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a r X i v : h e p - t h / 9 8 0 7 2 1 7 v 1 2 9 J u l 1 9 9 8

Dirac Fermions and Domain Wall Defects in2 + 1 Dimensions

C.D. Fosco

andA. Lopez

Centro At omico Bariloche, 8400 Bariloche, Argentina

February 1, 2008

Abstract

We investigate some properties of a system of Dirac fermions in2 + 1 dimensions, with a space dependent mass having domain wall

like defects. These defects are dened by the loci of the points wherethe mass changes sign. In general, they will be curves lying on thespatial plane. We show how to treat the dynamics of the fermionsin such a way that the existence of localized fermionic zero modeson the defects is transparent. Moreover, effects due to the higher,no-zero modes, can be quantitatively studied. We also consider therelevance of the prole of the mass near the region where it changessign. Finally, we apply our general results to the calculation of theinduced fermionic current, in the linear response approximation, inthe presence of an external electric eld and defects.

fosco@cab.cnea.edu.arlopez@cabtep2.cnea.edu.ar

1

http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1http://arxiv.org/abs/hep-th/9807217v1

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1 Introduction

In this paper we study some properties of a system of Dirac fermions in 2+1dimensions with a space dependent mass that corresponds to domain walllike defects. Dirac fermions models in 2 + 1 dimensions appear as effectivetheories for various systems of interest in particular in condensed matterphysics. Consider for instance a tight binding model for spinless electrons ona two dimensional square lattice, in the presence of an external magnetic eldof half of a magnetic ux quanta per plaquette. The nearest neighbor hoppingamplitude is one, the second neighbor hopping amplitude is t/ 4, and there isa staggered potential ( 1)x 1 + x 2 , where x1 and x2 are the coordinates on the

plane. It has been shown [ 2] that, in the continuum limit, the effective theoryfor this system corresponds to two massive Dirac fermions whose masses areproportional to ( t). In the low energy sector, the heaviest particle canbe neglected and the theory to be considered is a Dirac fermion with massproportional to M = t. Now imagine that in the system there is adomain wall (or stacking fault) created in the process of crystal growth bysome specially prepared circumstances. At the level of the effective Diractheory this can be obtained by replacing M by M in half of the plane[3]. Thus the mass becomes position dependent with a prole M (x1, x2) =M ((x2) ( x2)) for a sharp wall at x2 = 0. Exactly the same effect canbe obtained when the system, even without any defect, has an odd numberof lattice rows in one direction. If the system is build up periodically thepresence of the extra row produces the same effect as a domain wall.

In this work we study the general case in which the mass of the Diracfermions changes sign along an arbitrary curve in the spatial plane, and withan arbitrary prole near the regions where the sign change is observed. Dueto the Callan and Harvey mechanism [ 4], localized chiral zero modes appearon the defects. This property has been extensively applied, following an ideaby Kaplan [5], to the problem of dening a lattice action for chiral fermions,thus providing a possible way of overcoming the kinematical obstructionposed by the Nielsen-Ninomiya theorem [ 6]. The proposal evolved into thesomewhat more abstract, but nevertheless equivalent, overlap formalism [ 7,8, 9]. In Kaplans formulation, one deals with step-like rectilinear defects forthe mass in an odd number of spacetime dimensions. Based on those ideasand techniques, we shall here study the contribution to the effective actionof the zero modes appearing along a curve of arbitrary shape in the spatial

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plane in 2 + 1 dimensions, quantifying the effect of the extended bulk statesand of the particular shape of the domain wall prole on the dynamics of the modes. Localized non chiral zero modes are also taken into account. Asan example of an application, we use the linear response approximation toderive an expression for the induced current in the presence of an externaluniform electric eld.

The paper is organized as follows: In section 2 we introduce an eigenmodeexpansion appropriate to the presence of domain wall defects, consideringrst the simplest case of a rectilinear domain wall, and then extending it tothe more general situation of a defect of arbitrary shape. We apply this tothe particular example of a circular defect. In section 3, the eigenmodes ex-

pansion is applied to the problem of evaluating the vacuum fermionic currentin the presence of an external eld, to understand the consequences of havingdefects to the spatial distribution of the fermionic current. The results forthe concrete example of the induced current due to a constant electric eldare presented and discussed. Section 4 is devoted to our conclusions.

