported) with others. Adopting suchan approach means that we have to ignore the exchangesymmetry. This is what we want to avoid here.We therefore adopt and generalize the approach pre-sented in Ref. [13]. The basic idea is that, since the diffi-culty is due to the fact that a Bloch state does not havea deterministic position expectation value, we replace itwith a wave packet state which has a central wave-vector k c and give rises to a deterministic position expectationvalue z c : (cid:12)(cid:12)(cid:12) ˜Ψ k c , z c (cid:69) = ˆ d k a ( k , t ) | Ψ k (cid:105) , (8)where we assume that | a ( k , t ) | is narrowly distributed around k c and satisfies ˆ d k | a ( k , t ) | = (cid:68) ˜Ψ k c , z c (cid:12)(cid:12)(cid:12) ˜Ψ k c , z c (cid:69) = 1 , (9) ˆ d k | a ( k , t ) | k = k c . (10)We choose the time-dependence of a ( k , t ) to make k c traverses a path C while keeping z c fixed. By applyingEq. (2), we can then determine the Berry phase acquiredby the wave-packet state. In the end, the width of thedistribution | a ( k , t ) | will be set to zero so that the wave-packet state approaches to the Bloch state. We will showthat it yields a well-defined limit.We still need to define z c . It is easy to see that (cid:104) ˜Ψ k c , z c | z | ˜Ψ k c , z c (cid:105) does not yield a deterministic expec-tation value. This is because z loses its associationwith k in the antisymmetrized wave function, and (cid:104) ˜Ψ k c , z c | z | ˜Ψ k c , z c (cid:105) = (cid:104) ˜Ψ k c , z c | z i | ˜Ψ k c , z c (cid:105) is nothing butthe center-of-mass position. Since electrons, all but one,have definite wave-vectors in | ˜Ψ k c , z c (cid:105) , the center-of-massposition has infinite uncertainty. To obtain a determin-istic position, we define z c as the position of the electronassociated with the wave-vector k by: z c = Re (cid:68) ˜Ψ k c , z c (cid:12)(cid:12)(cid:12) z (cid:12)(cid:12)(cid:12) ˜ ϕ k c , z c (cid:69) , (11)where | ˜ ϕ k c , z c (cid:105) is the unsymmetrized form of | ˜Ψ k c , z c (cid:105) , i.e., | ˜Ψ k c , z c (cid:105) ≡ ˆ P | ˜ ϕ k c , z c (cid:105) .We can show that z c does have a deterministic value.To see this, we substitute Eq. (8) into Eq. (11), and have z c =Re ˆ d k ˆ d k (cid:48) a ∗ ( k (cid:48) , t ) a ( k , t ) × (cid:2) − i ∂ k (cid:104) Ψ k (cid:48) | ϕ k (cid:105) + i (cid:10) Ψ k (cid:48) (cid:12)(cid:12) e i k · z (cid:12)(cid:12) ∂ k u k (cid:11)(cid:3) =Re ˆ d k [ a ∗ ( k , t ) i ∂ k a ( k , t )+ i | a ( k , t ) | (cid:10) Ψ k (cid:12)(cid:12) e i k · z (cid:12)(cid:12) ∂ k u k (cid:11) ] , (12)where we define u k ( z ) ≡ e − i k · z ϕ k ( z ) . To obtainthe last expression, we make use of Eq. (5) which re-duces to (cid:104) Ψ k (cid:48) | ϕ k (cid:105) = (cid:104) Ψ k (cid:48) | Ψ k (cid:105) = δ ( k − k (cid:48) ) for the cur-rent case. Moreover, one can show that e i k · z | ∂ k u k (cid:105) has the same center-of-mass periodicity as | Ψ k (cid:105) , thus (cid:10) Ψ k (cid:48) (cid:12)(cid:12) e i k · z (cid:12)(cid:12) ∂ k u k (cid:11) ∝ δ ( k − k (cid:48) ) .Writing the amplitude a ( k , t ) as the form a ( k , t ) = | a ( k , t ) | e − i γ ( k ,t ) , and setting the width of the distribu-tion | a ( k , t ) | to zero, we obtain z c = ∂γ (cid:16) ˜ k , t (cid:17) ∂ k c − Im (cid:10) Ψ ˜ k (cid:12)(cid:12) e i k c · z (cid:12)(cid:12) ∂ k c u ˜ k (cid:11) (13)with ˜ k ≡ { k c. , k , . . . } . Equation (13) is the many-bodygeneralization of Eq. (2.8) of Ref. [13].We then apply Eq. (2) to determine the Berry phaseof transporting k c . Following a procedure similar to thatfor determining z c , we obtain i (cid:42) ˜Ψ k c ( t ) , z c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d ˜Ψ k c ( t ) , z c d t (cid:43) = ∂γ (cid:16) ˜ k , t (cid:17) ∂t . (14)Using the relation ∂γ (cid:16) ˜ k , t (cid:17) ∂t = d γ (cid:16) ˜ k , t (cid:17) d t − ˙ k c · ∂γ (cid:16) ˜ k , t (cid:17) ∂ k c (15)and Eq. (13), we determine the phase acquired by trans-porting k c along C : γ ( C ) = ˆ t t d t d γ (cid:16) ˜ k , t (cid:17) d t − ˙ k c · z c − ˆ C Im (cid:10) Ψ ˜ k (cid:12)(cid:12) e