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Department of Physics and Astronomy

arXiv:cond-mat/9310024v1 13 Oct 1993Lattice Magnetic WalksThomas Blum†Department of Physics and AstronomyUniversity of PittsburghPittsburgh, PA 15260USAYonathan ShapirDepartment of Physics and AstronomyUniversity of RochesterRochester, NY 14627-0011USAAbstractSums of walks for charged particles (e.g. Hofstadter electrons) on a square lattice inthe presence of a magnetic field are evaluated.

Returning loops are systematically addedto directed paths to obtain the unrestricted propagators.Expressions are obtained forspecial values of the magnetic flux-per-plaquette commensurate with the flux quantum.For commensurate and incommensurate values of the flux, the addition of small returningloops does not affect the general features found earlier for directed paths. Lattice Green’sfunctions are also obtained for staggered flux configurations encountered in models ofhigh-Tc superconductors.1

I. IntroductionThe problem of electrons confined to a lattice in the presence of a transverse magneticfield is a classic problem of mathematical physics with many applications in condensed-matter physics.The early works of Hofstadter [1], Wannier [2], and Azbel [3] focusedon the spectral properties of non-interacting electrons as a function of the energy and theparameter α = φ/φo where φ is the magnetic flux-per-plaquette and φo = hc/e is the fluxquantum.For commensurate α = p/q the spectrum has q subbands with special scalingproperties as q grows larger, eventually turning into a Cantor set for irrational α. Hence,the behavior of the spectrum and the wave function is very sensitive to the exact value ofα and the energy E; (the intricate structure of the bands as a function of α and E hasbeen dubbed the “butterfly”).Lately the renewed interest in interacting electrons in two dimensions, in the contextof the Quantum Hall Effect and various theories for high-Tc superconducting materials forinstance, has led to more recent investigations of this general problem.Over time, theseworks have yielded an extraordinary richness using a variety of techniques including pathintegrals [4], renormalization group [5], sophisticated algebraic methods [6] closely relatedto “quantum groups,” and so on.More recently the study of non-interacting electrons in a magnetic field has beenapproached from a new direction — that of “localized” wave functions. [7-10] If the energyof the electron does not lie within a quasiband, its eigenfunction is not a bulk (extended)eigenstate.

However, such a wave function may be localized at inhomogeneities, such as theedge of the lattice (i.e., surface gap states) or at isolated impurities in an otherwise orderedlattice. These states decay exponentially as one moves away from the inhomogeneity intothe bulk.The effect of the magnetic field on this exponential decay (and particularlythe associated “localization length”) has been studied in the so-called “directed-pathsapproximation.” [7,10]These calculations are based on the transfer matrix approach (reviewed below) which2

is applicable only in this approximation.While it can be shown that this approximationyields a good description in the extremely localized limit in the absence of any field, thequestion is more delicate when the magnetic field is present.Indeed, to the next order,paths with small returning loops should be added.In the absence of a magnetic field,this inclusion leads to a simple renormalization of the “localization length.” On the otherhand, one finds already in the directed-paths approximation that in the presence of thefield, the combined effects of lattice periodicity and the magnetic flux yield a complexinterference pattern with a very sensitive dependence of the localization length upon theflux. [10] Therefore, the effects of including even small returning loops with their enclosedflux are unpredictable and potentially may be very drastic.Including the returning loops, as explained below, becomes even more important ifthe energy to begin with is not extremely far from the band edge and the wave functionsare consequently only moderately localized.We thus set out in this work to go beyondthe directed-paths approximation and to calculate systematically the full Green’s functionswhich include the sum over all paths.These will be given as formal expressions whichmay be expanded systematically in various ways.Notable among these is the expansionin terms of the path length starting with the directed (the shortest) ones; it reveals thatthe generic features found in the directed-paths limit are not drastically changed by thesmall returning loops.

In this paper we restrict ourselves to the square lattice, but usingthis approach, the sums for other lattices may be calculated as well.The expression wederive may also be useful to analyze properties of Hofstadter electrons within the band ofbulk eigenstates.Nonuniform flux configurations have been studied recently in the context of theoriesof strongly interacting electrons relevant to high-Tc cuprates. The most important amongthem are the staggered flux configurations [11].We consequently obtain here the latticeGreen’s function for the square lattice with staggered flux as well.The layout of this paper is as follows.In section 2, we review the model and the3

“directed-paths-only” transfer-matrix approach. The heart of the paper is section 3, wherewe consider the addition of “backward excursions” to the transfer-matrix method.Here,we find recursion relations and other expressions for the various quantities involved.Insection 4, we examine some special cases thereof.Section 5 demonstrates how the fullGreen’s function can be obtained from the quantities derived in sections 3 and 4.

Section6 considers the staggered flux configuration, and section 7 contains a discussion of theresults. We also include an appendix, listing some additional results.II.

The Model and the ApproachOne can model non-interacting electrons confined to a square lattice with aperpendicularly-applied magnetic field by the following tight-binding hamiltonian:H = ǫXia†iai + VX⟨ij⟩a†iajeiγij,(2.1)where ǫ is the on-site energy and V measures the overlap of wave functions centered atneighboring sites. The energy spectrum is the same as the one obtained using the “Peierlssubstitution,” which replaces ¯h⃗k with ⃗p −e ⃗A/c in the dispersion relation obtained fromthe tight-binding approximation in the non-magnetic case.

The phase γij = −γji denotesthe Aharonov-Bohm phase an electron acquires as it moves from site i to site j; physically,it corresponds the line integral of the vector potential from i to j. Some freedom exists inthe choice of gauge: To characterize a constant magnetic field, one selects the phases suchthat the directional sum of the phases γij around each elementary plaquette is 2πφ/φowhere φ is the magnetic flux-per-plaquette and φo the flux quantum (hc/e).Many of the physically interesting quantities are contained in or are derivable fromthe Green’s function associated with eq.

(2.1).One can express the (real-space) Green’sfunction G(i, f, E, φ) between sites i and f for an electron of energy E as a sum over pathsconnecting the two sites:G(i, f; E, φ) =XpathsY⟨jk⟩V expiγjkE −ǫ,(2.2)4

where the product is over steps comprising a given path.Each path carries a weight vL,where v =VE−ǫ and L is the length of the path.Since the total number of paths on asquare lattice grows as 4L, the sum should converge provided v < 14.The model has aphase transition from localized to extended states occurring at v = 14. [5]In the strong localization regime (v ≪14), the electronic wave function decays quiterapidly as one moves away from some initial position.In this case, G(i, f; E, φ) is domi-nated by the shortest paths — those “directly” connecting the initial and final sites.

