IIsing model as Wilson-Ma jorana Fermions
Ulli Wolff ∗ Institut f¨ur Physik, Humboldt Universit¨atNewtonstr. 1512489 Berlin, Germany
Abstract
We show the equivalence of the 2D Ising model to standard free Euclideanlattice fermions of the Wilson Majorana type. The equality of the looprepresentations for the partition functions of both systems is establishedexactly for finite lattices with well-defined boundary conditions. The honey-comb lattice is particularly simple in this context and therefore discussedfirst and only then followed by the more familiar square lattice case.
The two-dimensional Ising spin model can probably be called the prototype exactlysolved model in statistical physics and lattice field theory. The break-throughsolution was achieved by Onsager [1] who has computed the partition function bythe transfer matrix approach in 1944. Since then a large number of alternative –usually less complicated – strategies to derive the same result have been presented,like for instance [2]. Without any attempt toward completeness we shall onlymention below a few more selected references which are more or less close to ourapproach. Finally, we then hopefully will be able to sufficiently justify the presentaddition to this literature.On a very naive level one may find the two-valuedness of the elementary spinsreminiscent of fermionic states that can be empty and occupied. A much moreconcrete such link was established in the paper of Schultz, Mattis and Lieb [5].The transfer matrix of the model is first expressed in terms of a tensor product ∗ e-mail: uwolff@physik.hu-berlin.de A more extended recent collection of references is found in [3] for example. See also [4]. a r X i v : . [ h e p - l a t ] M a y f Pauli algebras attached to the sites of a one dimensional row of the lattice.By performing a Jordan-Wigner transformation [6] the Pauli matrices are tradedfor anticommuting fermion operators. The transfer matrix in this form inher-its the nearest neighbor bilinear structure of the original model and can hencebe diagonalized by Fourier expansion on the lattice. Simpler analogous steps inthe Hamiltonian limit of the transfer matrix (‘continuous time’) have later beendiscussed in [7] and [8].An alternative to representing fermions by operators with canonical anticom-mutation relations is given by path integrals over anticommuting Grassmann ‘num-bers’ [9]. Such a representation has been derived from the operator transfer matrixin [10]. Even earlier, Samuel [11] has used Grassmann integrals to directly ‘draw’the high temperature series of the Ising model to all orders which constitutes anequivalent representation of the model. In either case the resulting integrals areGaussian (free fermions), can be performed, and thus furnish an exact solution.Samuel’s work is by far the closest to our work. The differences distinguishingthe presentation at hand are however the following. We demonstrate the equiv-alence of the Ising model with free Euclidean Majorana fermions of the Wilsontype [12] that is one of the standard choices in lattice field theory. The criticalpoint corresponds to zero mass and the Euclidean relativistic invariance in thecontinuum/scaling limit is manifest in this standard framework. All phase factorsin the matched expansions arise naturally from the fermion nature combined withhalf-angle spin rotation phases. The Fermi-Bose equivalence proven here will be anexact identity between arbitrary finite lattice partition functions with well-definedperiodic or antiperiodic boundary conditions in each of the two lattice directions.In section 2 the loop representation of the Ising model is defined and matchedto the fermionic model for the honeycomb lattice. In section 3 the same program isimplemented for the standard square lattice which is technically more complicated.Some conclusions and remarks on more than two dimensions are offered in section4. In two appendices we report details on the evaluation of the spin weights andon the actual evaluation of the free fermion partition functions. In particular, thelattice