i k c · z (cid:12)(cid:12) ∂ k c u ˜ k (cid:11) · d k c . (16)The first term is integrable (provided z c is fixed) andvanishes when C is a closed path. We thus exclude itfrom the definition of the Berry phase, and define theBerry connection as: A k = − Im (cid:10) Ψ k (cid:12)(cid:12) e i k · z (cid:12)(cid:12) ∂ k u k (cid:11) , (17)where we relabel k c as k . The Berry phase is just a lineintegral of the Berry connection.For numerical calculations, it is more convenient tocalculate the Berry phase induced by a small but discretechange of the wave-vector. By using the trapezoidal ruleto estimate the line integral and approximating ∂ k u k asa first-order divided difference, we determine the Berryphase for k → k (cid:48) : φ k (cid:48) k = 12 (cid:2) arg (cid:10) ϕ k (cid:48) (cid:12)(cid:12) e i q · z (cid:12)(cid:12) Ψ k (cid:11) − ( k (cid:10) k (cid:48) ) (cid:3) , (18)with q ≡ k (cid:48) − k .Equation (17) and (18) are the definitions of the Berryconnection and Berry phase for a many-body system, re-spectively. The definitions are directly descended fromthe original definition Eq. (1). In our derivation, we onlymake use of the aforementioned three properties of thewave function. As we will show later, CFL wave func-tions considered in this paper do have these properties.Therefore, the definitions are also applicable for CFL sys-tems. C. Representations of the Berry phase
Motivated by the dipole picture of CFs [17], we alsodefine another Berry phase (connection). According tothe dipole picture, a CF is a bound state of an electron and quantum vortices, and the position of the quantumvortices z v c is displaced from that of the electron by [15,16] z v c = z e c + n × k c , (19)where z e c ≡ z c is the position of the electron, and n denotes the unit normal vector of the system plane. Ob-viously, the phase determined by Eq. (16) depends onwhich position is fixed when k c is transported. The Berryconnection Eq. (17) and the Berry phase Eq. (18) are de-fined by assuming that z e c is fixed. Hereafter, we willlabel A k ( φ k (cid:48) k ) as A e k ( φ e k (cid:48) k ) to explicitly indicate theassumption. On the other hand, if we assume that z v c isfixed, we should replace z c in Eq. (16) with z v c − n × k c ,and obtain another Berry connection A v k : A v k = A e k − k × n . (20)The corresponding Berry phase for k → k (cid:48) is φ v k (cid:48) k = φ e k (cid:48) k + ( k × k (cid:48) ) · n . (21)Then, which one is the Berry connection (phase) of theCF? The answer depends on how we define the position ofa CF. By definition, the k -space Berry connection is theconnection of transporting k while keeping the positionfixed. If one chooses to define the CF position as the posi-tion of the quantum vortices (electron), then A v k ( A e k )should be regarded as the CF Berry connection. Onecan even adopt other definitions of the CF position, andobtain other definitions of the Berry connections. Math-ematically, all these definitions are equivalent. They arejust different representations of the same physics.Nevertheless, for a reason to be discussed in the nextsubsection, we will call the v-representation as the CFrepresentation, and interpret φ v k (cid:48) k and A v k as the Berryphase and Berry connection of CFs. D. Interpretation of Geraedts et al.’s result
Finally, we would like to comment on the definitionof the Berry phase introduced by Geraedts et al. inRefs. [18, 19]. It is obviously not a definition descendedfrom the original definition of the Berry phase. It shouldbe more appropriately called as a scattering phase. In-deed, the transition amplitude for k → k (cid:48) induced by asingle-body scalar potential V ( z i ) = e i q · z i is proportionalto U k (cid:48) k = (cid:42) Ψ k (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i e i q · z i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ψ k (cid:43) (22)with k (cid:48) = k + q . It is exactly the matrix element cal-culated by Geraedts et al. Their results thus provide a“first-principles” determination of the scattering ampli-tude.We can actually