We calculate the ground state current densities for 2+1 dimensional free fermion theories with local, translationally invariant boundary states. Deformations of the bulk wave functions close to the edge and boundary states both may cause edge current divergencies, which have to cancel in realistic systems. This yields restrictions on the parameters of quantum field theories which can arise as low energy limits of solid state systems. Some degree of Lorentz invariance for boosts parallel to the boundary can be recovered, when the cutoff is removed

Leitner, Marianne

Nahm, Werner

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arXiv:0808.1497v2  [math-ph]  16 Feb 2010Boundary states and edge currentsfor free fermionsMarianne Leitner, Werner NahmDublin Institute for Advanced StudiesSchool of Theoretical Physics10 Burlington Road, Dublin 4, Irelandleitner@stp.dias.ie, wnahm@stp.dias.ieAbstractWe calculate the ground state current densities for 2+1 dimensionalfree fermion theories with local, translationally invariant boundary states.Deformations of the bulk wave functions close to the edge and boundarystates both may cause edge current divergencies, which have to cancel inrealistic systems. This yields restrictions on the parameters of quantumfield theories which can arise as low energy limits of solid state systems.Some degree of Lorentz invariance for boosts parallel to the boundarycan be recovered, when the cutoff is removed.1 IntroductionThe prediction of a relativistically invariant quantum field theory in 2+1dimensions as low energy limit of a solid state system [1] was confirmedby investigation of graphene [3]. Modification of the graphene structuree-print archive:http://lanl.arXiv.org/abs/math-ph/0808.14972BOUNDARY STATES AND EDGE CURRENTS FOR FREE FERMIONSmay allow to obtain more general quantum field theories [2]. Since solidstate systems typically have edges, the study of quantum field theories withboundaries acquired additional interest. An important tool in this con-text is the bulk-boundary OPE. In particular, some of its singularities yielddivergent observables and constitute obstructions to any experimental re-alization. The coefficient function of the identity in such an OPE is justthe corresponding vacuum expectation value in the presence of the bound-ary. Integrated expectation values of current operators can be interpretedas measurable charge or spin currents and should remain finite in realis-tic systems. Here we study this obstruction in the simplest case of freefermions, where vacuum expectation values can be interpreted as integratedcontributions of the particles in the Dirac sea.It is of course well known that such effects can prevent the lifting of aquantum mechanical system to a quantum field theory, even without thepresence of a boundary. In our case the Dirac sea of a single fermion yieldsa half-integral value of the Hall conductivity [4], [5], [6]. Consequently oneneeds an even number of fermions, though some of them may be shifted toinfinite mass, which results in a Chern-Simons term instead. For bound-ary states we shall find more intricate obstructions, which depend on thenumerical parameters of the boundary state.2 Boundary conditions from self-adjoint extensionsWe consider spinor fields ψ acted upon by the Dirac operator(1) H := −iσ1 ∂∂x− iσ2 ∂∂y+mσ3on the open half plane {x = (x, y) ∈ R2| x > 0}, as in [7]. This operatoris a closable symmetric operator on the domain D(H) = C∞0 (R+ × R,C2)of smooth functions with compact support vanishing in a neighbourhoodof x = 0. The possible boundary conditions are classified by its self-adjointextensions. The latter can be obtained by the standard von Neumann theory.Here we give a