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The Ising Model As Fermionic Field Theories

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The Ising Model As Fermionic Field Theories The Ising Model as Fermionic Field Theories Tuan Nguyen MIT Department of Physics (Dated: May 17, 2019) The high temperature picture of the Ising model is a sum over loops in 2D and surfaces in 3D. However, free loop/surface theories have an overcounting problem due to self-intersection of oriented objects. Thus, a fermionic structure can be realized to remove overcounting, allowing us to write a fermionic representation of the 2D and 3D Ising model. In 2D, this manifests as a free majorana field theory and in 3D, it is hypothesized to become a fermionic string theory. A standard method for deriving the 2D fermionic Ising theory is presented, with a discussion of the critical point of the fermionic representation coinciding with massless fermions. The conformal theory at criticality is then explored for the 2D case, and extended to the 3D case for the hypothetical fermionic string theory. I. INTRODUCTION take a particularly simple form Z −ηAη¯ The 2D Ising model was analytically solved by On- d¯η dη e = A (1) sager using a transfer matrix formalism, deriving his fa- and more generally mous result for the free energy of a rectangular 2D Ising Z model. [1] After Berezin developed the integration of an- d¯η dη e−η¯iAij ηj = det(A) (2) ticommuting Grassmann variables, he realized it could be used to simplify Onsager's transfer matrix method. [2] Since then, both the 2D and 3D Ising models near criticality have been successfuly described by fermionic III. HIGH TEMPERATURE EXPANSION OF 2D field theories. [3,4] ISING MODEL Modern study of critical exponents has shifted towards investigating the conformal field theory at criticality. [5] Plechko considers a generic anisotropic triangular lat- The Ising-fermion relationships in two and three dimen- tice, with hamiltonian [3] sions provide a powerful tool for calculating the critical X exponents by finding the scaling dimensions of the spin − βH[fσg] = βJ1σm;nσm+1;n and energy density operators. mn + βJ3σm;nσm+1;n+1 + βJ1σmnσm+1n (3) where σ = ±1. II. GRASSMANN VARIABLES Using the fact that σ2 = 1, we can write eβJiσaσb = cosh(βJ ) + sinh(βJ )σ σ To start, we introduce the calculus of Grassmann vari- i i a b ables. As anticommuting variables, two Grassmann vari- = cosh(βJi)[1 + tiσaσb] ables ηi and ηj must satisfy where ti = tanh(βJi) With the following substitutions ηiηj = −ηjηi (σ1; σ2; σ2)mn = (σm;n; σm+1;n; σm+1;n+1) the main consequence of which is that η2 = 0. Addition- ally, any function of Grassmann variables f(fηig) has the 2 α0 = 1 + t1t2t3 property f(fηig) = 0. [6] α = t + t t The fundamental difference between Grassmann vari- 1 1 2 3 ables and commuting variables is that Grassmannian in- α2 = t2 + t1t3 tegration acts identically to differentiation. [6] Namely α1 = t3 + t1t2 Z we can rewrite the partition function as [3] dη 1 = 0 Z = [2 cosh(βJ ) cosh(βJ ) cosh(βJ )]N Q Z 1 2 3 dη η = 1 Y Q = Spσ α0 + α1σ1σ2 + α2σ2σ3 + α3σ1σ3 mn (4) mn From anticommutivity, we see that exp[f(η)] = 1 + f(η). Thus, Gaussian integrals over Grassmann variables where Sp[··· ] is the normalized spin average. 2 IV. THE 2D ISING MODEL AS A FREE MAJORANA THEORY We now aim to include Grassmann variables and eliminate the spin variables. Consider the equality Z Z α p α d¯η dη 1 + (α σ σ + α σ σ + α σ σ )ηη¯ = d¯η dη 1 + η p1 σ 1 + χ(η +η ¯)σ 1 +η ¯p2 σ 1 1 2 2 2 3 3 1 3 χ 1 2 χ 3 with χ = α1α2/α3 Using (1), Q can be written as [7] YZ nh i o Q = Spσ d¯η dη exp α0 + α1σ1σ2 + α2σ2σ3 + α3σ1σ3 ηη¯ mn mn Z Y α1 p α2 = Sp d¯η dη eα0ηη¯ 1 + η p σ 1 + χ(η +η ¯)σ 1 +η ¯p σ σ χ 1 2 χ 3 mn Z Y α0ηη¯ (1) (2) (3) = Spσ d¯η dη e Bm;nBm+1;nBm+1;n+1 mn Z L h i Y α0ηmnη¯mn Y (3) (3) (2) (1) (2) (1) = d¯ηmn dηmn e Spσ BL;n :::B1;n × B1;nB1;n ::: BL;nBL;n mn n (i) where in the last line we use mirror factorization to reorder the Bmn factors. (3) (2) (1) We then spin average Bm;nBm;nBm;n for each (m; n) (which equals a commuting exponential), reducing Q to a fully Grassmannian expression. [7] Z Y nXh