interpret Geraedts et al.’s resultsin light of the picture of independent CFs. They ob-serve that the matrix element has an i ( − i ) factor fora discrete wave-vector change in the clockwise (anti-clockwise) sense [24]. It means that the matrix elementhas a form like U k (cid:48) k ∝ i ( k (cid:48) × k ) e iΦ k (cid:48) k , (23)and Φ k (cid:48) k is interpreted as the Berry phase. For aCF system, the potential will induce a density modu-lation, which in turn induces a modulation of the effec-tive magnetic field. The scattering amplitude induced bythe modulated effective-magnetic-field does have a factor ∝ i ( k (cid:48) × k ) , as shown in Eq. (23) of Ref. [25]. Differentfrom what is assumed in Ref. [25], the scattering ampli-tude carries an extra phase Φ k (cid:48) k , indicating the presenceof a Berry phase when the transition k → k (cid:48) occurs.It is then interesting to ask how the Berry phase in-ferred from the scattering amplitude is related to theBerry phases we have defined. Our numerical calcula-tion (See Sec. IV C) suggests that Φ k (cid:48) k ≈ φ v k (cid:48) k , (24)i.e., the Berry phase inferred from the scattering am-plitude is actually the Berry phase defined in the v-representation. For the dipole picture, it means that asfar as the impurity scattering is concerned, the positionof a CF should be defined as the position of its quantumvortices.On the other hand, one expects that CFs are scatterednot only by the fluctuation of the effective magnetic field,but also by the scalar potential itself. As a result, thescattering amplitude should in general have the form [25]: U k (cid:48) k = [ V ( k (cid:48) , k ) + i ( k (cid:48) × k ) V ( k (cid:48) , k )] e iΦ k (cid:48) k . (25)The presence of the first term will interfere the deter-mination of Φ k (cid:48) k . Indeed, we observe that the Berryphase inferred from the scattering amplitude by assum-ing a vanishing V ( k (cid:48) , k ) always deviates from π , and thedeviation becomes more pronounced when N is scaled up(see Table II). The approach becomes unreliable when thetwo terms in the prefactor of Eq. (25) are comparable inmagnitude. In this case, the phase carried by the prefac-tor cannot be easily distinguished from the Berry phaseto be determined. One encounters such a situation whenstudying the effect of the Landau level mixing [26]. Inthe study, Pu et al. adopt the wave function Ψ mix k = (1 − β ) Ψ k + β Ψ unproj . k , (26)where Ψ k is a CFL wave function projected to the LLL(either Eq. (27) or Eq. (47)), Ψ unproj . k is the unprojectedform of Ψ k (i.e., Eq. (27) without applying ˆ P LLL ), and β controls the strength of the Landau level mixing. Oneexpects that V dominates in the scattering amplitude of the unprojected wave function Ψ unproj . k , and V > ( V < ) for particles (holes). From Eq. (25), the pref-actor contributes a phase ( π ) for each step of trans-porting a particle (hole) [26]. On the other hand, forthe projected wave function Ψ k , V dominates, and theprefactor contributes a phase ± π/ . In between for amixed wave function, the phase depends on the relativestrength of V and V . Without a reliable way of sub-tracting the phase from the scattering amplitude, thedetermined “Berry phase” may fluctuate widely. This isindeed observed in Ref. [26]. In contrast, our definitionEq. (18) does not suffer from the difficulty. It is easyto see that the Berry phase of Ψ unproj . k is always zero.For the mixed wave function, it is reasonable to expectthat the Berry phase accumulated by transporting a CFaround the Fermi circle is a value between zero and thatyielded by Ψ k , i.e., π . III. BERRY PHASE OF THE STANDARD CFLWAVE FUNCTION