modification, which is less general but better adapted to thepresent problem. Since D(H) is dense in the Hilbert space H := L2(R+ ×R) ⊗ C2, the image of an arbitrary vector ψ ∈ H in the domain of a self-adjoint extension is uniquely determined by the unextended operator, so theproblem is reduced to the determination of such domains. Let H∗ be theadjoint of the unextended H. By von Neumann theory, every self-adjointextension of H corresponds 1-1 to an isometry V : N+ → N− between thedefect spacesN± = {ψ ∈ D(H∗)|H∗ψ = ±iµψ},M. LEITNER, W. NAHM 3where µ is a fixed but arbitrary positive number. Indeed, V gives rise to theself-adjoint extension HV with domainD(HV ) = D(H¯) + (I − V )N+.(2)The choice µ = 1 is standard, but we shall see that the limit of large µallows a more direct physical interpretation. Applied to (1), N± is given byk integrals over elements of the one-dimensional spacesN±(k) = C e−λx+iky(1s±k)(1 +O(µ−1)),wheres±k =i(k + λ)m± iµ , and λ =√µ2 + k2 +m2.For µ → ∞ the defect spaces consist of wave functions which become sup-ported on an arbitrarily small neighborhood of the boundary and factorizein a universal term exp(−µx) and a spinor of the formg(y)(1±1)with g ∈ L2(R).Thus isometries N+ → N− become tensor products of a unitary map V :L2(R) → L2(R) with the map between one dimensional subspaces of C2which takes (1,−1) to (1, 1). From (2) we read off that any ψ ∈ D(HV )satisfiesψ1|x=0 = (1− V )g,ψ2|x=0 = (1 + V )g.If 1 is not an eigenvalue of V , thenψ2|x=0 = iΓ ψ1|x=0,(3)where Γ is the self-adjoint operatorΓ = −i(1 + V )(1− V )−1.The physical necessity of such a boundary condition can be understood asfollows. Consider the conserved current densitiesjµ(x) = limx′→xψ(x′)† σµ ψ(x), .One needs a boundary condition which implies∫j1(0, y) dy = 0, and thisindeed follows from (3).We only study local boundary conditions, for which j1(0, y) vanishes iden-tically and also impose translational invariance. Thus Γ has to commutewith d/dy and with multiplication by functions of y. This implies that it isa real constant γ, and we obtain the boundary conditionψ2(0, y) = iγ ψ1(0, y).4BOUNDARY STATES AND EDGE CURRENTS FOR FREE FERMIONSwhich we shall use in the sequel. To include the case V = 1, we consider γas an element of the projective real line, including the value γ =∞.For any γ the Hamiltonian commutes with the PT transformation ψ(x, y) 7→σ3ψ∗(x,−y). The space reflection (x, y) 7→ (x,−y) lifts by ψ(x, y) 7→σ1ψ(x,−y) to a map between systems with parameters (m,γ) and (−m,−γ−1).Because of this duality we only need to consider positive m. Arbitrary val-ues of m will only be considered at the very end of the calculation. It turnsout that the results allow a continuous extension to m = 0, which is the caserelevant for graphene.The CPT map ψ(x, y) 7→ σ2ψ(x,−y) relates the positive energy statesfor (m,γ) to the negative energy states for (m,γ−1). A different type ofduality is given by ψ(x, y) 7→ σ2ψ(−x, y), (m,γ) 7→ (−m,γ−1), which relatesboundary states of complementary half planes.3 EigenfunctionsThe Dirac operator commutes with −i∂/∂y and consequently with(i∂/∂x)2 = H2 + (∂/∂y)2 −m2.We denote the real eigenvalues of these hermitean operators by k and l2.We take l ∈ R+ for positive l2 and l = iλ with λ ∈ R+ for negative l2. Thecorresponding eigenvalues E of H satisfy E2 = k2 + l2 +m2.According to the sign of l2 the spectrum of H has two parts, which willbe refered to as bulk and edge. Bulk eigenfunctions ulk with given k, l2 havethe formulk(x, y) = exp(ilx+ iky)χ+ + exp(−ilx+ iky)χ−with constant spinors χ±. The equation Hulk = Eulk and the boundarycondition reduce to (m−E, ±l−ik)χ± = 0 and (χ++χ−)2 = iγ(χ++χ−)1.For each l > 0 there is a unique solution up to normalization, such