2 Eigenmodes expansions in the presence of domain wall defects

A functional integral over fermionic (Grassmann) elds usually carries aquadratic action, in terms of some operator, which depends on the exter-nal elds. They are regarded as sources at this stage, either because theyhave no dynamics (as in our case), or because it would be desirable to post-pone their quantisation. The formal integration over spacetime dependentGrassmann elds may be converted into one over the (Grassmannian) coef-cients for the expansion of the elds in a complete set of eigenfunctions of some operator. A judicious choice of eigenfunctions may simplify the task of evaluating the functional integral, or provide a better starting point for anapproximate treatment.

In the present case, we deal with Dirac fermions in 2 + 1 Euclidean di-

mensions, in the presence of an external Abelian gauge eld A , and a spacedependent parity breaking mass M ( x), with x = ( x1, x2). The Euclideanaction S is dened by

S = d3x (x) [ + ie A + M ( x)] (x) (1)3

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where x = ( x, x3), x3 being the Euclidean time coordinate. Hermitian -matrices are in an irreducible (2 2) representation of the Euclidean Diracalgebra:

{ , } = 2 . (2)

By denition, the mass M ( x) has a domain wall defect whenever itchanges sign. As these defects proceed from the regions where the surfacespanned by the function M ( x) crosses the plane M ( x) = 0, we see that theyshall consist of (possibly disconnected) curves. Let

x( ) , [ 1, 2] (3)

be a parametrisation of one of the connected components of the curve. Then,of course we shall have

M ( x( )) = 0 , [ 1, 2] . (4)

This curve can in principle have an arbitrarily complicated structure, depend-ing of course on the assumptions made about the function M ( x). We shallassume that the mass is regular enough as to dene a curve which is a dif-ferentiable manifold. We will see that the phenomenon of localization in thegeneral case may be understood in a simple way, if one denes a convenientsystem of coordinates in the neighborhood of each point on the curve. For

each coordinate patch, the physics can be shown to be equivalent to the oneof a mass term depending on only one coordinate. Thus, in each coordinatepatch, the defect will look like rectilinear. Our strategy will be to considerrst the simpler situation of a rectilinear defect, and then to generalize it tothe case of a wall dening an arbitrary curve, through the use of the abovementioned system of coordinates.

2.1 Rectilinear defect

This is indeed the simplest situation, consisting of the mass being a functionof only one of the spatial coordinates, say x2 1, and changing sign along thestraight line x2 = 0:

S = d3x (x) D (x) (5)1 For a step-like defect, this is the 2+1 analog of the congurations relevant to Kaplans

formulation [ 9].

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where we denedD = D + M (x2) . (6)

and D = + ieA .It is easy to show that in order to be able to disentangle the dynamics of

the fermions into two pieces: one depending on x2 only, and the other onewith support in the plane x2 = 0, the most general gauge eld congurationwe can consider is

F 2 = 0 , = 1 , 3 , (7)

which corresponds to having no magnetic eld ( F 21 = 0), and no electriceld perpendicular to the defect ( F 23 = E 2 = 0). These are gauge-invariant

conditions, which in the gauge A2 = 0 (which we shall adopt) leads to 2A = 0 , = 1 , 3 . (8)

In this gauge, we immediately see that the operator D may be rewritten asfollows:

D = ( a+ d)P L + ( a+ d)P R (9)

where a, a are operators acting on functions of x2,

a = 2 + M (x2) , a = 2 + M (x2) , (10)

P L,R are projectors along the eigenspaces of the matrix 2:

P L =12

(1 + 2) , P R =12

(1 2) , (11)

and d is the two dimensional Euclidean Dirac operator corresponding to thetwo coordinates x1 and x3, which we denote collectively by x, namely

d = 1( 1 + ieA1(x)) + 3( 3 + ieA3(x)) . (12)

Expression ( 9) suggests the possibility of getting rid of the dependenceon x2 for the elds, by a suitable expansion in the modes of some operator,

and obtaining in a way a dimensional reduction from the three dimensionalspacetime to the two dimensional one corresponding to x. As D itself is notHermitian, we use instead the positive Hermitian operator H

H = DD , (13)

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which, by using the explicit expression ( 9) for D leads to

H = ( h d2)P L + ( h d2)P R , (14)

with h = aa and h = aa . The appearance of the two conjugate Hermitianoperators h, h suggests an expansion for the x2 dependence of the fermionicelds in terms of their eigenfunctions, namely