In this section, we evaluate the Berry phase of the stan-dard CFL wave function. First, we introduce the explicitform of the CFL wave function on the torus geometry.Next, we show its center-of-mass translational symmetryunder the magnetic translation operator. Then, we ana-lytically determine the Berry curvature of the standardCFL wave function.To unify notations, we use the symbols a i ≡ a ix + i a iy , a ∗ i ≡ a ix − i a iy and a i ≡ ( a ix , a iy ) to denote an electron-related variable in its complex form, complex conju-gate and vector form, respectively, with the subscript i indexing electrons. Symbols without a subscript (e.g. a ≡ { a i } ) denote a list of the variables for all electrons,and symbols in the upper case (e.g. A ≡ (cid:80) i a i ) denotesums of the variables over all electrons. a · b ≡ (cid:80) i a i b i de-notes the inner product of two lists of variables. The unitof length is set to the magnetic length l B ≡ (cid:112) (cid:126) /e | B | . A. CFL wave function
The standard CFL wave function for a system on atorus with a filling fraction ν = 1 /m ( m is an even in-teger) can be written as (omitting the Gaussian factor e − (cid:80) i | z i | / ) [27] Ψ CF k ( z ) = ˆ P LLL det (cid:104) e i ( k i z ∗ j + k ∗ i z j ) / (cid:105) J ( z ) , (27) J ( z ) = ˜ σ m ( Z ) (cid:89) i The wave function is an eigenstate of the magneticcenter-of-mass translation operator defined by [29] ˆ T ( a ) = (cid:89) i e a · ∂ z i + i( n × z i ) · a = e ( aZ ∗ − a ∗ Z ) (cid:89) i e a · ∂ z i (31)and a ∈ { n L /N, n ∈ Z } , where L is a period of thetorus. To show that, we apply ˆ T ( a ) to the wave functionEq. (30), and have ˆ T ( a ) ϕ CF k ( z ) e − (cid:80) i | zi | = e i K ∗ a − a ∗ ( Z + nL ) × e i k ∗ · z ˜ σ m ( Z + i K + nL ) × e − (cid:80) i | zi | (cid:89) i It is obvious from the above discussions that the stan-dard CFL wave function have all the properties outlinedin Sec. II B. Therefore, the definition of the Berry phaseEq. (18) can be applied. It turns out that the standardCFL wave function has a simple structure which makespossible an analytic determination of the Berry phase.To determine the Berry phase, we need to determinethe matrix element (cid:10) Ψ CF k (cid:12)(cid:12) e − i q · z (cid:12)(cid:12) ϕ CF k (cid:48) (cid:11) = (cid:68) Ψ CF k (cid:12)(cid:12)(cid:12) ˆ P LLL e − i q · z (cid:12)(cid:12)(cid:12) ϕ CF k (cid:48) (cid:69) = e − | q | (cid:10) Ψ CF k (cid:12)(cid:12) ˆ t ( − q ) (cid:12)(cid:12) ϕ CF k (cid:48) (cid:11) , (37)where k (cid:48) ≡ { k + q , k , . . . } , and we define the operator ˆ t i ( k α ) ≡ exp (cid:18) i k α ∂ z i + i2 k ∗ α z i (cid:19) . (38)It is easy to verify the relation ˆ t i ( k α ) ˆ t i ( k β ) = e i( k α × k β ) · n ˆ t i ( k α + k β ) . (39)On the other hand, apart from an unimportant co-efficient, the unsymmetrized wave function ϕ CF k can bewritten as ϕ CF k (cid:48) ( z ) = ˆ t ( k + q ) (cid:89) i ≥ ˆ t i ( k i ) J ( z ) . (40)Using Eq. (39), we have ˆ t ( − q ) ϕ CF k (cid:48) ( z ) = e − i ( q × k ) · n (cid:89) i ˆ t i ( k i ) J ( z ) ≡ e − i ( q × k ) · n ϕ CF k ( z ) . (41)Inserting the relation into Eq. (37), we obtain (cid:104) Ψ CF k | e − i q · z | ϕ CF k (cid:48) (cid:105) = e i ( k × q ) · n − | q | (cid:104) Ψ CF k | Ψ CF k (cid:105) . (42)Applying Eq. (18), we determine the Berry phase inthe e-representation: φ e k (cid:48) k = 12 ( q × k ) · n . (43)The Berry phase in the CF (v-)representation can bedetermined by applying Eq. (21): φ v k (cid:48) k = − 12 ( q × k ) · n . (44)The Berry connections corresponding to the Berryphases are A e / v k = ± ( k × n ) / . (45)They give rise to the Berry curvatures (in the unit of /qB ) Ω e / v k ≡ ( ∇ k × A e / v k ) · n = ∓ . (46)It indicates that the Berry curvature in the momentumspace is a constant, exactly the one suggested in theuniform-Berry-curvature picture of CFs [14–16]. IV. BERRY PHASE OF THE JK WAVEFUNCTION In this section, we evaluate the Berry phase and Berrycurvature of the JK wave function of the CFL. First, weintroduce the JK wave function on the torus geometry.Next, we describe the numerical implementation of thecalculations of the Berry phase. Then, we numericallyevaluate the Berry curvature and determine its distribu-tion in the whole wave-vector space. A. JK Wave function It is numerically hard to implement the LLL projec-tion in Eq. (27) since it expands the wave function to N ! terms. To address the issue, Jain and Kamilla introducean alternative