that forthe bulk part we just recover the spectrum of the Dirac operator on thewhole plane, with a gap ∆ = (−m,m). We normalize these eigenfunctionswith respect to the measure dk dl/(2pi2) and choose phases such thatulk(x, y) =(eiφ(l,k)(iρlk1)eilx+iky −(iρ∗lk1)e−ilx+iky)√E −m4E,M. LEITNER, W. NAHM 5where(4) eiφ(l,k) =1 + γρ∗lk1 + γρlkwith(5) ρlk :=k + ilm− E .Edge eigenfunctions [7] have the form Uk = exp(−λx + iky)χ. We nor-malize them with respect to the measure dk/pi and choose phases such thatUk(x, y) =√λ1 + γ2(i−γ)e−λx+iky.The equation HUk = EUk yieldsE =2γ1 + γ2k +1− γ21 + γ2mandλ =γ2 − 1γ2 + 1k +2γγ2 + 1m.Thus E,λ are one-valued functions of k, and for any k one has a unique edgestate whenever λ > 0 and no edge state otherwise. The latter inequalityrestricts the possible values of k,E. The limiting values for λ = 0 lie on themass shell. When we putη = sgn1− γ1 + γ,then positive energy half of the mass shell is touch for mη = 1 and thenegative energy half for mη = −1. One finds energies within the spectralgap ∆ of the bulk iff mγ > 0.The linear dispersion relation for the boundary equation implies that edgestates travel at the fixed velocityvedge =2γ1 + γ2.Note that vedge determines γ up to the replacement of γ by γ−1, which isthe one induced by CPT. The latter transformation changes the signatureη. Altogether, the boundary condition can be characterized by vedge and η,except for the fact that η is undefined in the limiting cases γ = ±1. Oneobserves that6BOUNDARY STATES AND EDGE CURRENTS FOR FREE FERMIONS1) for vanishing edge velocity, i.e. for γ = 0 and γ =∞, edge states haveE = m and E = −m, respectively, independently of k. This case isprobably unphysical, since all edge states have the same energy andthe thermodynamic partition function diverges for a system of infinitewidth. Since the width of the edge excitation increases linearly withk, the thermodynamic properties of a system of a finite system woulddepend on the sample size.2) The edge velocity is equal to the maximal bulk velocity for the CPTinvariant boundary conditions γ = ±1. These yield E = γk andλ = γm independently of k. For γm < 0 there are no edge states, forγm > 0 all edge states have the same dependence on x. This yieldsparticular divergences, so we will tacitly exclude γ = ±1 in most ofthe subsequent calculations.In (k,E)-space the half-line of edge states is tangent to the half-hyperboloidgiven by l ≥ 0 and E2 = k2+m2+ l2 at l = λ = 0. For γ2 < 1 this happensat positive energy and for γ2 > 1 at negative energy. The tangency hasconsequences for the behaviour at large distance from the boundary, as weshall see later.The portion of edge energies within ∆ gives rise to an edge current parallelto the boundary, with edge-conductivity equal to (in units of e2h ) [7](6) σedge ={sgn(m), if mγ > 0,0, otherwise.This applies whenever some subinterval of the gap is actually occupied,independently of the size of this subinterval.4 The current densitiesWhen no boundary exists, Fermi energies within the gap ∆ obviously yieldthe same theory. This means that any pair (γ,EF ) with EF ∈ ∆ yieldsa boundary state of the standard Lorentz invariant free fermion theory.We want to calculate the corresponding expectation values of the currentdensities. The bulk contribution does not depend on the choice of EF andedge contributions for energies within ∆ are finite and were calculated in[7]. Thus it suffices to consider EF = −m.The required vacuum expectation values are〈jµ〉 = jµbulk + jµedge,M. LEITNER, W. NAHM 7wherejµbulk =∫jµlk|bulkdl dk2pi2,jµedge =∫Θ(−m− E) jµk |edgedkpi,andjµlk(x)|bulk = u†lk(x)σµ ulk(x),jµk (x)|edge = U †k(x)σµ Uk(x).Here the variable E in ulk is the negative roots of k2 + l2 +m2.Note that j1lk|bulk and j1k |edge vanish identically. On the other hand eqs.