fermion spectra are plotted. The honeycomb lattice can be spanned by two triangular sublattices A and B ,see figure 1. Each site x ∈ A has three nearest neighbors x + ˆ e a ∈ B , a = 0 , , making 120 degree angles with each other fulfillˆ e a · ˆ e b = 12 (3 δ ab − . (1)The sites of A are labeled by integers x , x in the form A (cid:51) x = x f + x f , f = ˆ e − ˆ e , f = ˆ e − ˆ e , f i · f j = 32 ( δ ij + 1) . (2)All sites in B can now be generated as neighbors x + ˆ e of a unique x ∈ A .Figure 1: The honeycomb lattice with L = L = 4. Sites of A ( B ) carry full blue(empty red) dots. Periodic boundary conditions are indicated by open links withequal indices.A simple way to impose periodic boundary conditions to obtain a finite systemis to identify points x with x + L f and x + L f with integer L i . We then have V = 2 L L independent sites in total, half in A and half in B .We adopt the convention to take coordinates x i ∈ [0 , L i ) and label links by thepairs ( x, a ) with x ∈ A , a ∈ { , , } . Which links close ‘around the boundary’ ? We use lattice units a = 1 with respect to these nearest neighbor links. In [13] a more ‘rectangular’ periodicity is introduced that results in helical boundary condi-tions for the honeycomb fields. Clearly, also this case can be handled along the lines presentedhere. x + ˆ e a are in B and in our coordinate system associated with (gen-erated by) x + ˆ e a − ˆ e back in A , or in other words, with x, x + f , x + f which arefolded back into the ranges [0 , L i ) by standard modulo operations. A site y ∈ B on the other hand is generated by y − ˆ e = z ∈ A and its neighbors y − ˆ e a are z, z − f , z − f . We attach Ising spins s ( x ) ∈ { +1 , − } to all sites and write the partition functionof the Ising model on the honeycomb lattice as Z = (cid:88) s e β (cid:80) a,x ∈A s ( x ) s ( x +ˆ e a ) = 2 V (cosh β ) V/ Z r . (3)The reduced partition function Z r is given by Z r = 2 − V (cid:88) s (cid:89) a,x ∈A (1 + ts ( x ) s ( x + ˆ e a )) , t ≡ tanh β. (4)We read off the loop graph representation of Z r : • we multiply out the big product and for each term draw lines on the linkswhere the tss term is picked and leave empty links with factors one, • after averaging each term over s , nonzero contributions arise only from graphswhere each site is surrounded by an even number of lines, • as each site has only 3 neighbors only zero or two lines are allowed at sites, • therefore each nonzero contribution to Z r can be seen as configuration ofmultiple non-intersecting closed loops, • Z r is given by the sum over all different such loop gas configurations weightedwith a factor t per line segment.Symbolically we may write Z r = (cid:88) Λ t | Λ | , (5)where the sum runs over the loop gas configurations on the lattice and | Λ | meansthe total number of links making up all the closed loops contained in Λ.Up to here we have tacitly assumed periodic boundary conditions. For eitheror both of the directions f , f in which we close the torus, we may also takeantiperiodic boundary conditions. We designate the 4 possible cases that arise by4its ε , ε with ε i = 0 standing for periodic and ε i = 1 for antiperiodic. To detectif antiperiodicity leads to negative amplitudes we define winding numbers q i [Λ] = number of occupied links( x, i ) | x i = L i − (mod 2) . (6)Then, for generalized boundary conditions, the sign ( − ε q + ε q has to be in-cluded in the sum in (5). The generalized Ising partition function with dynamicalboundary conditions, in which we sum over the four cases with weights ρ ( ε ), reads Z Iρ = (cid:88) Λ t | Λ | Φ ρ [Λ] with Φ ρ [Λ] = (cid:88) ε ρ ( ε )( − ε q [Λ]+ ε q [Λ] . (7)This enlarged ensemble, including a sum over boundary conditions, will be foundto be a convenient starting point for the finite size equivalence to be derived. We consider a two component Grassmann-valued field ξ α ( x ) , α = 1 , S = 12 (cid:88) x ¯ ξ ( x ) ξ ( x ) − κ (cid:88) a,x ∈A ¯ ξ ( x ) P (ˆ e a ) ξ ( x + ˆ e a ) (8)written in the hopping parameter form. For a unit vector n , P ( n ) is the Wilsonprojector P ( n ) = 12 (1 − n µ γ µ ) . (9)The 2 × { γ µ , γ ν } = 2 δ µν . (10)Note that µ, ν = 0 , ξ the field ¯ ξ ( x ) is not independent but given by¯ ξ = ξ (cid:62) C (11)with the charge conjugation matrix C defined by γ (cid:62) µ = −C γ µ C − . (12)In any representation one may prove antisymmetry, C = −C (cid:62) , and our normaliza-tion will be C = +1. We note that (8) contains a simple sum over all links. Theyare unoriented – as in the Ising model – because¯ ξ ( x ) P (ˆ e a ) ξ ( x + ˆ e a ) = ¯ ξ ( x + ˆ e a ) P ( − ˆ e a ) ξ ( x ) (13)5olds due to (12).To study the so called naive continuum limit, we substitute ξ ( x + ˆ e a ) (cid:39) (1 + ˆ e a · ∂ ) ξ ( x ). Using the identities (cid:88) a ˆ e a = 0 , (cid:88) a ˆ e a,µ ˆ e a,ν = 32 δ µν (14)we find S (cid:39) κ (cid:88) x ∈A ¯ ξ ( γ µ ∂ µ + m ) ξ with m = 2 κ (2 / − κ ) . (15)Hence, by rescaling ξ we have a canonical Majorana fermion of mass m . A smallpositive mass (in lattice units) appears as the hopping parameter κ approachesthe critical value κ c = 2 / Z M = (cid:90) Dξ e − S = (cid:90) Dξ (cid:40)(cid:89) x (cid:18) −
12 ¯ ξξ (cid:19)(cid:41) (cid:89) a,x ∈A [1 + κ ¯ ξ ( x ) P (ˆ e a ) ξ ( x + ˆ e a )] . (16)The integration over two Grassmann components per site factorizes Dξ = (cid:81) x d ξ and the local measure d ξ is taken such that (cid:90) d ξ ξ α ¯ ξ β = δ αβ ⇒ (cid:90) d ξ ( − ) 12 ¯ ξξ = 1 . (17)Moreover, to arrive at the factorized form (16), the nilpotency of the Grassmannbilinears has been used, including the fact that P (ˆ e a ) are one-dimensional projec-tors.A moment of thought will reveal now, that upon executing the Grassmannintegrations site by site and using (17) the same loop gas structure arises as from(4). For each loop λ the successive projectors P appear multiplied up and anover-all factor w ( λ ) = − tr[ P ( n ) P ( n ) · · · P ( n N )] . (18)arises with n , n , . . . , n N being the unit vectors ± ˆ e a met on the links along theloop. The minus sign is the usual fermionic one: We order the commuting bilinearsfor the sequence of links in a loop schematically as ¯ ξP ξ ¯ ξP ξ · · · ¯ ξP ξ . Successiveinner pairs ξ ¯ ξ are at the same site and integrate to δ αβ . The first ¯ ξ and the last ξ similarly close the trace but come in the ‘wrong’ order, hence a factor −
1. Thegeometric factor w is evaluated in detail in appendix A. For periodic boundaryconditions in both directions, for example, we are led to Z M = (cid:88) Λ [ κ cos( π/ | Λ | (2 δ q [Λ] , δ q [Λ] , − . (19)6ome further explanations are in order: • Closed loops on the honeycomb lattice have as many 60 degree bends as theycontain links. This results in the same powers of κ per link and cos( π/
6) perbend (half-angle between n i and n i +1 , see (56)). • For loops not winding around the torus, each Fermi sign is paired with thespin minus from 2 π rotation (see appendix A). If Λ contains loops windingaround one or both periodic directions (nonzero q i ), the rotation is lackingand a minus sign is left.For the superposition of boundary conditions with weight η ( ε ) the Majorana par-tition function becomes Z Mη = (cid:88) Λ [ κ cos( π/ | Λ | Φ Mη [Λ] (20)with Φ Mη [Λ] = (2 δ q [Λ] , δ q [Λ] , − (cid:88) ε η ( ε )( − ε q [Λ]+ ε q [Λ] . (21)We see that this loop gas coincides with (7) if the following matching conditionshold tanh β = t = κ cos( π/
6) = √ κ (22)and Φ ρ [Λ] = Φ Mη [Λ] . (23)The Φ × depend on the graph Λ only through the winding numbers q i [Λ] and theirequality translates into a relation between ρ and η as follows. We may view ε i and q i as conjugate binary Fourier variables and invert ρ ( ε ) = 14 (cid:88) q Φ ρ ( q )( − ε q + ε q . (24)If we impose (23) this implies ρ ( ε ) = 14 (cid:88) q (2 δ q , δ q , − (cid:88) ε (cid:48) η ( ε (cid:48) )( − ε (cid:48) q + ε (cid:48) q = 2¯ η − η ( ε ) , ¯ η = 14 (cid:88) ε η ( ε )(25)or the particularly symmetric form ρ ( ε ) + η ( ε ) = 2 ¯ ρ = 2¯ η. (26) It is this pairing that in our language makes two dimensional fermions special with anessentially positive loop representation.