projection method [21]. The projectionmethod is adopted by Refs. [18–20] for numerically eval-uating the Berry phase. The wave function (JK wavefunction) has the form [27] Ψ JK k ( z ) =det [ ψ i ( k j )] × ˜ σ m ( Z + i K ) (cid:89) i For both Geraedts et al.’s definition and our definition,the calculation of the (Berry) phase involves the evalua-tion of a matrix element (cid:104) Ψ k | ˜Ψ k (cid:48) (cid:105) . For Geraedts et al.’sdefinition, ˜Ψ k (cid:48) is defined by (cid:101) Ψ k (cid:48) = e − i q · z Ψ k (cid:48) , (50)where we drop the superscript JK for brevity. We notethat Geraedts et al.’s original definition uses the factor ρ q = (cid:80) i e − i q · z i . The two forms are equivalent except foran unimportant factor.For our definition, after inserting ˆ P into the matrixelement of Eq. (18), we can write (cid:101) Ψ k (cid:48) as a form likeEq. (47) but with the determinant modified to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − i q · z ψ ( k + q ) ψ ( k ) . . . ψ ( k N ) e − i q · z ψ ( k + q ) ψ ( k ) . . . ψ ( k N ) ... ... . . . ... e − i q · z N ψ N ( k + q ) ψ N ( k ) . . . ψ N ( k N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (51)i.e., the column with respect to the transported wave-vector ( k ) is modified by the “momentum boost opera-tor” e − i q · z i .We implement a Metropolis Monte-Carlo algorithmsimilar to that detailed in Ref. [19]. Specifically, the over-lap | D | and phase φ of the matrix elements are evaluatedby | D | e iφ ≡ (cid:104) Ψ k | (cid:101) Ψ k (cid:48) (cid:105) (cid:113) (cid:104) Ψ k | Ψ k (cid:105)(cid:104) (cid:101) Ψ k (cid:48) | (cid:101) Ψ k (cid:48) (cid:105) = N (cid:80) (cid:48) | Ψ k | (cid:101) Ψ k (cid:48) Ψ k (cid:114) N (cid:80) (cid:48) | Ψ k | · (cid:12)(cid:12)(cid:12) (cid:101) Ψ k (cid:48) Ψ k (cid:12)(cid:12)(cid:12) , (52)where N denotes the normalization factor of | Ψ k | , and (cid:80) (cid:48) stands for lattice summation of z i ’s [14, 19]. TheMarkov chains of our Monte-Carlo simulation sample z i ’sby assuming a probability density ∝ | Ψ k ( z ) | . By using aMarkov chain, we can determine the phases and overlapswith respect to both the definitions simultaneously, sincethe two definitions are only differed by ˜Ψ k (cid:48) . Path a) b) c) o l d Ref. [19] 0.813(7) 0.965(6) -0.058(6)This work 0.821(1) 0.964(2) -0.050(1)new 1.110 0.906 0.068(1)Table I. The phases for N = 13 . For Geraedts et al.’s def-inition (old), both the results presented in Ref. [19] and ourown calculation are shown. The results with respect to ourdefinition (new) are shown in the last row. The numericaluncertainties for paths a and b are not shown because theyare too small. C. Numerical results To test our numerical implementation, we first cal-culate the Berry phases along the paths calculated inRef. [19]. The numerical results are presented in Table. I.For Geraedts et al.’s definition, the results presented inRef. [19] and our own calculation results coincide wellwithin numerical uncertainties. The results with respectto our definition are also shown.Berry phases with respect to both definitions for a fewrepresentative paths are shown in Table. II. An imme-diate observation is that the calculation with our defini-tion is much more robust numerically, as evident fromthe magnitudes of the overlap. With our definition, theoverlap is always close to one and improves when N isscaled up. For Geraedts et al.’s definition, the overlap isnowhere close to one and further deteriorates for larger N , and even nearly vanishes for steps along directionsperpendicular to the Fermi circle, resulting in poor statis-tics and undeterminable results. Moreover, our definitionyields directly interpretable results, i.e., no subtraction ofthe extraneous ± π/ phases noted in Ref. [18] is needed.It is interesting to observe that the two different defi-nitions actually lead to similar qualitative conclusions aslong as φ v is interpreted as the CF Berry phase. Withour definition, the Berry phase of adiabatic transport ofa CF around the Fermi circle is converged to π (path a, N = 