(x2, x) =n

n (x2)(n )L (x) + n (x2)

(n )R (x)

(x2, x) =n

(n )L (x)n (x2) +

(n )R (x)

n (x2) (15)

where the subscripts L, R denote the chirality dened by the matrix 2 and

hn (x2) = 2n n (x2) , hn (x2) = 2n n (x2)

n |m = n,m , n |m = n,m . (16)

To write eq.( 16) we have used that (n )L,R (x) = P L,R (n ) , and (n )L,R (x) =

(n ) P R,L . Note that the two dimensional fermionic elds (n ) and their Diracadjoints are independent variables for different values of the discrete indexn. We have made explicit the property that h and h are positive (the n areassumed to be real and we x their sign, by convention, to be positive) andhave the same spectrum, with the only possible exception of the zero modesn = 0, since, for any n with n = 0, there also exists one eigenvector of hwith identical eigenvalue

hn (x2) = 2n n (x2) h [1

nan (x2)] = 2n [

1n

an (x2)] (17)

where the factor 1 n is introduced to normalize the eigenvectors of h. (Of course the reciprocal property for the eigenstates of h also holds.)

We remark that, depending on the specic form of M (x2), one of thezero modes may not appear in the sum ( 15). In the Callan and Harveymechanism, the defect is step-like, what gives rise to only one zero mode,with exponential localisation [ 4]. This issue will be made more explicit whenconsidering concrete expressions for the mass function.

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the effective action. Rather than considering defects of the most generalpossible kinds, we plan to study domain walls of a localized type, namely,defects having a typical width . This width is dened by the extension of the region around a zero of the fermion mass, where it has an appreciablevariation. The mass is assumed to have an approximately constant absolutevalue outside the defect. This kind of behaviour is displayed in Figure 1.The analysis of the different contributions to the fermionic effective actionhas to take the nite width of the defect into account. To incorporate thelocalized modes, we expand the functional form for the mass around thedefect in a Maclaurin series keeping a few terms, and then solve for theeigenmodes of h and h corresponding to that expansion. This procedure

will only make sense if the modes so obtained are concentrated in a regioncontained in the band of width . So we shall only add in the sum overmodes (21) those having a spatial dispersion smaller than . This sum willgive an approximate estimation of the contribution of the localized modes tothe effective action. Non-localized modes correspond to Dirac fermions thatlive effectively in 2 + 1 dimensions, and shall have a smaller contribution tothe effective action than the localized modes, as a simple argument shows:The action for a non-localized mode is given by the corresponding term in thesum (18). The property that distinguishes it from the action for a localizedmode is that the former corresponds to an unbounded state of the operatorsh, h, while the latter is due to a bounded state. These operators can bewritten more explicitly as,

h = 2 + M 2(x2) 2M (x2)h = 2 + M 2(x2) + 2M (x2) (24)

which, far from the defect (for a defect like the one of Fig.1) may be replacedby

h h 2 + 2 , (25)since the derivative of the mass obviously tends to zero. We conclude thatnon-localized modes have an action, which has to be larger or equal than ,the absolute value of the mass outside the defect.

If the mass function is approximately linear near to the center of thedefect, we may keep only the rst term in a Maclaurin expansion, arrivingto the operators a and a

a = 2 + M (x2) = 2 + M (0) x2

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a = 2 + M (x2) = 2 + M (0) x2 (26)

in terms of which we may dene a and a, which are identical to the usual(harmonic oscillator) creation and destruction operators:

a =1

2M (0)a , a =

1

2M (0)a

[a, a] = 1 . (27)

We note that, as far as this linearization is reliable, we may of course writeM (0) in terms of and :

M (0) = 2

. (28)

The eigenvalues and eigenvectors of h and h are immediately found. Wesee that there is a zero mode for a but none for a,

a0(x2) = 0 0(x2) = N 0 e12 |M

(0) |x 22 (29)

with N 0 a normalisation constant. As the Hamiltonians h and h are likenumber operators, the eigenvalues n = 2|M (0) |n are evenly spaced.Then the effective action becomes

= (0)L +

n =1(n ) (30)

where(n ) = Trln d + 2|M (0) |n . (31)

In order to determine the number N of modes that we may include in thesum above, we recall that for the harmonic oscillator states corresponding tothe label n, the dispersion (n )x is

(n )x = n + 12M (0) . (32)The condition that modes are localized inside the defect becomes (N )x = ,so

N = [M (0) 2 12

] . (33)

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As N 0, we see thatM (0) 2 1

2(34)

or

12

, (35)

which must be interpreted as a sufficient condition to fulll in order to haveat least one localized state. Its physical interpretation is quite transparent:the thinner the defect, the bigger should the mass jump be in order to localizea mode inside the defect.