Q = d¯ηmn dηmn exp (α0ηm;nη¯m;n − α1ηm;nη¯m−1;n − α2ηm;nη¯m;n−1 − α3ηm;nη¯m−1;n−1) mn mn (5) io − α1ηm;nηm−1;n − α2η¯m;nη¯m;n−1 We introduce the discrete derivatives @1ηm;n = ηm;n − ηm−1;n and @2ηm;n = ηm;n − ηm;n−1, and define fermionic ¯ fields (m; n) = ηmn and (m; n) =η ¯mn. Thus, in the continuum limit we can write (5) as a fermionic field theory [3] Z Z ¯ n 2 h ¯ ¯ ¯ Q = D D exp d x m (x) (x) + λ1 (x)@1 (x) + λ2 (x)@2 (x) (6) ¯ ¯ ¯ − α3 (x)@1@2 (x) + α1 (x)@1 (x) + α2 (x)@2 (x) m = α0 − α1 − α2 − α3 λ1 = α1 + α3 & λ2 = α2 + α3 ¯ ¯ Ignoring the second order kinetic term α3 (x)@1@2 (x), by an appropriate linear transformation of (x) and (x) and a rescaling of x,(6) can be written as a free majorana theory [3] Z nZ h io Q = DΨDΨ¯ exp d2x mΨ(¯ x)Ψ(x) + Ψ(¯ x)@Ψ(¯ x) + Ψ(x)@¯Ψ(x) ( " T #) (7) Z Z Ψ¯ Ψ¯ = DΨDΨ¯ exp d2x iσ m + σ @ + σ @ Ψ 2 1 1 2 2 Ψ α − α − α − α m = 0 1 2 3 p 1=2 (2 α0α1α2α3) 1 @ = (@ + i@ ) 2 1 2 1 @¯ = (@ − i@ ) 2 1 2 3 V. CRITICALITY OF 2D ISING MODEL By Fourier transforming (5) and integrating with (2), one arrives at exactly the Onsager expression for a triangular lattice [3] ZZ 0 1 dq dq h 2 2 2 2 −βfQ = ln (α0 + α1 + α2 + α3) − 2(α0α1 − α2α3) cos(q) 2 2π 2π (8) 0 0 i − 2(α0α2 − α1α3) cos(q ) − 2(α0α3 − α1α2) cos(q + q ) Expanding to quadratic order in q and q0 and rescaling conformal theory by calculating the two point function the momenta, we get the singular part of the free energy of the energy momentum tensor T . as Looking at the action (6), we can easily read off the 1 Z two point functions for the fermion operators −βf = d2q ln α2 + q2 sing 8π2 (9) 1 1 const hΨ(z )Ψ(z )i = (12) = α2 ln 1 2 z − z 8π α2 1 2 α − α − α − α α = 0 1 2 3 ≡ m 1 p 1=2 Ψ(¯¯ z )Ψ(¯¯ z ) = (13) (2 α0α1α2α3) 1 2 z¯1 − z¯2 Thus, the mass m of the fermionic theory is propor- 1 tional to jT c − T j, and the fermions become massless at By N¨oether's theorem, T = − Ψ@Ψ. Therefore, criticality. 2 Wick's Theorem give us [8] 1h VI. 2D CONFORMAL FREE FERMION hT (z )T (z )i = hΨ(z )@Ψ(z )i h@Ψ(z )Ψ(z )i THEORY 1 2 4 1 2 1 2 i − hΨ(z1)Ψ(z2)i h@Ψ(z1)@Ψ(z2)i Because a field theory becomes conformally invariant at criticality, the Ising-fermion relationship gives us an- 1 = 4 other method of calculating the critical exponents of the 4(z1 − z2) Ising model as a conformal fermion theory. For two di- c = 4 mensions, it is convenient to work in complex coordinates 2(z1 − z2) z andz ¯ Consider the two point functions for the spin operator 1 which tells us the central charge c = σ and the energy density operator [5] 2 From knowledge of the central charge and the scaling 1 hσ σ i = (10) dimension of , we can identify the free fermion theory 0 r d−2+η 1 jrj as the minimal unitary model M , with ∆ = . (4;3) σ 8 Checking (10) and (11), we indeed get the correct critical 1 1 h0ri = (11) exponents η = and ν = 1 for the 2D Ising model (which jrjd=2−2/ν 4 we can calculate directly from the Onsager solution). In a conformally invariant theory, scaling the axes as x ! λx will also scale the operators as σ ! λ−∆σ σ and −∆ ! λ , where ∆O is the scaling dimension of an operator O. In the case of the 2D free fermion theory, VII. THE 3D ISING MODEL AS A FERMIONIC the energy density is proportional to the mass term ( / STRING THEORY ΨΨ).¯ Thus, a simple dimensional analysis reveals that ∆ = 1. [5] A similar analysis potentially allows us to probe the The spin operator σ is nonlocal in the fermion theory critical exponents for the 3D Ising model. Dotsenko [5], and thus requires analysis of operator product ex- shows that the 3D Ising model can be represented as pansions to directly calculate its scaling dimension. A a fermionic string theory. Consider placing four fermions α simpler method is to calculate the central charge of the µ (x) on every link of the lattice. The corresponding 4 fermionic theory is given by the partition function [4] strings, requiring us to take special care when a surface crosses over itself. [4] Z n X If a surface S has lines of self-intersection with total Z = D exp ¯(x) (x) length l, an additional sign factor of Φ[S] = (−1)l=a must links be included.
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