110 ), whereas with Geraedts et al.’s definition, it in-volves guesswork to reach the same conclusion. We alsofind that the Berry phase for transport around a unit pla-quette outside the Fermi sea (path b2) nearly vanishes.This is consistent with Geraedts et al.’s observation thatthe phase is independent of the area of the trajectory en-closing the Fermi sea. The consistencies may not be acoincidence. See Sec. II D for an interpretation.The distribution of the Berry curvature, both insideand outside the Fermi sea, can now be determined be-cause of the improved numerical robustness. To deter-mine the Berry curvature, we transport a CF or a holealong the edges of a unit plaquette (see path b in Ta- Path a) b) c) N 13 38 110 36 (b1) (b2) φ v /π old . 82 0 . 72 0 . U.D. U.D. . new . 11 1 . 03 1 . 01 0 . ∗ . ∗ . | D | min old . ∗ . ∗ . ∗ . ∗ . ∗ . ∗ new . 94 0 . 98 0 . 99 0 . 99 0 . 99 0 . Table II. CF Berry phases φ v and minimal overlaps | D | min along different paths for the JK wave function. The paths areindicated by arrowed solid lines comprised of steps with min-imal changes of the quantized wave vectors. Three kinds ofpaths are considered: a) the Fermi circle; b) a unit plaquetteinside (b1) or outside (b2) the Fermi sea; (c) a closed pathinside the Fermi sea. Both results for our definition (new)and Geraedts et al.’s definition (old) are shown. | D | min is theminimum overlap among steps along a path. For the paths in-side the Fermi sea, a hole is transported, and resulting Berryphases are shown with inverted signs. The values markedwith ∗ have been scaled by a factor of N . U.D. indicates anundeterminable result due to a vanishing overlap. . . | k | l B Ω v ( | k | )( / q B ) particlehole CFJK Figure 1. The Berry curvature Ω v ( | k | ) as a function ofthe CF wave number | k | for the half-filled CFL ( m = 2 ).The Berry curvature for the JK wave function is numericallydetermined by transporting a CF (hole) outside (inside) theFermi sea consist of 109 CFs, shown as filled (empty) dots.The inset bar plot shows its distribution on the 2D plane ofthe momentum space. The Berry curvature for the standardCFL wave function is equal to one, shown as the solid line. ble. II), and the Berry curvature for the plaquette is de-termined by Ω v = φ vB /S , where S = 2 π/N φ is the areaof the unit plaquette. The result is shown in Fig. 1.We see that the Berry curvature has a continuous distri-bution inside the Fermi sea and vanishes outside. Thedistribution is obviously not the singular one implied bythe Dirac interpretation [20, 30]. V. UNIFORM BACKGROUND OF THE BERRYCURVATURE It is evident that for both the wave functions, the Berrycurvature is a constant in most of the region of the k -space except the one occupied by the Fermi sea. In otherwords, the Berry curvature has a uniform background.The values of the background for the two wave functionsare different: for the standard CFL wave function and for the JK wave function, respectively. In this section,we show that the values can be determined analyticallyby inspecting the quasi-periodicities of the wave func-tions in the k -space [31]. It turns out that the two wavefunctions have different quasi-periodicities, resulting inthe different values of the uniform background.We first examine the standard CFL wave functionEq. (30). By using Eq. (29), it is easy to show that it hasthe quasiperiodicity in the k -space: Ψ CF k (cid:12)(cid:12) k → k + L × n = ξ N φ ( L ) e L ∗ ( i k + L )Ψ CF k , (53)where k → k + L × n corresponds to i k → i k + L inthe complex form.As a result, we can define a super-Brillouin zone (SBZ)spanned by K α = L α × n , α = 1 , with L and L being the two edges of the torus. From Eq. (53), byapplying either the original definition Eq. (1) [32] or thegeneralized definition Eq. (17), we find that the Berryconnection has the quasi-periodicity: A e k + K = A e k + 12 ( K × n ) , (54)where K is one of reciprocal lattice vectors of the SBZ.The total Chern number of the SBZ can be determined by C tot = (2 π ) − ¸ A e k · d k with the circuit integral