7y setting for example ρ ( ε ) = δ ε,ε (cid:48) we obtain for fixed boundary conditions (foreither the Ising or the Majorana system) Z I ( β, L i , ε ) + Z M ( κ, L i , ε ) = 12 (cid:88) ε Z I ( β, L i , ε ) = 12 (cid:88) ε Z M ( κ, L i , ε ) (27)with β, κ related by (22). Obviously, by taking derivatives, we may relate internalenergy, susceptibility, etc. We have checked our formulas by exact summation onsome small lattices. We remark that all Z × here are even in β or κ . This is shownby flipping the signs for all fields on one of the two sublattices.Combining (22) with (15) the fermion mass (close to crititicality) reads m = 2 t c − tt , t c = 1 √ , β c = 12 ln (cid:16) √ (cid:17) . (28)Needless to say, the critical coupling of the Ising model on a honeycomb latticehas been well-known before, see references in [3]. We see that here the phase with κ > κ c or m < The Ising model is clearly most popular on the square lattice that we discuss now.It will turn out, however, that the loop representation and the Majorana form isa bit more complicated.
The formulas analogous to those in section 2.2 are rather obvious so that wehere start immediately from the loop gas form which looks identical to (5) and(7) with just a re-definition of Λ. As before Λ is an arbitrary collection of line-carrying links such that an even number of lines touch any site of the torus. Thisallows for zero, two and, in contrast to the honeycomb lattice, also four linksaround a site. Because of the latter possibility, to be called crossings from hereon, the configuration does in general not decompose into simple disjoint loops.By some abuse of language it is however customary to still talk about a loop gasconfiguration. The definition (6) can also be taken over unchanged if we substitutethe orthogonal directions µ = 0 , i = 1 , q µ [Λ] now are the correspondingmodulo two winding numbers. Here the normalization of Z matters; we fix it by demanding Z × = 1 for β = κ = 0 . .2 Majorana Wilson fermion The loop gas of a single species of Majorana Wilson fermions on the square latticehas been discussed in [14]. Attempting to ‘draw’ the Ising loop gas we notice twoproblems: • crossings cannot occur with only two Grassmann components per site, •
90 degree bends come with weight factors cos( π/
4) = 1 / √ ξ µ ( x ) either of which has two spinor components. Now ξ has hoppingterms in the zero direction only and ξ implements the perpendicular hops. OurAnsatz for a bilinear action is S = (cid:88) x s ( ξ µ ( x )) + κ (cid:88) x,µ ¯ ξ µ ( x ) P (ˆ µ ) ξ µ ( x + ˆ µ ) (29)with unit vectors ˆ µ pointing in the positive µ direction. To determine the on-siteterm s we postulate (cid:90) d ξ e − s ξ ,α ¯ ξ ,β = (cid:90) d ξ e − s ξ ,α ¯ ξ ,β = δ αβ (30)to connect straight sections, and (cid:90) d ξ e − s ξ ,α ¯ ξ ,β ≡ (cid:90) d ξ e − s ξ ,α ¯ ξ ,β = √ δ αβ (31)to cancel the corner weights. A short calculation shows that this is uniquelyachieved by the quadratic form s = 12 ( ¯ ξ ξ + ¯ ξ ξ ) − √ ξ ξ . (32)A novelty arises for the empty sites. They now contribute minus signs to the loopamplitude because we find (cid:90) d ξ e − s = − . (33)The integral with all four Grassmann components is now determined and reads (cid:90) d ξ e − s ξ α ¯ ξ β ξ γ ¯ ξ δ = δ αβ δ γδ . (34)We find that at crossings the ‘vertical’ pair gets connected by spin contractionas well as the ‘horizontal’ one, see figure 2. Hence in this case we now do findseparate closed loops, with intersections (including self-intersections of the sameloop) allowed. The unusual sign of κ will turn out to be convenient later. ε now reads Z M ( κ, L µ , ε ) = (cid:90) Dξ e − S = (cid:88) Λ κ | Λ | ( − ε q [Λ]+ ε q [Λ] (2 δ q [Λ] , δ q [Λ] , − . (35)The sign (33) has been absorbed here into Dξ = (cid:81) x ( − d ξ ( x )) to adhere to thenormalization Z M (0 , L µ , ε ) = 1. This however implies now extra signs at all non-empty sites, i.e. connections as well as the crossings (34). In addition the hoppingterms come with factors ( − κ ). In this way for a graph without crossings, whichvisits the same number | Λ | of links and sites, these signs cancel. For each crossingthere first seems an extra minus left over. If a crossing is a self-intersection, thisextra sign cancels however with an extra 2 π rotation collected along the corre-sponding line, which, if it does not run around the torus, then contributes a totalplus sign. If the crossing is between separate loops, there always is an even numberof them.Figure 3: Examples of self-intersection and intersections of separate loops.We try to visualize this in figure 3. In this way the number of crossings n + [Λ] doesnot appear in the final weight, which is essential to be able to match the Ising loopgas. The condition for this is given bytanh β = κ (36)10or the square lattice. The relation (27) between partition functions holds un-changed. The factor in the last bracket in (35) has the same reason as discussedfor the honeycomb lattice. We finally re-emphasize that the loop configurationsΛ in (35) are the same as those described in section (3.1). The exact evaluationof the partition function (35) by performing the Gaussian Grassmann integral isdiscussed in appendix B.2.If both L and L are even, also the square lattice is bi-partite and partitionfunctions are even in β or κ respectively. If we also allow for odd lattice lengthsthe generalized relation Z M ( − κ, L µ , ε ) = Z M ( κ, L µ , ε ( L ) ) (37)can be proven with ε ( L ) µ = ε µ + L µ mod 2 , (38)i. e. a swap between periodic and antiperiodic for odd L µ directions.For an easier interpretation of the continuum limit of the action ( 29) we diag-onalize s by changing to new fields η and χξ = i √ κ ( η + χ ) , ξ = i √ κ ( η − χ ) . (39)Introducing the forward, backward and symmetric difference operators ∂ µ , ∂ ∗ µ and˜ ∂ µ , the action reads S = 12 (cid:88) x ¯ η (cid:18) m η + γ µ ˜ ∂ µ − ∂ µ ∂ ∗ µ (cid:19) η + 12 (cid:88) x ¯ χ (cid:18) m χ + γ µ ˜ ∂ µ − ∂ µ ∂ ∗ µ (cid:19) χ + (cid:88) x ¯ η (cid:18) γ ˜ ∂ − γ ˜ ∂ − ∂ ∂ ∗ + 12 ∂ ∂ ∗ (cid:19) χ (40)with m η = 2 κ (cid:104) √ − − κ (cid:105) , m χ = − κ (cid:104) √ κ (cid:105) . (41)The standard Ising critical point appears at m η = 0 ⇔ κ = κ c = tanh β c = √ − ⇒ β c = 12 ln (cid:16) √ (cid:17) (42)While the field η is critical here and acquires long range correlations we have a largenegative mass in lattice units m χ = − (cid:0) √ (cid:1) . Hence the coupling to this fieldonly contributes small lattice corrections to the Euclidean symmetric Majorana‘particles’ described by η . As before small positive m η ( κ < κ c ) corresponds tothe symmetric Ising phase with the ferromagnetic one being on the other side atnegative m η ( κ > κ c ). 