We can then exhibit the contribution to the effective action of thesemodes. The zero mode contribution is simply the effective action for achiral fermion in 1 + 1 dimensions, a result we recall, for example, fromreference [11],

L =e2

8 d2xA a ( + i ) 2 ( i ) A . (36)where a is a regularization parameter, which must be xed to the value 1 forthe regularization to be gauge invariant.

Regarding the massive modes contributions, we apply the result presentedin the Appendix for the massive determinant in the quadratic approximation,with a mass given by n , to obtain the n mode contribution:

(n )(A) =12 d

2k(2)2

A ( k)(n ) (k)A (k) (37)

where(n ) (k) =

(n )(k)( k k k2

) , (38)

and

(n )(k) =e2

1 2

2nk2

(1 +42nk2

)12 ln

(1 + 42n

k 2 )12 + 1

(1 + 42n

k 2 )12 1

(39)

Then the effective action reads

(A) = L (A) +

12 d

2k(2)2

A ( k)v(k)( kk k2

)A (k) (40)

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wherev (k) =

n(n )(k) (41)

If a low momentum expansion for (n ) is used, then we have for v thefollowing expression:

v(k) =n

(n ) (k) =e2

12 |M (0) |

M (0) 2 12

n =1

1n

k2 . (42)

Moreover, for cases where the condition >> 1 (the number of local-ized modes is >> 1) is satised, the sum over n can be approximated by a

logarithm, leading to

v e2

12 |M (0) |[ln() + ]k2 , (43)

where is Eulers constant.In order to understand the nature of the error involved in the assumption

of taking a linear prole for the mass near the defect, we recall the theexpression for the error in a Maclaurin expansion to rst order may be writtenas

E (x2) =12

M ()x22 (44)

where 0 x2, and we assumed that M has a continuous second deriva-tive. A rough estimate of the value of the radius of the interval in whichwe may use the linear approximation is by imposing the condition of havingan error much smaller than the value of the linear term, which leads to

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posteriori the utility of the truncation of the sum over modes according totheir dispersion: it ensures the additivity of the corresponding contributionsto the action.

2.2 Defects of arbitrary shape

We shall consider here the case of a wall having the shape of an arbitrarycurve lying on the plane. For a curve C dened by a parametric equation(3), we introduce the (normalized) tangent, normal and binormal vectors,denoted by e1, e2 and e3, respectively 2. To that end, it is better to usea natural parametrisation for the curve, dened in terms of the arc length

parameter ss( ) =

1d dx( )d dx(

)d

(46)

s x(s) , s [0, L] (47)

where L is the total length of the curve. The tangent e1 is then obtainedas a derivative of the parametric equations of the curve with respect to thelength parameter

e1 =dds

x(s) . (48)

It is natural to dene an orientation for the curve: we shall assume that sincreases when the region of positive mass is left to the right and the oneof negative mass is to the left of the curve. The normal can, as usual, bedened by

e2 = ||de1(s)

ds|| 1

de1(s)ds

(49)

with a choice of sign which we x by the convention that the normal shouldpoint from the side of positive to the one of negative mass. As the curve isplane, however, we can do more simply than that and dene the normal asfollows:

e j2 = jk ek1 (50)

which is explicitly orthogonal to e1. It is immediate to realize that e3 can betaken as a constant and pointing in the third direction, with a sense such that

2 Of course, being the curve plane, it is torsionless ( e 3 is constant), or, equivalently, theosculating plane coincides with the plane x 3 = 0 for all the points on the curve.

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det ei j = +1. We now proceed to dene coordinates in a neighborhood of the defect. Through each point on C, we draw a curve tangent to the normalvector, and another one tangent to the binormal. Of course the conditionof being tangent only denes the curves in a small neighborhood, but thisis enough for our purposes. We then perform a Lie dragging of the curve Calong those elds. This denes a complete set of coordinate lines around thecurve of the defect.