(4, 5) yield(7) j2lk(x)|bulk =kE− 1E<(fge−2ilx).whereg = m− E + γ(k − il),f = (k − il)g∗.The first term on the r.h.s. of eq. (7) is an odd function of k, so thatits contribution to the vacuum expectation value of j2 vanishes for anysymmetric regularization. For the second term we also will find a partialcancelation.We substitute k ∈ (−∞,∞) by v = earcsinh(k/a) ∈ (0,∞), which yieldsfor negative E(8) −dkE=dvv.Moreover−fgkE=(a24(v − v−1)2 + l2)(D−11 +D−12 ) dv.HereD1 =a2(γ + 1)(v − v3)(v − v4),D2 =a2(γ−1 + 1)(v − v3)(v + v4),wherea =√l2 +m28BOUNDARY STATES AND EDGE CURRENTS FOR FREE FERMIONSandv3 =il +m√l2 +m2γ − 1γ + 1,v4 =−il +m√l2 +m2.The two denominators D1 and D2 are formally related by a reflection acrossthe boundary, with (m,γ) 7→ (−m,γ−1).One obtains a partial fraction expansion−fgdkE= {P1(v) + P2(v) + P3(v) + P4(v)} dvwithP1(v) =a2,P2(v) = −a2v−2.To regularize the momentum integrals we have to introduce a cutoff. In asolid state context the underlying physics selects a stationary Lorentz frame,in which a symmetric cutoff in k is natural. For this cutoff the dv integralsof these two terms cancel against each other due to the symmetry k 7→ −k,v 7→ 1/v. The remaining terms areP3(v) =ilγ + 1γ − 1v−1,P4(v) = −il 4γγ2 − 1(v − v3)−1.For a symmetric cutoff Λ of the v-integration one finds∫<(P3(v)e−2ilx)dldv2pi2= − 12piγ + 1γ − 1δ′(x) ln Λ.To calculate the P4 integral we need∫ (v − il +m√l2 +m2γ − 1γ + 1)−1dv = ln(Λ)− ln(1 + γ1− γ√m+ ilm− il).where the principal value of the logarithm has to be taken, so thatln(1 + γ1− γ√m+ ilm− il)= ln(√m+ ilm− il)+ ln∣∣∣∣1 + γ1− γ∣∣∣∣− piiΘ(γ2 − 1).M. LEITNER, W. NAHM 9To calculate the dl integral over the logarithmic part we use2∫ ∞0il ln(√m+ ilm− il)cos(2lx) dl =∫ ∞−∞il ln(√m+ ilm− il)e2ilx dl.The argument of ln has branching points at l = ±im. We may move theintegration path towards the cut with branching point l = im. Since thediscontinuity of the logarithm is ipi, this yields an elementary integral. Al-together we obtainj2(x)bulk(9)=− 12piγ2 + 1γ2 − 1δ′(x) ln Λ +γpi(γ2 − 1) ln∣∣∣∣1 + γ1− γ∣∣∣∣ δ′(x)+γ2pi(γ2 − 1)(12x2+mx)e−2mx − γpi(γ2 − 1)12x2Θ(γ2 − 1).For the contribution from the edge functions, we have according to (3)dkγ2 − 1γ2 + 1= dλ.We have to integrate over those λ for which λ > 0 and E < −m. This yieldsj2(x)edge =γpi(γ2 − 1)[12x2Θ(γ2 − 1)−(12x2+mγx)e−2mx/γΘ(γ)].Note that j2edge vanishes for γ ∈ (−1, 0) and is non-singular for γ > 1.For γ2 > 1 both the bulk and the edge current densities decay alge-braically at large distance from the boundary. In their sum these termscancel, an interesting effect which reflects the tangency of the edge stateand bulk state manifolds in (k,E) space, which was discussed in the previ-ous section.In all cases the singular part of the vacuum expectation value has theform 〈j2(x)〉singular= − 12piγ2 + 1γ2 − 1δ′(x) lnΛ− |γ|4pi(γ2 − 1)1x2.Remarkably, this equation is invariant under y reflection together with γ 7→−γ−1, such that it remains true for m < 0 and is in fact independent of m.The regular part of the expectation value is continuous inm and vanishes form = 0. For infinite m the contribution becomes localized at the boundaryand is given by the δ′ terms in (9). It may be interesting to rederive thisresult from a Chern-Simons Lagrangian.10BOUNDARY STATES AND EDGE CURRENTS FOR FREE FERMIONS5 ConclusionA single free fermion yields a divergent edge current on a half plane. ForN fermions with boundary conditions γn the cut-off dependent edge currentvanishes forN∑n=1γ2n + 1γ2n − 1= 0.One still has a dipolar edge currentN∑n=1γnpi(γ2n − 1)ln∣∣∣∣1 + γn1− γn∣∣∣∣ δ′(x),which in general does not vanish. In addition there is a 1/x2 