[12]Note that on a lattice there exists in general more than one shortest path connecting twosites. The presence of a magnetic field gives rise to interference among various constituentpaths.Since it is the interference phenomena that interests us and since the differencein the phases accumulated between two paths is proportional to the area they enclose, wefind it convenient to study the geometry in which directed paths can enclose the largestpossible areas.

For this reason, we begin our study with paths directed along the diagonalof a square lattice; we will call the directed axis t and the transverse axis x. (See Figure1.

)Summation of paths directed along a lattice axis (rather than the diagonal) is alsopossible but slightly less convenient.One can employ the transfer-matrix approach to sum over directed paths. [7] If thevector |vt−1⟩represents a walk or walkers at column t−1, then the matrix T(t, φ) operatingon |vt−1⟩|vt⟩= T(t, φ) |vt−1⟩(2.3)provides all possible extensions of these walks by one forward step and weights thesepossibilities appropriately.Consecutive operation by transfer matrices hence sums overall possible walks|vt⟩= T(t, φ) T(t −1, φ) .

. .

T(1, φ) |v0⟩. (2.4)Summing over paths weighted by different phase factors yields the desired interferenceeffects.5

The transfer matrix connecting column t −1 to column t is:T(t, φ) = v0e−itφπ0...00eitφπeitφπ0e−itφπ...0000eitφπ0...000..................000...0e−itφπ0000...eitφπ0e−itφπe−itφπ00...0eitφπ0,(2.5)where the bonds are viewed as being directed from lower to higher t.We exploit thegauge degree-of-freedom to furnish a transfer matrix that depends only on t (and not onx).This choice is called the diagonal staggered gauge [7].As required, summing thephases as one proceeds clockwise around any elementary plaquette results in 2πφ; we haveset φo = hc/e = 1 for convenience.Note that in addition we have applied periodic boundary conditions along the x axis. (However, since we approach the problem from the localization side, we may always takethe lateral extent xmax of the contributing paths to be smaller than the vertical size of thesystem L⊥, and so the sum over paths will not be sensitive to the boundary conditions.

[13]) With periodic boundary conditions, the eigenvectors of all T’s are plane waves:|vk⟩=1√L⊥eik⊥ei2k⊥ei3k⊥...eiL⊥k⊥. (2.6)The corresponding eigenvalues are λk⊥(t, φ) = 2v cos(k⊥−tφπ).

The transverse momentatake on values k⊥= 2πm/L⊥, where L⊥is the vertical size of the lattice and m =0, 1, ..., L⊥−1. The eigenvalues might also be called Gdirk⊥(t −1, t, φ) for they correspondto the Green’s function for directed walks (Fourier transformed in the transverse direction)joining columns t −1 and t.Note that just as in the continuum model, one finds plane waves in the “transverse”direction, leaving an effective one-dimensional problem (an harmonic oscillator in the con-tinuum case).The reduction to a one-dimensional problem does not depend on the6

restriction to directed walks since the same transfer matrix also yields a “backward” stepfrom column t to t −1. Henceforth, we need only consider eigenvalues, as all matrices ofinterest are simultaneously diagonalizable by the plane-wave eigenvectors.In the presence of a magnetic field, the exponential decay of the electron wave functionaway from its central position is modulated by a very intricate interference pattern.Inthe strongly localized regime, many of its properties can be uncovered by investigating theproduct obtained from the directed paths onlyGdirk⊥(0, To, φ) =To−1Yt=0Gdirk⊥(t, t + 1, φ) =ToYt=1λk⊥(t, φ) =ToYt=1h2v Cti,(2.7a)where for convenience, we have introduced the notationCt = cos(k⊥−tφπ);(2.7b)suppressing the φ and k⊥dependences as most of the calculations considered herein involvea single φ and a single k⊥.

Gdirk⊥(0, To, φ) supplies the Green’s function for directed walksthat begin at the origin and end in column To. The properties of this product have beenthe focus of a recent investigation.

[10] The transition amplitude between two sites, say(0, 0) and (xo, To), may be obtained from them through:G(0, 0; xo, To) =1L⊥Xk⊥Gdirk⊥(0, To; φ)e−ik⊥xo. (2.8)In the absence of any field, the directed Green’s function is simply a product of cosines:Gdirk⊥(0, To, 0) =h2v cos(k⊥)iTo.

(2.9)When the applied flux-per-plaquette is a rational multiple of the flux quantum (i.e.φφo = pqwith p and q relatively prime integers), the interference-induced modulation is periodic;in fact, the pattern on sites of a superlattice with lattice constant q times larger thanthe original lattice mimics precisely the simpler decay pattern found in the absence of7

an applied field. [7-10]If we restrict our attention to sites on the superlattice in thecommensurate case, thenGdirk⊥(0, To, pq ) ="qYt=12v Ct#To/q.

(2.10)One can then use the following property:qYt=1Ct =2−q+1 cos(qk⊥),if q is odd;2−q+1 sin(qk⊥),if q is even;(2.11)to see the relation between the commensurate case on the superlattice and the nonmagneticcase. Note that in the commensurate case, v−ToGdirk⊥=0(0, To, pq ) ∝2To/q.The work of Fishman, Shapir and Wang [10] addresses the behavior of this quantityin the incommensurate case (i.e.

when q →∞).In this case, the structure becomesaperiodic. For generic irrational, it has been found that ln|v−ToGdir| increases as [ln(To)]2;while for an algebraic irrational ln|v−ToGdir| increases as ln(To).III.

Allowing Backward ExcursionsThe main motivation for the present work is to consider the consequences of liftingthe “directed paths only” restriction from the previous analysis, thereby broadening itsscope beyond the strongly localized regime. We have opted to pursue the transfer-matrixmethod; however, this choice requires reconciling that approach, so ideally suited to di-rected paths, with our current concern of including returning loops.

