Grassmann techniques applied to classical spin systems

We review problems involving the use of Grassmann techniques in the field of classical spin systems in two dimensions. These techniques are useful to perform exact correspondences between classical spin Hamiltonians and field-theory fermionic actions. This contributes to a better understanding of critical behavior of these models in term of non-quadratic effective actions which can been seen as an extension of the free fermion Ising model. Within this method, identification of bare masses allows for an accurate estimation of critical points or lines and which is supported by Monte-Carlo results and diagrammatic techniques.


Introduction
Classical an quantum spin models such as Ising model play an important role in the field of statistical physics as they allow for an accurate understanding of critical phenomena in general. Many techniques [ 1,2] were developed in order to deal with the difficulty of estimating the partition function and other thermodynamical properties in the critical region in dimension more than one. An exact mathematical description of the two-dimensional (2D) Ising model relies on the Jordan-Wigner transformation [ 3] which maps the product of Boltzmann weights onto a fermionic action of free fermions with a mass vanishing at the second order critical temperature given in dimensionless units T c = 2/ ln(1+ √ 2) ≃ 2. 2691851. Also a method based on the correspondence between the Ising model and dimer problems [ 4] uses the notion of Pfaffians, which are directly connected to integrals over Grassmannian objects.
Both fermions and Grassmann variables are therefore closely tied to the Ising model.
A direct introduction of Grassmann variables as an alternative tool to solve the Ising model was done long ago in the 80's by Bugrij [ 5] and Plechko [ 6] (see also a later discussion by Nojima [ 7]). It is based on a simple integral representation of the individual Boltzmann weights and which has the property to decouple the spins.
The price to pay is a non-commutativity of terms arising from this representation.
In order to deal with this particular representation, Bugrij used two families of Grassmann variables which commute with each other, then identified the resulting functional integral of the partition function with a determinant. From another point of view, Plechko introduced symmetries which order the non-commuting quantities so that the sum over the spins can be performed exactly. In this paper we review the process of how to generalize Plechko's method for Blume-Capel model [ 8], which is the simplest model beyond Ising, to spin-S Ising models and how to construct an exact fermionic action for each case. This would provide a natural extension of the exact fermionic quadratic action found for the spin S = 1/2 Ising model. In particular, we will build on previous work on the Blume-Capel (BC) case S = 1 [ 8] where a line of second-order critical points is terminated by a tricritical point. This is the next case beyond the Ising model and which possesses a rich critical behavior.
This model was used to qualitatively explain the phase transition in a mixture of He 3 -He 4 adsorbed on a 2D surface [ 9]. Below a concentration of 67% in He 3 , the mixture undergoes a λ transition and the two components separate through a first order phase transition with only He 4 superfluid. On a 2D lattice, He atoms are represented by a spin-like variable, according to the following rule: an He 3 atom is associated to the value 0, whereas a He 4 is represented by a classical Ising spin taking the values ±1. Within this framework, all the lattice sites are occupied either by an He 3 or He 4 atom. In addition to nearest-neighbor interactions, the energy includes a term ∆ 0 mn S 2 mn , with S 2 mn = 0, 1, to take into account a possible change in vacancies number. ∆ 0 can be viewed as a chemical potential for vacancies, or as a parameter of crystal field in a magnetic interpretation of atomic physics. It would be in particular interesting to have a fermionic description of the BC model in order to obtain more information about the kind of interaction fermions living on the 2D lattice have compare to the Ising free fermion case.
So one of the main question is how to obtain a generic fermionic action for a spin-S model and what does this method teaches us for the BC model in particular.
We explain in the next section the main ideas of this method.