Denoting by u1, u2 and u3 the parameters labeling points along each oneof the integral lines, we may then write the action corresponding to the regionaround the defect as

S C =

d3u det[ei j ]

ei D i + M (u1) (51)

where the spin connection term vanishes because the coordinate system hasbeen dened by dragging. We then note that det[ ei j ] = +1, that ei i is thedirectional derivative along the integral curve , and that ei Ai is the gaugeeld component along the same direction. In order to arrive to a situationentirely similar to the one of the case of the rectilinear defect, we need tochose a convenient gauge. The analog of the condition A2 = 0 for that casewould be here to use a gauge such that the component of A along the normalto the curve vanishes. We note that a condition like that does not forbid thepresence of a non vanishing magnetic ux through the surface enclosed bythe curve, since that requires only the existence of a non vanishing tangentcomponent for the gauge eld. We then arrive to the expression for the action

S C = d3u 1Du 1 + 2 u 2 + 3Du 3 + M (u2) . (52)We conclude this section with an example of an application: the case of

a circular defect. We assume the defect is a circle of radius R, and that themass is positive inside the circle. The convenient choice of coordinates forthis case corresponds to using polar coordinates.

With the natural parametrisation, in terms of the arc length s, we have:

x(s) = R( cos(sR

), sin(sR

) ) , 0 s < 2R . (53)

The tangent and normal vectors are, of course,e1 = ( sin(

sR

), cos(sR

))

e2 = (cos(sR

), sin(sR

)) . (54)

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Local coordinates are dened as follows: u1 = r , where r is the radial (plane)distance to the origin and the polar angle (of course, u1 reduces to s whenr = R). u2 is equal to r R and u3 = x3. In terms of r , and x3, the action(52) for a defect of width becomes

S C = R +

R drr

2

0d dx3 1r 1( + ieA) + 2 r + 3Dx 3 + M (r R) .

(55)We may now consider, for example, a gauge eld which corresponds to a

static uniform magnetic eld B( x) through the surface. Near to the defect,we may take

A =

2R (56)where is the total ux = BR 2. Note that, as we are only consideringthe elds near to the defect, the dependence of A on the radius may beignored, evaluating it at R. This will hold true as long as the radius of thedefect is sufficiently large. The action for the chiral zero mode localized onthe defect will then be

S L = R 2

0d dx3 (0)L ( 1R 1( + ie 2R ) + 3 3)(0)L (57)

which corresponds to chiral fermions in the presence of a constant gauge eld.

This problem is equivalent to the one of free fermions with twisted boundaryconditions, and the dynamics of the zero mode is then periodic in the ux.Non zero modes might also be incorporated, but only if their dispersion issmall enough as to ignore de dependence of the gauge eld on the radialdirection.

3 Fermionic current density

In this section we present an evaluation of the expectation value of thefermionic current in the presence of defects, and under the inuence of an

external electric eld. From what we said in the previous section, it shouldbecome evident that what we need is just to study the case of one singledefect, and then the more general situations are easily understood from theknowledge of the results that follow for this case. We shall consider the vac-uum expectation value j (x1, x2, x3) ( = 1 , 3) of the fermionic current in

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the presence of an external gauge eld such that F 2 = 0. That expectationvalue may of course be written as

j (x) = etr [ S (x, x )] (58)

where S (x, y) is the fermionic propagator, the inverse of the operator deningthe quadratic form in the fermionic action:

S (x, y) = x, |D 1 |y, (59)

whereD = ( a+ d)P L + ( a+ d)P R . (60)

The simple algebraic identity

D 1 = ( DD) 1D (61)

leads to

D 1 = ( h d2) 1P L (a d) + ( h d2) 1P R (a d) . (62)

The x2 dependence of the propagator is easily extracted by using the com-pleteness of the eigenfunctions of h and h, obtaining

S (x, y) =

n =0 ,m =1n (x2)m (y2) x1, x3, |(2n d2) 1n P L |y1, y3, nm

n =0 ,m =0n (x2)m (y2) x1, x3, |(

2n d

2) 1 dP R |y1, y3, nm

+

n =1 ,m =0n (x2)m (y2) x1, x3, |(

2n d

2) 1n P R |y1, y3, nm

n =1 ,m =0n (x2)m (y2) x1, x3, |(

2n d

2) 1 dP L |y1, y3, nm . (63)