We explain first howthe formal expressions can be obtained in an elegant and minimal way; we then proceed toa more detailed derivation which starts from the directed paths and adds systematicallylonger and longer returning loops. The expressions gained by this latter method will formthe basis for our conclusions on the effects of adding small loops to fully directed paths.With these ends in mind, let us begin our investigation of “directed paths with backwardexcursions,” which we define next.Let ˜λk⊥(t, φ) be the eigenvalue of a transfer matrix ˜T(t, φ) which allows any amountof backward excursion, that is, any number of loops of any length starting from t −1 and8

ending at t — provided only the last step bridges columns t −1 and t. (See, for example,Figure 1.) The eigenvalue ˜λk⊥(t, φ) might also be called Gb.e.k⊥(t −1, t, φ) for it denotes theGreen’s function linking columns t−1 and t that admits “backward excursions.” The full,unrestricted Green’s function can then be calculated from Gb.e.k⊥(s, t, φ) as will be shown insection V.In view of the fact that all of the terms comprising ˜λk⊥(t, φ) contain a factor 2vCtcorresponding to the last step, it is convenient to define ˆλk⊥(t, φ) = ˜λk⊥(t, φ)/2vCt withthis last piece stripped off.

Note that the walks contributing to ˆλk⊥(t, φ) begin and end incolumn t −1, never reaching column t. Again we could use the Green’s function notationˆλk⊥(t, φ) = Gb.e.k⊥(t −1, t −1; φ) which is more descriptive but a bit more cumbersome, andso we will put offits use until section V.We have found that ˆλk⊥(t, φ) obeys the recursion relation:ˆλk⊥(t, φ) = 1 +h2vCt−1i2 ˆλk⊥(t −1, φ)+h2vCt−1i2 ˆλk⊥(t −1, φ)2+ ....(3.1a)This relation can be understood in the following way:In the expansion, the first termon the right-hand side (the 1) indicates the option of making no backward steps.In thesecond term, one of the 2vCt−1’s denotes a step back to column t−2, then the ˆλk⊥(t−1, φ)designates any amount of backward excursion beyond t −2 which eventually returns tocolumn t −2 (including again the option of no excursion at all) and the second 2vCt−1returns the walker to column t −1.Finally this whole process might be repeated anynumber of times as indicated by the expansion.The expansion is readily summed (atleast formally) as a geometric series.ˆλk⊥(t, φ) =1 −4v2C2t−1 ˆλk⊥(t −1, φ)−1,(3.1b)Next, consecutive ˆλ’sˆλk⊥(t −1, φ), ˆλk⊥(t −2, φ), . .

.can be substituted into eq.9

(3.1b) to arrive at the following continued-fraction expression:ˆλk⊥(t, φ) =11 −4v2C2t−11 −4v2C2t−21 −4v2C2t−3etc..(3.2)Terminating this continued fraction at the C2t−s term and re-expressing ˆλ as a ratio ofpolynomials, one finds that the numerator and denominator have essentially the same formexcept that the terms containing Ct−1 are absent in the numerator. More specifically, theprocedure leads to:ˆλk⊥(t, φ) =Dk⊥(t, φ; 2, s)Dk⊥(t, φ; 1, s) + O(v2s+2),(3.3a)whereDk⊥(t, φ; r, s) = 1 −4v2sXj=rC2t−j + 16v4 Xj=rsXk=j+2C2t−jC2t−k−64v6 Xj=rXk=j+2sXl=k+2C2t−jC2t−kC2t−l + ....(3.3b)The problem has thus been reduced to that of understanding the properties of thisparticular function Dk⊥(t, φ; r, s).

Some of its more obvious and useful properties are:Dk⊥(t, φ; r, s) = Dk⊥(t + n, φ; r + n, s + n),(3.4a)Dk⊥(t, φ; r, s) = Dk⊥(t, φ; r + 1, s) −4v2C2t−rDk⊥(t, φ; r + 2, s),(3.4b)Dk⊥(t, φ; r, s) = Dk⊥(t, φ; r, s −1) −4v2C2t−sDk⊥(t, φ; r, s −2). (3.4c)Much is encoded in eq.

(3.3). However, since our motivation is to extend the analysisof fully directed walks to include returning loops, we present a methodical derivation whichdoes just that.

Along the way useful expansions for ˆλk⊥(t, φ) are obtained.Returning then to the directed paths, a first step toward including returning loopswould permit in addition to one forward step connecting columns t−1 and t, one backwardstep followed by two forward steps. The modified eigenvalues λmk⊥(t, φ) become:λmk⊥(t, φ) = λk⊥(t, φ) +λk⊥(t −1, φ)2 λk⊥(t, φ).

(3.5)10

Let us analyze the effect of this addition on the Green’s function for one plaquette on thesuperlattice in the case of commensurate flux.As in eq. (2.7a) for directed paths, theGreen’s function is a product of these eigenvalues, and this product can be separated intotwo products as follows:Gmk⊥0, q, pq=" qYt=12vCt#×" qYt=11 + 4v2C2t−1#.

(3.6a)The first product is Gdirk⊥(0, q, pq ) which we considered in eqs. (2.9) and (2.10); the secondproduct is also readily handled; together they yield:Gmk⊥0, q, pq= Gdirk⊥0, q, pq× (1 −α)−2qh1 −(−1)q2αq cos(k⊥q) + α2qi,(3.6b)whereα = 1 + 2v2 −√1 + 4v22v2≈v2 −2v4.

(3.6c)From this expression, one can see that the main effect is a renormalization of v withoutany dependence on φ. The only structural change arises from the cos(k⊥q) term; however,it is of the order v2q times smaller than the leading behavior.Next, let us consider theinclusion of longer backward excursions.As one moves further out of the strongly localized regime, one should account for pathswith increasingly longer backward excursions, culminating in ˜λk⊥(t, φ) which consists ofcontributions of all lengths.

˜λk⊥(t, φ) can be written as:˜λk⊥(t, φ) =∞Xj=0˜λ(j)k⊥(t, φ),(3.7)where ˜λ(j)k⊥(t, φ) designates the contribution of length (2j + 1) which therefore carries aweight (2v)2j+1. The first three are:˜λ(0)k⊥(t, φ) = (2v)Ct(3.8a)˜λ(1)k⊥(t, φ) = (2v)3CtC2t−1(3.8b)˜λ(2)k⊥(t, φ) = (2v)5CthC4t−1 + C2t−1C2t−2i.

(3.8c)11

We provide expressions for ˜λ(j)k⊥(t, φ) for j = 0, 1, . .

., 5 in the appendix.They becomecumbersome rather quickly as j increases. For j ≥1, ˜λ(j)k⊥(t, φ) can be shown to obey thefollowing recursion relation:˜λ(j)k⊥(t, φ) ="jXℓ=1˜λ(j−ℓ)k⊥(t, φ) ˜λ(ℓ−1)k⊥(t −1, φ)#˜λ(0)k⊥(t −1, φ).