11he two fields swap their roles at m χ = 0 ⇔ κ = κ (cid:48) c = tanh β (cid:48) c = −√ − ⇒ β (cid:48) c = − β c ± i π m η = − (cid:0) − √ (cid:1) in this case. In [11] Samuel has employed Grassmann variables to reproduce the low tempera-ture expansion of the Ising model (Bloch walls) in powers of z = exp( − β ). Asthe 2-dimensional model on the square lattice is self-dual, this coincides with thehigh temperature tanh β expansion (finite size effects are disregarded in [11]).In a first step we adapt Samuels notation for the Grassman fields by replacing( η h x , − η h o , η v x , − η v o ) → ( η , η , η , η ) , (44)and temporarily assume gamma matrices γ = τ , γ = τ in terms of Paulimatrices. The action (3.4) in [11] for the Ising case now translates to A = − z (cid:88) xµ ¯ η µ ( x ) P (ˆ0) η µ ( x + ˆ µ ) − (cid:88) xµ ¯ η µ η µ + (cid:88) x ¯ η (1 + C − ) η , (45)where spin summations are implicit again. To bring the hopping terms into thesame form as in (29), we perform a spinor rotation η → Rη with R = exp( iπτ / R † γ R = γ . In terms of these fields the total action now reads A = − z (cid:88) xµ ¯ η µ ( x ) P (ˆ µ ) η µ ( x + ˆ µ ) − (cid:88) x ¯ ηη + (cid:88) x ¯ η (1 + C − ) Rη . (46)Using now C = iτ and R = (1 + iτ ) / √ We have given an exact mapping between the Ising model and free MajoranaWilson fermions for finite honeycomb and square lattices. The critical point occursat vanishing mass m or, equivalently, the critical hopping parameter κ c . Althoughtrivial, we mention that the equivalence with free fermions immediately explainsthe value ν = 1 for the correlation length exponent as this scale is given by the We take z h = z v = z for simplicity. Z (2) symmetry corresponds to κ > κ c or negative mass.The question of extensions to three dimensions comes to mind. There is a latticewith coordination number three, the so-called hydrogen-peroxide lattice [15]. Theloop expansion of the three dimensional Majorana Wilson fermion worked out in[14] for the cubic lattice can be adapted to this case by just eliminating a fraction ofthe links. Then the graphs ‘drawn’ by the free fermions would indeed coincide withthose of the tanh β expansion of the Ising model, namely a gas of non-intersectingclosed loops. However, as explicitly worked out in [14], any such fermion graphcontaining non-planar loops comes with spin phase factors in the group Z (8) –related to cubic lattice rotations – which oscillate in an essential way. Therefore,the graph weights cannot be matched in this case. A Spin factor
The calculation in this appendix closely follows the arguments given in appendixB of [14], but is generalized here beyond the square lattice.We consider a single closed loop λ of length N on a 2 dimensional lattice to beassociated with a sequence of lattice unit vectors n i , i = 1 , , . . . , N which connectnearest neighbors and add to zero N (cid:88) i =1 n i = 0 . (47)For a given starting point x on the lattice, all points recursively given by x i = x i − + n i are nearest neighbor lattice sites until the loop closes at x N = x .The spin factor associated with such a loop λ is given by the traced product ofWilson projectors (18) where the additional Fermi minus is included. Note that w is invariant under cyclic changes of the n i and under inversions due to (12). Henceneither the starting point along the loop nor the chosen orientation matters for w ,which thus is a