2. Description of the fermionization for general classical spin-S models Let us consider the following Hamiltonian on a 2D lattice of size L × L where J is the Ising coupling constant and ∆ 0 the splitting crystal field or represents a chemical potential in the BC case. In particular for ∆ 0 large and positive, it favors small spin values. This crystal field can be replaced by any potential V (S 2 mn ) depending on the square of the local spin. Spins S mn take 2S + 1 values with S mn = −S, −S + 1, · · · , S. The partition function is the sum over all possible spin configurations Z = Tr exp(−βH). Z contains products of the Boltzmann weights exp(KS mn S m+1n ) (where S mn and S m+1n are neighboring spins and K = J/k B T ) which take q + 1 = S(S + 1) + 1 different values if S is an integer, and q + 1 = (S + 1/2)(S + 3/2) values if S is half-integer. Since there are q + 1 possible values for each Boltzmann weight, we can project each of them onto a polynomial function of degree q in the variable S mn S m+1n : where the q + 1 constants u k are determined by solving the linear system of q + 1 equations satisfied by the above relation. To see on specific examples how it works, let us consider first the Ising case, S = 1/2. Since S is half integer, we have q = 1.
Therefore exp(KS mn S m+1n ) = ch(K/4) + 4 sh(K/4)S mn S m+1n , u 0 = ch(K/4), u 1 = 4 sh(K/4). (3) In the Blume-Capel model, since S is integer, we have q = 2 and it is straightforward For S integer the first coefficient u 0 is always unity, and from equation (2) we can write u k = u 0 We will set for convenience in the following u k≥q+1 = 0 since the polynoms are all finite. Our purpose is to transform the partition function Z which is a sum over spin variables into a multiple integral over Grassmann variables. For this let us introduce q pairs of Grassmann variables [ 8,6,10] (a α mn ,ā α mn ) on each site for the horizontal direction and q additional pairs (b α mn ,b α mn ) for the vertical direction. Here α takes the values 1, . . . q. There are therefore 4q Grassmann variables at each site of the lattice.
In particular the Ising model is represented by two pairs of Grassmann variables per site which can afterward be reduced to only one pair [ 6]. For each couple of terms appearing in the partition function, we introduce the following integral representa- From the last expression, we introduce the link factors A α mn = 1 + a α mn S mn ,Ā α m+1n = 1 + x αā α mn S m+1n , B α mn = 1 + b α mn S mn , andB α mn+1 = 1 + x αb α mn S mn+1 , so that the partition function can be written as can be moved freely with the other terms since they correspond to commutative scalars after integration. In particular, it is convenient to rearrange the products over α in order to put together the link factors of different α with the same site indices (m, n) using the mirror ordering symmetry introduced in Plechko's method [ 6] in the context of the 2D Ising model, and which is still relevant in the spin-S case: where the arrows indicate that the product is ordered, i.e. increasing label α in the first product from left to the right and in the second one from right to the left. For convenience, we will use the notation for objects on the horizontal links and for the ones on vertical links. Then the partition function can be rewritten as At this stage of the algebra, we use the mirror and associative symmetries which were used for solving the Ising model [ 6,10] and which are still valid here to rearrange the operators O and P. In principle boundary terms should be treated separately in order to obtain the exact finite size partition function depending on boundary conditions [ 6] but they are not relevant in the thermodynamical limit L → ∞ we are interested in here. Here we consider instead the simple case of free boundary conditions, and we obtain the exact expression after rearrangement of the O and P operators: Now, from this expression, the spins can individually be summed up from S Ln to S 1n for any given n. We will need to introduce the following weights W mn which include all the dependence on the individual spins S mn where we have defined the following 4q sets of Grassmann variables c α mn in the following order: The sum over S mn = ±1 in equation (12) can be performed by noticing that only products involving an even number of S mn give a non-zero contribution. We also define the scalars (we remind that ∆ = −β∆ 0 ) and the ordered products with q (4q) mn = c 1 mn · · · c 4q mn the term of highest degree in Grassmann variables. Using these quantities, it is easy to show that the partial Boltzmann weights (12) are given by Then the fermionic representation of the partition function can be expressed as a multiple integral over Grassmannian variables only For small values of S, the weights W mn can be exponentiated so that a fermionic action can be defined. Indeed, since the first term of W mn is the pure scalar α 0 and the others products of pure commutating Grassmannian objects, it is tempting to exponentiate the sum (16) to obtain directly a fermionic action. This comes from the simple observation that for any Grassmann variable a, we have 1 + a = e a . Of course, the