(3.9)Again we find it convenient to strip offthe last step of the walk and study ˆλk⊥(t, φ).After generating ˆλ(j)k⊥(t, φ)’s from the recursion relation, we have observed that they couldbe expressed in the following way:ˆλ(j)k⊥(t, φ) =X{ni|Pji=1ni=j}"∞Yi=1ni + ni+1 −1ni+14v2 C2t−ini#,(3.10a)where ni = 0, 1, . .

., j for i = 1, 2, . .

., j and whereni + ni+1 −1ni+1=1,if ni = 0 and ni+1 = 0;0,if ni = 0 and ni+1 > 0;(ni + ni+1 −1)!(ni+1)! (ni −1)!

,otherwise. (3.10b)The restriction on the ni’s in eq.

(3.10a) ensures that the contributing walks have theappropriate length.Note first of all that each cosine term appears an even number oftimes, making the round trip possible. Moreover, the terms are “connected;” for instance,if a term contains a factor Ct−3, it must also contain factors Ct−2 and Ct−1.Finally,the numerical component (the degeneracy) factorizes into terms which depend only on thenumber of steps in two neighboring columns (ni and ni+1).Expresson (3.10) may have certain practical advantages over eq.

(3.3) in that itselects out walks of a particular length and that it is not written in the form of a quotient.Nevertheless, we proceed now to use it in a second derivation of eq. (3.3).

Performing thesum of ˆλ(j)k⊥(t, φ) over jˆλk⊥(t, φ) =∞Xj=0ˆλ(j)k⊥(t, φ)(3.11a)12

actually simplifies matters as it eliminates the restriction on ni’s, leading toˆλk⊥(t, φ) =X{n1,n2,...}"∞Yi=1ni + ni+1 −1ni+14v2C2t−ini#. (3.11b)Next carry out the sums over ni consecutively.

Using repeatedly the following sum:Xj=0j + k −1kxj = 1 +x(1 −x)k+1 ,(3.12)leads to:ˆλk⊥(t, φ) = 1 +(2v)2C2t−11 −4v2C2t−1 +(2v)4C2t−1C2t−21 −4v2C2t−11 −4v2C2t−1 −4v2C2t−2+(2v)6C2t−1C2t−2C2t−31 −4v2C2t−1 −4v2C2t−21 −4v2C2t−1 −4v2C2t−2 −4v2C2t−3 + 16v4C2t−1C2t−3+ O(v8). (3.13)Finally finding common denominators for the first two terms, then the first threeterms, etc., yields expressions like:ˆλk⊥(t, φ) =1 −4v2C2t−21 −4v2C2t−1 −4v2C2t−2+ O(v6),ˆλk⊥(t, φ) =1 −4v2C2t−2 −4v2C2t−31 −4v2C2t−1 −4v2C2t−2 −4v2C2t−3 + 16v4C2t−1C2t−3+ O(v8), (3.14)and so on.

Hence, one arrives at:ˆλk⊥(t, φ) = Dk⊥(t, φ; 2, s)Dk⊥(t, φ; 1, s) + O(v2s+2),(3.15)which is identical to eq. (3.3).IV.

Special CasesIn this section, we consider the evaluation of ˆλk⊥(t, φ) in three special cases: 1) whenv is small (which provides a straightforward extension of the directed-paths-only analysisin the strongly localized regime); 2) when the magnetic flux-per-plaquette is a rational13

multiple of the flux quantum; and 3) when φ is small (weak magnetic field, for which oneshould be able to make some correspondence with the continuum model).iv.a. When v is smallThe first special case we shall treat is when v is small.Under such conditions, it isreasonable to view the various quantities as power series in v.We have already seen inthe previous section thatˆλ(0)k⊥(t, φ) = 1(4.1a)ˆλ(1)k⊥(t, φ) = (2v)2C2t−1(4.1b)ˆλ(2)k⊥(t, φ) = (2v)4hC4t−1 + C2t−1C2t−2i,(4.1c)ˆλ(3)k⊥(t, φ) = (2v)6hC6t−1 + 2C4t−1C2t−2 + C2t−1C4t−2 + C2t−1C2t−2C2t−3i,(4.1d)and so on.

From these, we see that ˆλk⊥(t, φ) can be re-expressed asˆλk⊥(t, φ) = exp(2v)2C2t−1+ O(v4). (4.2a)ˆλk⊥(t, φ) = exp(2v)2C2t−1 + (2v)4h12C4t−1 + C2t−1C2t−2i+ O(v6).

(4.2b)ˆλk⊥(t, φ) = exp(2v)2C2t−1 + (2v)4h12C4t−1 + C2t−1C2t−2i+(2v)6h13C6t−1 + C4t−1C2t−2 + C2t−1C4t−2 + C2t−1C2t−2C2t−3i+ O(v8). (4.2c)Expressions (4.2a), (4.2b), and (4.2c) account accurately for backward excursions of length2, 4, and 6, respectively.Note that the expansion of the ln[ˆλk⊥(t, φ)] (the argument of exp{} above) looksrather like the expansion of ˆλk⊥(t, φ) itself (eq.

(3.11b)) except that coefficients have anadditional 1/n1 factorlnˆλk⊥(t, φ)=X{n1,n2,...}"1n1∞Yi=1ni + ni+1 −1ni+14v2C2t−ini,(4.3)14

and, of course, the n1 = 0 term is absent.We can verify this expression with repeateduse of the relationXj=11jj + k −1kxj = −ln(1 −x),if k = 0;1k(1 −x)−k −1k,if k ̸= 0,(4.4)which leads tolnˆλk⊥(t, φ)= lnhDk⊥(t, φ; 2, s)i−lnhDk⊥(t, φ; 1, s)i+ O(v2s+2),(4.5)which is simply the logarithm of eq. (3.3a).The form given above is particularly convenient for calculating Gb.e.k⊥(0, To, φ), theGreen’s function for long walks that allow backward excursions, because Gb.e.k⊥(0, To, φ)factors into Gdirk⊥(0, To, φ) and a product of ˆλ’sGb.e.k⊥(0, To, φ) =ToYt=1˜λk⊥(t, φ) = Gdirk⊥(0, To, φ) ×ToYt=1ˆλk⊥(t, φ),(4.6a)and the exponential form converts that product into a sum:Gb.e.k⊥(0, To, φ) = Gdirk⊥(0, To, φ)× exp ToXt=1h(2v)2C2t−1 + (2v)4 12C4t−1 + C2t−1C2t−2+ .

. .i.

(4.6b)Let us return to the example considered in the previous section (eqs. (3.5) and (3.6))in which we have examined the effect of short backward excursions (of length 2 only)on the interference pattern generated on the superlattice when the flux-per-plaquette iscommensurate (φ = pq ).We are now in a position to extend that analysis.SubstitutingTo = q and φ = pq into eq.