alongthe boundary of the SBZ [31]. Using Eq. (54), it is easy toshow that the integral is equal to − A/ π = − N φ . Sincethe Berry curvature is a constant in most of region of theSBZ, C tot is equal to ¯Ω e A/ π in the limit of A → ∞ ,where ¯Ω e is the value of the uniform background of theBerry curvature. We thus obtain ¯Ω e = − , (55)for the standard CFL wave function. For the CF repre-sentation, we have ¯Ω v = 1 . (56)Similarly, it is easy to show that Ψ JK k has an approxi-mated quasi-periodicity: Ψ JK k (cid:12)(cid:12) k → k + L × n ∝ exp(i mL ∗ k / JK k , (57)where we ignore the small change of ¯ k in the limit of N →∞ . The presence of m in the phase factor is notable. It originates from the i mk j factor in the argument of ˜ σ -function in Eq. (48) [27, 33]. The quasi-periodicity ofthe Berry connection is modified to A e k + K = A e k + m K × n ) . (58)By applying the same analysis as that for the standardCFL wave function, we obtain that the uniform back-ground of the Berry curvature for the JK wave functionis − m in the e-representation and − m in the CF rep-resentation.We summarize the values of the Berry curvature back-ground as follows: ¯Ω v = (cid:40) , (CF)2 − m, (JK) . (59)Note that the result is solely determined by the quasi-periodicities of the wave functions. The fact that the twowave functions have different quasi-periodicities meansthat they must have different Berry curvatures. VI. DISCUSSION AND SUMMARY In summary, we have (a) derived the definition of theBerry phase applicable for CFL systems; (b) analyticallydetermined the Berry phase of the CFL with respect tothe standard CF wave function, and found that it yields auniform Berry curvature; (c) numerically calculated theBerry phase with respect to the JK wave function, anddetermined the distribution of the Berry curvature in thewhole momentum space; (d) analytically shown that theBerry phases with respect to the two wave functions mustbe different because of their different quasi-periodicities.For both the wave functions, we find that a CF adia-batically transported around the Fermi circle acquires aBerry phase π in the CF representation. Since the Berryphase can be interpreted as the intrinsic anomalous Hallconductance [34] (in the unit of − e / πh for σ xy [35]),both the wave functions can correctly predict the Hallconductance of CFs for a particle-hole symmetric half-filled Landau level [11], in both its magnitude and sign.The result is actually consistent with the Dirac theory.However, microscopically, both the Berry curvature dis-tributions with respect to the two wave functions are not the singular one implied by the Dirac picture. We thusexpect that the effective theories with respect to the twowave functions are not the Dirac theory when physicsaway from the Fermi level is concerned.On the other hand, one may question the physical rel-evance of these subtle differences between different effec-tive theories. Indeed, up to now, most of predictions ofdifferent effective theories are focused on the effects ofthe π -Berry phase, and indistinguishable. It doesn’t helpthat the HLR theory, which has no π -Berry phase, can0also correctly predict the Hall conductance of CFL byconsidering the effect of scattering by the fluctuation ofthe effective magnetic field [25]. The situation is actuallytypical, as for other theories in a similar stage when dif-ferent pictures compete and seem to provide equally goodexplanations for a limited set of observations. For the CFtheory, we would like to argue that: (a) a wave functionmust have one and only one correct effective theory, un-less different effective theories can be shown equivalent;(b) microscopic details of different effective theories arerelevant because they may lead to different physical pre-dictions. One such example is shown in Ref. [36], whichindicates that different ways of modulating CF systemscan induce different asymmetries in geometric resonanceexperiments [37] as a result of the “subatomic” dipolestructure of the CF. 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