singularity,which yields an unphysical divergent particle transport in the neighborhoodof the edge, unlessN∑n=1|γn|γ2n − 1= 0.Since the bulk system and the boundary at x = 0 are Lorentz invariantunder boosts in the y direction, one should be able to act with Lorentztransformations on the boundary conditions. In the context of relativistcquantum mechanics this is straightforward. The action on the parametersγn can be determined from eq. (3) or more easily from eqs. (8, 9), accordingto which boundary states are characterized by a Lorentz invariant signatureηn and their fixed travel velocityvn =2γn1 + γ2n.This translates into a rapidity θn given by tanh θn = vn, or equivalently byηn exp θn =1 + γn1− γn ,to which the rapidity of the Lorentz boost is added. When one moves fromone-particle states to field theory, control of divergencies requires a cutoffwhich breaks Lorentz invariance. It may or may not be restored, when thiscutoff is taken to infinity. One would expect that a restoration of Loretzinvariance implies CPT invariance. Due to the unbroken PT invariance thisshould yield invariance under charge conjugation and consequently an ab-sence of net boundary currents. In our case we will find an unexpectedlyM. LEITNER, W. NAHM 11subtlety, however. In terms of edge velocities and signatures the two diver-gency cancellation conditions can be rewritten asN∑n=1ηn exp(nθn) = 0 =N∑n=1ηn exp(−nθn),wheren = sgnvn.For N = 1 there is no solution, in agreement with the restriction to an evenN coming from the bulk physics. For N = 2 one needs γ2 = −γ−11 , thusinvariance under charge conjugation. More generally, pairs of edge statesrelated by charge conjegation do not contribute to the breaking of Lorentzinvariance or to net boundary currents, as expected. There is an unexpectedremainder of Lorentz invariance in more general situation, however. Whenall vn of the remaining states have the same sign, the cancellation conditionsare invariant under sufficiently small boosts which do not change the signof edge velocities. When vn of different signs occurs, no aspect of Lorentzinvariance is recovered when the cutoff is removed. In any case, breaking ofLorentz invariance by edge states does not seem to be an argument againstthe realization of such systems in solid state physics, where this invarianceis only a low energy phenomenon.It will be interesting to see to what extent the γ values can be controledexperimentally. Systems without time reversal invariance would be partic-ularly interesting, but are certainly very difficult to realize.References[1] G. W. Semenoff, Condensed-Matter Simulation of a Three-DimensionalAnomaly, Phys. Rev. Lett. 53 (1984) 2449.[2] F. D. M. Haldane, Model for a Quantum Hall Effect without LandauLevels: Condensed-Matter Realization of the Parity Anomaly, Phys.Rev. Lett. 61 (1988) 2015.[3] K. S. Novoselov et al., Two-dimensional gas of massless Dirac fermionsin graphene, Nature 438 (2005) 197.[4] R. Jackiw, Fractional Charge and Zero Modes for Planar Systems in aMagnetic Field, Phys. Rev. D 29 (1984) 2375.[5] A. N. Redlich, Parity Violation and Gauge Invariance of the EffectiveGauge Field Action in Three Dimension, Phys. Rev. D 29 (1984) 2366.[6] M. Leitner, Zero Field Quantum Hall Effect in (2+1)-DimensionalQED, Adv. Theor. Math. Phys. (2008).12BOUNDARY STATES AND EDGE CURRENTS FOR FREE FERMIONS[7] M. Gruber and M. Leitner, Spontaneous Edge Currents for the DiracEquation in Two Space Dimensions, Lett. Math. Phys. 75.1 (2006) 25.Received February 16, 2010

Boundary states and edge currents for free fermions

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Boundary states and edge currents for free fermions

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