function of the unoriented loop.For the evaluation of w we note each pair of n i , n j can be rotated into eachother. Using the spinor representation this allows us to write P ( n i +1 ) = R − i P ( n i ) R i (48)with R i = exp (cid:16) α i γ γ (cid:17) with cos( α i ) = n i · n i +1 . (49)This is used, starting from the rightmost factors in the product, P ( n N − ) P ( n N ) = P ( n N − ) R − N − P ( n N − ) R N − = cos( α N − / P ( n N − ) R N − (50)13here we have used P exp( αγ γ ) P = cos( α ) P . Upon iteration we arrive at w ( λ ) = − (cid:40) N − (cid:89) i =1 cos( α i / (cid:41) tr[ P ( n ) R R · · · R N − ] . (51)If we define the additional rotation R N to achieve P ( n ) = R − N P ( n N ) R N , (52)then the total rotation R λ = R R · · · R N − R N (53)has the direction n N as a fixed point P ( n N ) = R − λ P ( n N ) R λ (54)which implies R λ = exp (cid:32) γ γ N (cid:88) i =1 α i (cid:33) = ± w ( λ ) = − R λ N (cid:89) i =1 cos( α i / . (56)For simple non-selfintersecting contractable closed loops in the plane the angles α i add up to 2 π and thus R λ = − π rotation that a spinor receives as it is transported once around the closed loop.Note that this does not occur for loops closing around the torus in one or bothdirections. Bends along the loops are suppressed by weight factors cos( π/
6) = √ / π/
4) = 1 / √ B Exact dispersion of Wilson fermions
B.1 Honeycomb lattice
We switch to sublattice Majorana fields χ ( x ) = ( χ A ( x ) , χ B ( x )) ≡ ( ξ ( x ) , ξ ( x + ˆ e )) (57)with the 4-component field χ attached to sublattice A . We write down a Fourierrepresentation χ ( x ) = 1 L L (cid:88) p ˜ χ ( p )e i ( p x + p x ) . (58)14ome straightforward algebra yields the action (8) in terms of ˜ χS = 12 L L (cid:88) p { ¯˜ χ ( − p ) ˜ χ ( p ) − κ [ ¯˜ χ A ( − p ) M + ˜ χ B ( p ) + ¯˜ χ B ( − p ) M − (cid:102) χ A ( p )] } (59)with M ± ( p ) = (cid:88) j =0 P ( ± ˆ e j )e ± ip j ( p ≡ . (60)The momenta to be summed over depend on the lattice size L i and (anti)periodicity ε i . A possible choice would be p i = (2 π/L i )( ε i / n i ) , n i = 0 , . . . , L i −
1. Thespecial values p i = 0 are allowed for ε i = 0 and p i = π occurs if L i + ε i iseven. If all components are of this type, p and − p are identical and so are ˜ χ ( p )and ˜ χ ( − p ). The Grassmann integral for such momenta then leads to a Pfaffianof the quadratic form defined by (59). The remaining momenta come in pairsassociated with independent Grassmann fields and contribute determinant factorsto the partition function, one per pair. We divide up the set of all momenta asfollows, B ( L i , ε i ) = B ( L i , ε i ) ∪ B + ( L i , ε i ) ∪ B − ( L i , ε i ) , (61)where B contains momenta made of components 0 or π only, while B + contains one member of each of the remaining pairs ± p with the partner momenta in B − .This implies for the cardinalities |B | + 2 |B + | = L L to hold.In any case we may perform half of the Gaussian integrations - say over ˜ χ B -to obtain the reduced action S A = 12 L L (cid:88) p (cid:8) ¯˜ χ A ( − p ) { − κ M + M − } ˜ χ A ( p ) (cid:9) (62)which leads to 2 × M + M − = a + ib µ γ µ + icγ γ (63)and find in a short calculation a = 34 { cos( p ) + cos( p ) + cos( p − p ) } , (64) b = 12 { sin( p ) f + sin( p ) f + sin( p − p )( f − f ) } , (65) c = √ {− sin( p ) + sin( p ) + sin( p − p ) } . (66) p i differing by multiples of 