Whence, the expectation value of the current becomes

j (x) = e20(x2)tr[ x1, x3 | d 1P R |x1, x3 ]

+ e

n =12n (x2)tr[ x1, x3|(

2n d

2) 1 dP R |x1, x3 ]

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+ 2n (x2)tr[ x1, x3 |(2n d

2) 1 dP L |x1, x3 ]

n (x2)n (x2)tr[ x1, x3 |(2n d2) 1n P L |x1, x3 ]

n (x2)n (x2)tr[ x1, x3 |(2n d2) 1n P R |x1, x3 ] (64)

The rst term on the rhs is the contribution to the current coming from thechiral mode. Denoting by j L (x) and j (n ) the vacuum expectation values of the two dimensional chiral and vector like currents, respectively, we see thatthe above result for the current may be rewritten as

j (x1, x2, x3) = 20(x2) jL (x1, x3)

12

n =1(2n (x2) + 2n (x2)) + i(2n (x2) 2n (x2)) j (

n ) . (65)

We will now apply the result ( 65) to the determination of the current dueto a constant electric eld E along the defects. It is evident from ( 65) that thex2 dependence of the current is completely determined by the functions nand n . For example, the contribution carried by the chiral zero mode shall beconcentrated around the defects with a density 20, which is a Gaussian for alinear defect. A step-like defect will in turn have an exponential localization.The x2-dependent factors for the n = 0 modes yield contributions which aremore extended in x2, as the dispersion in x2 grows with n.

On the other hand, as the external electric eld is assumed to be constantin time and uniform in space, there only remains to know the x1 dependence,which of course may appear in the expectation values of the two dimensionalcurrents j L and j (n ) . It can be easily checked that, within the linear responseapproximation, the expectation value of every massive current j (n ) is zero,due to the existence of a gap 3 (the mass n ). The contribution of thechiral zero mode is obtained from the parity- conserving part of its effectiveaction, and yields exactly one half of the contribution corresponding to amassless Dirac fermion:

j L1 (x1) =e2

2EL , (66)

where L is the length of the system in the direction x1.3 The conductivity of a massive fermion in 1 + 1 dimensions is proportional to the value

of (0), which vanishes for mass different from zero. It is however non-zero for a masslessfermion.

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We conclude that the three dimensional current is:

j1(x1, x3) = 20(x2)e2

2EL . (67)

where, as we have already mentioned, the form of 20(x2) depends on theparticular shape of the defect.

4 Conclusions

Motivated by the fact that Dirac fermions with step like masses appear as

effective descriptions of various systems in particular in Condensed MatterPhysics, in this paper we have provided a formalism to study the general casein which the mass of the Dirac fermions changes sign along an arbitrary curvein the spatial plane, and with an arbitrary prole near the regions where thesign change is observed.

We showed that the dynamics of the fermions can be studied in sucha way that the contributions from the localized zero modes on the defects,and the ones of the higher no-zero modes can be clearly distinguished. Weprovided a criterion to evaluate which are the most relevant contributionsdepending on the prole of the mass near the region where it changes sign.

We applied this formalism to the simplest case of a rectilinear domain walland evaluate the vacuum fermionic current in the presence of an external eld,to understand the consequences of having defects to the spatial distribution of the fermionic current. We studied also a circular defect with a magnetic eldpiercing the enclosed surface and saw that the effective theory for the zeromode is equivalent to the one of fermions with twisted boundary conditionsin a circle.

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Appendix: Two-dimensional massive fermionicdeterminant in the quadratic approximation

We present here, for the sake of completeness, the calculation of the mas-sive fermionic determinant in two dimensions, in the presence of an externalAbelian vector eld [12] A , in the quadratic approximation (namely, therst non-trivial term in the usual perturbation expansion).

The generating functionals Z (A) and ( A), corresponding to massiveDirac fermions in the presence of an external Abelian vector eld A aredened by

Z (A) = exp[ (A)] = DD exp{ d2x( + ie A + m)}= det( + ie A + m) . (68)Thus the effective action is then given by

(A) = lndet( + ie A + m) = Tr ln( + ie A + m) , (69)

where the trace operates over both functional and Dirac space. In thequadratic approximation, becomes

(A) = 1

2

e2Tr ( + m) 1 A ( + m) 1 A , (70)

which, in momentum space representation, has the equivalent expression

(A) =12 d

2k(2)2

A ( k) (k)A (k) . (71)