(4.6b), we findGb.e.k⊥(0, q, pq ) = Gdirk⊥(0, q, pq )× exp(2v)2 q2 + (2v)4h7q16 + q8 cos 2πpqi+ . .

.,(4.7)15

where we have used the following relations for φ = pqqXt=1C2t−1 = q2for q ≥2,(4.8a)qXt=1C4t−1 = 3q8for q ≥3,(4.8b)qXt=1C2t−1C2t−2 = q14 + 18 cos 2πpq.for q ≥3. (4.8c)The exponential part represents the correction to the directed-paths-only approach.Wesee the same pattern here as before — that on the superlattice (To = q) the transversemomentum k⊥does not enter the corrections (i.e.

there are no structural changes) atorder v2 or order v4 provided q ≥3, suggesting that k⊥enters only at order v2q. Some ofthe sums needed to carry out this analysis to higher order are:qXj=1C2mt−j =q22m2mmif q > m,(4.9a)qXj=iC2mjC2nj+1 =q22m+2n−1" nXℓ=0 2mm + ℓ 2nn −ℓcos 2πpℓq−122mm2nn#,(4.9b)if q > m + n. This last calculation is really a mixture of two special cases — small v andcommensurate flux (φ = pq ).

We will now turn our attention to the commensurate case.iv.b. Commensurate fluxIn the case in which the flux-per-plaquette φ is a rational multiple ( pq ) of the flux quan-tum, the terms in the continued fraction expression (eq.

(3.2)) begin to repeat themselvesafter q terms. For instance, for φ = 13, one has:ˆλk⊥(t, 13) =11 −4v2C2t−11 −4v2C2t−21 −4v2C2t−3 ˆλk⊥(t, 13).

(4.10a)Simplifying the fraction yields:ˆλk⊥(t, 13) =1 −4v2C2t−2 −4v2C2t−3 ˆλk⊥(t, 13)1 −4v2C2t−1 −4v2C2t−2 −4v2C2t−3 ˆλk⊥(t, 13). (4.10b)16

Performing the same procedure for any φ = pq , we find:ˆλk⊥(t, pq ) =Dk⊥(t, pq ; 2, q −1) −4v2C2t−qDk⊥(t, pq , 2, q −2) ˆλk⊥(t, pq )Dk⊥(t, pq ; 1, q −1) −4v2C2t−qDk⊥(t, pq , 1, q −2) ˆλk⊥(t, pq ). (4.11)From here we conclude that ˆλk⊥(t, pq ) is the solution of the following quadratic equa-tion:Ak⊥(t, pq )hˆλk⊥(t, pq )i2+ Bk⊥(t, pq ) ˆλk⊥(t, pq ) + Ck⊥(t, pq ) = 0,(4.12a)whereAk⊥(t, pq ) = 4v2C2t−qDk⊥(t, pq ; 1, q −2),(4.12b)Bk⊥(t, pq ) = −Dk⊥(t, pq ; 1, q −1) −4v2C2t−qDk⊥(t, pq ; 2, q −2),(4.12c)Ck⊥(t, pq ) = Dk⊥(t, pq ; 2, q −1).

(4.12d)The solution of the quadratic equation is:ˆλk⊥(t, pq ) =−Bk⊥(t, pq ) −qB2k⊥(t, pq ) −4 Ak⊥(t, pq ) Ck⊥(t, pq )2 Ak⊥(t, pq ). (4.13)We have used this last expression to calculate ˆλk⊥(t, pq ) for relatively small q.Somealgebra and trigonometry reveal that:ˆλk⊥(t, 1) =1 −q1 −16v2C2t−18v2C2t−1,(4.14a)ˆλk⊥(t, 12) =1 −4v2 + 8v2C2t−2 −q(1 −4v2)2 −16v4 sin2(2k⊥)8v2C2t−2,(4.14b)ˆλk⊥(t, 13) = 1 −6v2 + 8v2C2t−3 −p(1 −6v2)2 −16v6 cos2(3k⊥)8v2C2t−3(1 −4v2C2t−1).

(4.14c)One can see a pattern arising from these expressions.For instance, we can see here andshow in general that the discriminant (B2 −4AC) is independent of t.17

Note that q = 1 corresponds to one flux quantum per plaquette or equivalently nofield at all.The small field limit, on the other hand, would require taking q to infinity.Fortunately, this limit is accessible by other means.iv.c. The small field limitThe third special case we consider here is when the magnetic field is small.Towardthis end, let us perform a Taylor expansion of ˆλk⊥(t, φ) around (k⊥−tφπ):ˆλk⊥(t, φ) =∞Xi=0Λ(i)k⊥(t, φ) φi.

(4.15)It is our goal in this section to calculate the i = 0, 1 and 2 terms Λ(i)k⊥(t, φ) above.Instead of expanding ˆλk⊥(t, φ) directly, we will begin by expanding ˆλ(j)k⊥(t, φ). Recall-ing eq.

(3.10a) for ˆλ(j)k⊥(t, φ), we clearly need an expansion of Cℓt−j around (k⊥−tφπ):Cℓt−j = Cℓt −jℓφπCℓ−1tSt + j2φ2π22hℓ(ℓ−1)Cℓ−2tS2t −ℓCℓti+ ...,(4.16)where St = sin(k⊥−tφπ).Below we present the expansions of ˆλ(j)k⊥(t, φ) to order φ2 forj = 0, 1, . .

., 6:ˆλ(0)k⊥(t, φ) =1(4.17a)ˆλ(1)k⊥(t, φ) =(2v)2hC2t −2πφCtSt + π2φ22(2S2t −2C2t ) −...i(4.17b)ˆλ(2)k⊥(t, φ) =(2v)4h2C4t −10πφC3t St + π2φ22(38C2t S2t −14C4t ) −...i(4.17c)ˆλ(3)k⊥(t, φ) =(2v)6h5C6t −44πφC5t St + π2φ22(332C4t S2t −76C6t ) −...i(4.17d)ˆλ(4)k⊥(t, φ) =(2v)8h14C8t −186πφC7t St + π2φ22(2246C6t S2t −374C8t ) −...i(4.17e)ˆλ(5)k⊥(t, φ) =(2v)10h42C10t−772πφC9t St + π2φ22(13348C8t S2t −1748C10t ) −...i(4.17f)ˆλ(6)k⊥(t, φ) =(2v)12h132C12t−3172πφC11t St+ π2φ22(73340C10t S2t −7916C12t ) −...i(4.17g)18

Next, we collect terms of the same order in φ and resum those series.Toward thatend, we define:Λ(0)k⊥=∞Xj=0Nj (2vCt)2j,(4.18a)Λ(1)k⊥= −2πStCt∞Xj=0Pj (2vCt)2j,(4.18b)Λ(2)k⊥= π2S2t2C2t∞Xj=0Qj (2vCt)2j −π22∞Xj=0Rj (2vCt)2j. (4.18c)From eqs.