2 π are identified, of course. D ( p ) = (1 − κ a ) + κ ( b µ b µ − c ) . (67)For momenta in B the contributions b µ and c vanish and the Pfaffian is given by P ( p ) = 1 − κ a ( p µ ∈ { , π } ) . (68)Note that P = D holds here, but the root of D has to be taken such, that P is apolynomial in κ . In total we arrive at Z M ( κ, L i , ε i ) = (cid:89) p ∈B ( L i ,ε i ) P ( p ) (cid:89) p ∈B + ( L i ,ε i ) D ( p ) . (69)The four eigenvalues λ = 1 + κρ (for each p ) of the quadratic form (59), whichcontrol the fermion two-point function, are given by ρ = ± (cid:113) a ± (cid:112) c − b µ b µ (4 sign combinations) . (70)The spectrum of complex ρ values is shown in figure 4. The spectral radius 3 / κ c = 2 /
3. The spectrum is invariant under ρ → ρ ∗ as one can show M ∗± = C M ∓ C − , and under ρ → − ρ that is related to the κ → − κ symmetry. The arc of eigenvalues tangent to the dashed line in figure 4approximates the imaginary spectrum of the continuum Euclidean Dirac operator γ µ ∂ µ .Upon expanding for small p , p one finds that a and b µ b µ depend on the com-bination p + p − p p while c only contributes to higher orders. To arrive at theFourier form (58) we actually expand p = p (cid:101) f + p (cid:101) f in the basis dual to (2) whichis defined by f i · ˜ f j = δ ij . Then by elementary steps we find p = 49 ( p + p − p p ) . (71)Hence this combination is Euclidean invariant in our basis and so is the free energyand the dispersion in the continuum limit. B.2 Square lattice
The action in momentum space can be written as S = 12 L L (cid:88) p (cid:40)(cid:88) µ ¯˜ ξ µ ( − p )[1 + κ e − ip µ γ µ ] ˜ ξ µ ( p ) − √
2[ ¯˜ ξ ( − p ) ˜ ξ ( p ) + ¯˜ ξ ( − p ) ˜ ξ ( p )] (cid:41) (72)16 Figure 4: Fermion spectrum ρ for L = L = 64 and antiperiodic boundaryconditions.By similar manipulations as in the previous subsection we may work out the char-acteristic polynomial whose zeros are the eigenvalues of the quadratic form definedby (72), C ( λ ) = (cid:88) i =0 c i (1 − λ ) i , (73)with c = 1 c = 2 κ [cos( p ) + cos( p )] c = 2 κ [1 + 2 cos( p ) cos( p )] − c = 2 κ ( κ − p ) + cos( p )] c = κ + 4[1 − κ cos( p ) cos( p )]While closed expressions for the eigenvalues can be given now, we found themnot very illuminating and content ourselves with figure 5 for κ = ±√ −
1. Thedashed vertical lines are tangent to approximate continuum spectra of γ µ ∂ µ . One17hows invariance of the spectrum under λ → λ ∗ due to the property (e − ip µ γ µ ) ∗ = C e − ip µ γ µ C − . The symmetry λ → − λ holds for even L µ or requires a simultaneouschange in the boundary conditions as in (38). -1 0 1 2 3-0.3-0.2-0.100.10.20.3 -2 0 2 4-2-1012 Figure 5: Eigenvalues λ for L µ = 64 and antiperiodic boundary conditions for κ = √ − κ = −√ − Z M the momenta are divided as in the previous subsection andthe Grassmann integrations again lead to Pfaffians and determinants. The resultis Z M ( κ, L µ , ε µ ) = (cid:89) p ∈B ( L µ ,ε µ ) P ( p ) (cid:89) p ∈B + ( L µ ,ε µ ) D ( p ) (74)with D ( p ) = C (0) = (1 + κ ) + 2 κ ( κ − p ) + cos( p )] (75)and for p ∈ B we find the Pfaffians P ( p ) = − (1 + κ ) for p = (0 , κ for p = ( π, , (0 , π )2 − (1 − κ ) for p = ( π, π ) . (76)For small p µ we obviously find the dependence on the relativistic invariant p + p at leading order in all terms. References [1] L. Onsager,
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