Here the tilde denotes the Fourier transform of a function of x, namely,

A (k) = d2x e ik x A (x) (k) = d2x e ik x (x) . (72)

Fourier transforming in ( 70), we can give a more explicit expression for asfollows

(k) = e2 d2 p

(2)2tr

1i( k+ p) + m

1

i p + m

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= e2

d2k

(2)2tr[( k+ p) p ]

[(k + p)2 + m2]( p2 + m2)

m2 d2k

(2)2tr( )

[(k + p)2 + m2]( p2 + m2). (73)

Gauge invariance implies the transversality of (k), which allows we to write in terms of only one independent scalar function

(k) = (k2)( kk k2

) . (74)

We follow a procedure analogous to the one followed in [10] for the mass-less case, to evaluate using dimensional regularization. Thus, in d dimen-sions, after taking the trace on both sides of the previous tensorial equation,

(d 1)(k2) = 2 e2(2 d) dd p

(2)d(k + p) p

[(k + p)2 + m2]( p2 + m2)

2e2m2d dd p

(2)d1

[(k + p)2 + m2]( p2 + m2). (75)

Letting d 2, the rst term on the right hand side of ( 75) contributesonly through its divergent part, which is identical to the one for the case

m = 0limd2

(2 d) dd p

(2)d p2

( p2 + m2)2=

12

. (76)

The second is regular at d = 2. Adding the contribution of both terms, wesee that

(k2) =e2

4e2m2 d

2 p(2)2

1[(k + p)2 + m2]( p2 + m2)

. (77)

The integral over p is evaluated by using the standard Feynman parametertrick, to obtain

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(k2) =e2

1 2

m2

k2(1 +

4m2

k2)

12 ln

(1 + 4m 2k 2 )12 + 1

(1 + 4m2

k 2 )12 1

(78)

In a low momentum expansion, the leading contributions are

(k2) =e2

k2

6m2

(k2)2

90m4. (79)

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References

[1] E. Fradkin, Field Theories of Condensed Matter Systems , Addison Wes-ley, Reading, 1991.

[2] See, for instance, A.W.W. Ludwig, M.P.A. Fisher, R. Shankar, and G.Grinstein, Phys. Rev. B50 , 7526 (1994).

[3] Boyanovsky D., Dagotto E., Fradkin E., Nuc. Phys B285 [FS19], 340(1987) .

[4] Callan C. G. and Harvey J. A., Nuc. Phys B250 , 427 (1985).

[5] D. B. Kaplan, Phys. Lett. B288 342 (1992).For a review of Kaplans formulation, see: K. Jansen,Phys. Rept. 273 : 1-54 (1996).

[6] L. H. Karsten, Phys. Lett. 104B , 315 (1981);L. H. Karsten and J. Smit, Nucl. Phys B183 , 103 (1981);H. B. Nielsen and M. Ninomiya, Nucl. Phys B185 , 20 (1981); Nucl.Phys B193 , 173 (1981); Phys. Lett. 105B , 219 (1981). For a review,see: Y. Shamir, Nucl. Phys. Proc. Suppl. 47 : 212-227 (1996).

[7] R. Narayanan and H. Neuberger, Nucl. Phys. B443 :305-385 (1995). Seealso: R. Narayanan, H. Neuberger and P. Vranas, Nucl. Phys. Proc.Suppl.47 : 596-598 (1996); R. Narayanan and H. Neuberger, Nucl. Phys.Proc. Suppl. 47 : 591-595 (1996); H. Neuberger,hep-lat/9511001 .

[8] S. Randjbar-Daemi and J. Strathdee, Nucl. Phys. B443 :386-416 (1995);S. Randjbar-Daemi and J. Strathdee, Nucl. Phys. B466 : 335-360 (1996).

[9] D.B. Kaplan and M. Schmaltz, Phys. Lett. B368 , 44 (1996).

[10] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena , second

edition, Clarendon Press, Oxford, 1993, section 30.2.[11] R. Jackiw and R. Rajaraman, Phys. Rev. Lett. 54 , 1219, 2060 (E),

(1985).

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[12] For the properties of this system, including a Maxwell dynamics for thegauge eld, see, for example:S. Coleman, Annals of Phys. 101 , 239 (1976).

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2

FIGURE 1

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Figure captions

Figure 1 The typical mass prole considered, showing the denitions of the defect height 2 and width .

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