(4.17a-g), we can conclude, for instance, thatPj=0, 1, 5, 22, 93, 386, 1586, . .

..(4.19)These four series are summed as follows:Xj=0Nj xj = 12xh1 −(1 −4x)1/2i,(4.20a)Xj=0Pj xj = 12h(1 −4x)−1 −(1 −4x)−1/2i,(4.20b)Xj=0Qj xj = 52(1 −4x)−5/2 −2(1 −4x)−2 −2(1 −4x)−3/2+ (1 −4x)−1 + 12(1 −4x)−1/2,(4.20c)Xj=0Rj xj = (1 −4x)−3/2 −(1 −4x)−1. (4.20d)Using these we arrive at:Λ(0)k⊥(t, φ) = 1 −p1 −16v2C2t8v2C2t,(4.21a)Λ(1)k⊥(t, φ) = −8πv2CtSt1 −16v2C2tΛ(0)k⊥(t, φ)(4.21b)andΛ(2)k⊥(t, φ) = 2π2v2U −3(2(S2t −C2t )+ S2t U −2h5 + Ui h1 −U 2i)Λ(0)k⊥,(4.21c)19

where U = (1 −16v2C2t )−1/2.Note that the linear term breaks the φ →−φ symmetry.Two comments should bemade: (i) φ →−φ with x →−x (or equivalently k⊥→−k⊥) is still a symmetry and (ii)most importantly all physical properties which depend on the absolute value |G| will beinvariant under φ →−φ or x →−x separately.This is consistent with the expectationthat no Lorentz force (and no parity symmetry breaking) should occur for states whichdo not carry current density. Such effects will be seen only if the dependence of G in thet direction goes like eik∥t so that the states have a nonvanishing current density in the tdirection.

[14]V. The Full Green’s FunctionIn this section, we show how to obtain the full Green’s function Gk⊥(0, To, φ), which isthe appropriately weighted sum of all walks connecting columns 0 and To. (Recall that the2D problem was reduced to 1D.) Thus far, we have calculated ˜λk⊥(t, φ) = Gb.e.k⊥(t−1, t, φ),which includes walks which advance from column t −1 to column t with any amount ofbackward excursion prior to the forward step.

In this section we will find it more convenientto use the Gb.e.k⊥(t −1, t, φ) notation. We have also already seen the product:Gb.e.k⊥(0, To, φ) =To−1Yt=0Gb.e.k⊥(t, t + 1; φ);(5.1)which supplies the Green’s function for the restricted walks that begin at the origin andend in the To column, again with the last step being the only one connecting the To −1column to the To column.

This is the Green’s function for the first time the walker landsin column To.Now we must allow the walker to go beyond To or to turn back and return later to To.Let us denote this sum by Gfullk⊥(To, To; φ); it is the Green’s function for all walks startingin column To and returning to that column. Note that Gfullk⊥(To, To; φ) obeys the followingrecursion relation:20

Gfullk⊥(To, To; φ) = 1 + 4v2C2To Gb.e.k⊥(To −1, To −1, φ) Gfullk⊥(To, To, φ)+ 4v2C2To+1 Gf.e.k⊥(To + 1, To + 1φ) Gfullk⊥(To, To; φ),(5.2a)where Gf.e. is the same Green’s function as we have already found except that the initialposition is to the right of the endpoint.

Therefore, the formal solution is:Gfullk⊥(To, To; φ) =h1 −4v2C2ToGb.e.k⊥(To −1, To −1, φ) −4v2C2To+1Gf.e.k⊥(To +1, To +1; φ)i−1. (5.2b)Finally, the full Green’s function Gfullk⊥(0, To; φ) is simply the product:Gfullk⊥(0, To; φ) = Gb.e.k⊥(0, To; φ) Gfullk⊥(To, To; φ),(5.3a)Gfullk⊥(0, To, φ) =To−1Yt=0Gb.e.k⊥(t, t + 1; φ)h1 −4v2C2ToGb.e.k⊥(To −1, To −1; φ) −4v2C2To+1Gf.e.k⊥(To + 1, To + 1; φ)i.

(5.3b)This last equation is the central equation of this paper: It gives the formal expressionfor the Green’s function of an electron on a lattice in a magnetic field. The dependence onx can be obtained by an inverse Fourier transform with respect to exp(ik⊥x).

In principle,all physical properties may be obtained from this expression.This is certainly true forE outside the spectrum.For the energies within the “butterfly,” one may still use thisformula for walks which do not reach the boundaries of the system (or wrap around thetorus for if periodic boundary conditions are assumed, other delicate questions then arise[13] if the two sizes L⊥and L∥are not integer multiples of q. )In the next section, we look at the Green’s function for charged particles propagatingwithin the staggered-flux configurations that have been studied in the context of high-Tcsuperconductivity.VI.

Staggered Flux Configurations21

Beginning with the work of Affleck and Marston [11], the notion of staggered fluxconfigurations was introduced and studied in the context of theories of high-Tc supercon-ductivity. It may, therefore, be useful to have expressions for electron propagation in suchflux configurations.

On a square lattice, such a configuration will be given by a chessboardarrangement of interpenetrating sets of plaquettes with (say) +φ on the white squares and−φ on the black ones. (See Figure 2.

)The most studied case is φ = π, for which thephase factors acquired around the plaquettes are ±1, so that time reversal symmetry isunbroken. The sum of walks in this particular case was obtained by Khveshchenko, Koganand Nechaev [15].In the present section, we obtain these sums for arbitrary staggeredflux ±φ.First, we must choose a gauge:this time we assign the phases only to the verticalbonds of a square lattice.Considering them as directed upwards, we assign a factorγ = eiφ/2 for bonds pointing from sublattice A to sublattice B (shown as dashed in Fig.2) and γ∗= e−iφ/2 to the bonds pointing from sublattice B to A (solid in Fig.

2).So astaggered arrangement of phases ±φ/2 is assigned to the (upward-pointing) vertical bonds.Next, we assign a factor α (α−1) to a step in the positive (negative) x directionand similarly β (β−1) for the y direction.A natural choice is α = eikx and β = eikywhich yields the Fourier transform of the generating functions.We define the followinggenerating functions: first, the local generating functions for a stepA →B :gA = α + α−1 + γ(β + β−1)B →A :gB = α + α−1 + γ−1(β + β−1),(6.1)one for each sublattice; then a global generating function for all paths of length 2L:Gstag2L (φ) = gLAgLB =h4(cos2 kx + cos2 ky + 2 cos φ2 cos kx cos ky)iL. (6.2)The probability amplitude to reach a given site (x, y) is the coefficient of αxβy (up to atrivial normalization) or the inverse Fourier transform:Pstag2l(φ) ∝1(2π)2Z π−πZ π−πdkx dky Gstag2L (φ) e−ikxx−ikyy,(6.3)22

where the system size is assumed to be infinite and we have thus gone over to integralsover the momenta. Note that in order to obtain the normalized Green’s function, gA andgB must each be multiplied by 12 and G2L by 2−2L.Of particular interest are the expressions for walks which return to their initial point.The weighted closed magnetic walks are given by the coefficient of α0β0 = 1 in the product:Gstag2L (φ) =h(α + α−1) + γ(β + β−1)iLh(α + α−1) + γ−1(β + β−1)iL,(6.4)where γ = eiφ/2.

Toward this end, we expand the above products as follows:h(α + α−1) + γ(β + β−1)iL=XL(A)x LL(A)x(α + α−1)L(A)xhγ(β + β−1)iL(A)y(6.5a)h(α + α−1) + γ−1(β + β−1)iL=XL(B)x LL(B)x(α + α−1)L(B)xhγ−1(β + β−1)iL(B)y (6.5b)where L(A)x(L(A)y) is the number of horizontal (vertical) steps initiated from the A sublatticeand likewise for the B sublattice. Consequently, L(A)x+L(A)y= L(B)x+L(B)y= L. Next, weexpand the remaining α products, introducing the variables ℓ(A)+x and ℓ(B)+x which representthe number of steps in the positive x direction initiated on the A and B sublattices,respectively.

The coefficients of α will thus be:L(A)xXℓ(A)+xL(B)xXℓ(B)+xL(A)xℓ(A)+xα2ℓ(A)+x −L(A)xL(B)xℓ(B)+xα2ℓ(B)+x −L(B)x . (6.6)To obtain the α0 term, the following relation must hold:2(ℓ(A)+x + l(B)+x ) = L(A)x+ L(B)x= Lx,(6.7)where Lx is, of course, the total number of horizontal steps.

Note that Lx must be even.Replacing ℓ(B)+x by Lx2 −ℓ(A)+x , we find that the coefficient of α0 is:L(A)xXl(A)+xL(A)xℓ(A)+xL(B)xLx2 −ℓ(A)+x=LxLx2. (6.8)23

Repeating the same procedure for the y direction and combining the results yields:Gstag2L (φ) =XL(A)xXL(B)x LL(A)xLxLx2 LL(B)xLyLy2γL(A)y−L(B)y(6.9a)LXLx=0LxXL(A)x=0 LL(A)xLxLx2LLx −L(A)xLyLy2γ2L(A)y−Ly(6.9b)=XLxLxLx22L −LxL −Lx2 XL(A)x LL(A)xLLx −L(A)xγLx−2L(A)x ,(6.9c)where we have used L(A)y= L −L(A)xand Ly = 2L −Lx.Using eq. (6.9) we have found Gstag2Lfor several L’s.

They are:Gstag0= 1(6.10a)Gstag2= 4(6.10b)Gstag4= 28 + 8 cos φ(6.10c)Gstag6= 256 + 144 cos φ(6.10d)Gstag8= 2716 + 2112 cos φ + 72 cos 2φ(6.10e)Gstag10= 31504 + 29600 cos φ + 2400 cos2φ(6.10f)Gstag12= 387136 + 411840 cosφ + 54000 cos 2φ + 800 cos 3φ(6.10g)Gstag14= 4951552 + 5752992 cos φ + 1034880 cos2φ + 39200 cos 3φ(6.10h)Gstag16= 65218204 + 80950016 cosφ + 18267200 cos2φ + 1191680 cos3φ+ 9800 cos 4φ(6.10i)Gstag18= 878536624 + 1148084928 cosφ + 307577088 cos2φ + 29070720 cos3φ+ 635040 cos4φ(6.10j)Gstag20= 12046924528 + 16407496800 cosφ + 5030575200 cos2φ + 625312800 cos3φ+ 24343200 cos4φ + 127008 cos5φ(6.10k)Gstag2Lcan also be obtained as the coefficient of v2L in the following expression:12πZ 2π0dk⊥n1 −8v2 −8v2 cos(φ/2) cos(2k⊥) + 16v4 sin2(φ/2) sin2(2k⊥)o−12 . (6.10ℓ)24

As we work toward procuring a more general expression, let us concentrate on thefollowing sum:LxXL(A)x LL(A)xLLx −L(A)xγLx−2L(A)X ,(6.11a)which is the latter portion of eq. (6.9c).

Introducing L(B)x= Lx−L(A)xthis may be writtenas:LXL(A)xLXL(B)x LL(A)x LL(B)xγL(B)x−L(A)x δhLx −L(A)x−L(B)xi= 12πZ 2π0dθXL(A)xXL(B)x LL(A)x LL(B)xγL(B)x−L(A)X e−iθ[Lx−L(A)x−L(B)x]= 12πZ 2π0dθh1 + γeiθiLh1 + γ−1eiθiLe−iθLx= 12πZ 2π0dθ eiθ(L−Lx)h2 cos θ + (γ + γ−1)iL= 2L2πZ 2π0dθ eiθ(L−Lx)hcos θ + cos φ2iL(6.11b)We may now use the identity:12πZ 2π0dθhµ +pµ2 −1 cos θincos mθsin mθ=n! (n + m)!P mn (µ)0(6.12)We identify cos φ2 = µ/pµ2 −1 which implies µ = ±i cot φ2 such that the integral becomes:2L2π1(µ2 −1)L2Z 2π0dθ eiθ(L−Lx)hµ +pµ2 −1 cos θiL.

(6.13)Therefore for the sumLxXL(A)x LL(A)xLLx −L(A)xγLx−2L(A)x= (−i)L2 (sin φ2 )L22LL! (2L −Lx)!P (L−Lx)L(i cot φ2 )(6.14)The final result is:Gstag2L= C(L)2LXLx=0LxLx22L −LxL −Lx21(2L −Lx)!P (L−Lx)L(i cot φ2 )(6.15a)25

withC(L) = (−i)L2 sin φ2 L2 2LL! (6.15b)We find agreement with the known cases [15] of φ = 0 for which eq.

(6.14) yields2LLxand φ = π for which it gives LLx2.One can generalize the expressions to walks that do not close. To reach a given point(xo,yo) the only difference will be in the summation over l(A)+x since what we need now isthe coefficient of αxo or2l(A)+x + l(B)+x−L(A)x+ L(B)x= xo.

(6.16)Therefore l(B)+x = Lx2 + xo2 −l(A)+x and the sum is:L(A)xXl(A)+x =0L(A)xl(A)+xL(B)xLx2 + xo2 −l(A)+x=LxLx2 + xo2. (6.17)Likewise (to get the coefficient of βyo) the summation in the y direction yields:γL(A)y−L(B)yLyLy2 + yo2,(6.18)and the final expression changes to:Gstag2L= C(L)2LXLx=0LxLx2 + xo22L −LxL −Lx2 + yo21(2L −Lx)!P (L−Lx)L(i cot φ2 ).

(6.19)Alternatively, one could choose α = eikx and β = eiky and study G2L(α, β) which can thenbe Fourier transformed back into real space.VII. DiscussionIn this paper we have derived expressions for the square-lattice Green’s function of acharged particle in a magnetic field.We have also obtained similar expressions for theso-called staggered flux configurations with arbitrary flux ±φ.

The results were obtained26

within the Peierls ansatz in which the magnetic field is represented by an extra phaseacquired by the nearest-neighbor hopping term.Our starting point was the directed-paths-only approach initiated in the study ofstrongly localized wave functions.From there, we have shown how to include system-atically the returning loops and obtain the Green’s function for walks with backwardexcursions, and then we have demonstrated how the full Green’s functions were realizedin terms of the former.Our main goal was to see if the inclusion of small returning loops would modify insome essential way the behavior found in the directed-paths approach.The analysis ofeqs. (3.5), (3.6) and (4.1) through (4.8) shows that the answer is negative in the sensethat the product over cosines with changing phases obtained for directed paths is modifiedinto a product of more complicated trigonometric expressions containing the same cosines.Of course, the behavior will change but the essential features:strong sensitivity to thevalue of the field, simple scaling in q for the commensurate case φ = p/q, and an aperiodicbut deterministic behavior for irrational φ, all are going to survive the addition of thesmall returning loops.

Repeating the analysis done in ref. 10. for the product of the newtrigonometric expressions would be a formidable challenge, since even for the former case,the analysis was a substantial and sophisticated task.Our expressions might aid in the study of other interesting issues as well: For example,the transmission through a slab of finite width in which the wave function does not decayall the way to zero, might also be studied.

An especially interesting question concerns theemergence of a Lorentz force (and the associated parity symmetry breaking) in the casewhich deals with such a current carrying state.Another interesting issue is how the continuum limit may be approached:Naively,one would expect that if the magnetic flux is small such that the magnetic length is muchlonger than the lattice spacing (but smaller than the system size) the lattice results willcoincide with those obtained in the continuum. In particular, the exponential decay should27

change into a Gaussian ∼exp{−cBr2} behavior. [4]Finally, these Green’s functions should be correct (analytically) for energies withinthe band.One should, of course, add a small imaginary part to the energy (and de-fine advanced and retarded Green’s functions) to deal with the singular behavior at theeigenenergies.In addition, the question of boundary conditions for walks hitting theboundaries should be handled with special care.In principle at least, all physical infor-mation could be extracted from these Green’s functions.

It will certainly not be straight-forward to extract physically relevant information from the formal expressions, but in viewof the importance of such information for many physical problems of interest (especiallythe many-body extensions studied in the context of the Fractional Quantum Hall Effect,high-Tc superconductivity, anyonic physics, etc. ), it may be worthwhile to pursue thosedirections.AcknowledgmentsAcknowledgment is made to the donors of The Petroleum Research Fund, administeredby the ACS, for support of this research.28

AppendixIn section 3 we have introduced the function ˜λk⊥(t, φ), the eigenvalue of the transfermatrix ˜T(t, φ) which allows any amount of “backward excursion” followed by a singleforward step.We called ˜λ(j)k⊥(t, φ) the contribution with path length (2j + 1). Using therecursion relation (3.9), we have generated ˜λ(j)k⊥(t, φ) for j = 0, 1, .

. ., 5.

They are:˜λ(0)k⊥(t, φ) = (2v)Ct(AI.1a)˜λ(1)k⊥(t, φ) = (2v)3CtC2t−1(AI.1b)˜λ(2)k⊥(t, φ) = (2v)5CthC4t−1 + C2t−1C2t−2i(AI.1c)˜λ(3)k⊥(t, φ) = (2v)7CthC6t−1 + 2C4t−1C22 + C2t−1C4t−2 + C2t−1C2t−2C2t−3i(AI.1d)˜λ(4)k⊥(x, φ) = (2v)9CthC8t−1 + 3C6t−1C2t−2 + 3C4t−1C4t−2 + 2C4t−1C2t−2C2t−3+ C2t−1C6t−2 + 2C2t−1C4t−2C2t−3 + C2t−1C2t−2C4t−3+ C2t−1C2t−2C2t−3C2t−4i(AI.1e)˜λ(5)k⊥(t, φ) = (2v)11CthC10t−1 + 4C8t−1C2t−2 + 6C6t−1C4t−2 + 3C6t−1C2t−2C2t−3+ 4C4t−1C6t−2 + 6C4t−1C4t−2C2t−3 + 2C4t−1C2t−2C4t−3+ 2C4t−1C2t−2C2t−3C2t−4 + C2t−1C8t−2 + 3C2t−1C6t−2C2t−3+ 3C2t−1C4t−2C4t−3 + 2C2t−1C4t−2C2t−3C2t−4 + C2t−1C2t−2C6t−3+ 2C2t−1C2t−2C4t−3C2t−4 + C2t−1C2t−2C2t−3C4t−4+ C2t−1C2t−2C2t−3C2t−4C2t−5i,(AI.1f)where Ct = cosk⊥−tφπ. These expressions can be seen to obey eq.

(3.10a).29

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Figure CaptionsFigure 1. The diagonal lattice with t and x axes labeled and an example of a directedpath with backward excursion.Figure 2.

The Staggered Flux Configuration.31

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