exponentiation of the sum (16) is more complicate. For example, for commuting objets a and b such as the q k mn s, we have 1 + a + b = exp(1 + a + b − ab). In this case the order of the polynomial object inside the exponential is bigger than in the original sum since the extra counter-term ab is necessary for the identity to be exact. These weights are moreover connected by nearest-neighbor interactions hidden in the variables c α mn . In the case of the Ising model, where the exponentiation can be done quite easily, the argument of the exponential is purely quadratic in the c α mn 's and therefore the partition function can be integrated out with the use of a determinant or a Bogoliubov transformation in the Fourier space. Moreover, the 4q = 4 Grassmann variables in this case can be reduced to 2 by partial integration of non relevant variables. In the BC model, the argument is a polynomial of degree 8 in Grassmann variables since there are 8 independent variables (4q = 8). In general we expect naturally the argument to be at most a polynomial of degree 4q in these variables, which can be reduced or not by partial integrations. Except for the case q = 1 however the partition function can not be expressed as a determinant, so that a full exact solution of the partition function can not be found this way. If the action is quadratic, the use of the following Gaussian integral [ 12], defined on Grassmann allows us to express the partition function as a determinant. Quadratic fermionic form in the exponential (18) is typically called action for a free-field theory. When the action is non-quadratic, the integral is not Gaussian and can not be expressed as a determinant, which yields in principle to a non integrable theory. However, physical information such as bare masses (see last section) can be extracted from these non-quadratic actions which represent generic theories of interacting fermions. Another simpler way of obtaining this BC fermionic action is possible [ 11] using the Z 2 symmetry of the spin variables S mn . Indeed the partition function is invariant if we perform the gauge transformation S mn → σ mn S mn with σ mn = ±1. In this case it is possible to simplify the process of the previous method and write an action containing only 2 pairs of variables per site instead of 4:

Fermionic action of the Blume-Capel model
where we have introduced the following constants: The fermionic integral (19) is the exact expression even for a finite lattice, provided we assume free boundary conditions for both spins and fermions. The other possible form for the partition function with periodic boundary conditions in both direction can be written in a similar way as the Ising model on a torus [ 6,13,14]. The partition function would be the sum of 4 fermionic integrals with periodic-antiperiodic boundary conditions for the fermions. In the expression (19), we can recognize the sum of the Ising action, which here appears as the Gaussian part of the total action [ 6,10]: and a non-quadratic interaction part, which is a polynomial of degree 8 in Grassmann variables (which can be seen if we expand the exponential inside the action): This allows us to rewrite the partition function as a fermionic field-theory in a compact form The BC model differs from the Ising model by the interaction term in the action (21) which is not quadratic. Therefore the BC model is not solvable in the sense of free fermions as a determinant of some matrix, unlike the 2D Ising model.

Mixed representation of the BC model
The coupling of Grassmann variables in equation (21) prevents us to integrate further and reduce the number of variables per site unlike the Ising model where the minimal action contains one pair only [ 18,7]. The minimal action of the Ising model admits an interpretation in term of Dirac representation of free fermions which become massless at the critical point. In a previous work we were able to reduce the number of Grassmann variables by partially introducing hard core bosons in the previous action, since terms such as η mn = a mnāmn or τ mn = b mnbmn may have an interpretation of local densities or occupation numbers. Variables η mn and τ mn are commuting and nilpotent, η 2 mn = τ 2 mn = 0. We can replace the quantities depending on a mnāmn and b mnbmn , especially in the interaction part, by their respective nilpotent variables, using, for this task, a general definition of Dirac distribution for any polynomial function f of a mnāmn or b mnbmn [ 11]: A natural definition [ 15] of the integrals involving commuting nilpotent variables is to impose the following rules (and similar forη mn ,τ mn ): dη mn (1, η mn ) = (0, 1) , dτ mn (1, τ mn ) = (0, 1) .
This change of variables allows us now to integrate over the a mn 's and b mn 's in the new action. One advantage is that after this operation there are only two fermionic variables per site, although two additional pairs of bosonic variables have been introduced. In fact we can integrate over one pair of bosonic variables [ 11], for examplē η mn ,τ mn , using the help of integration rules and Dirac function given by (23). At the end, it remains a mixed action made of one pair per site of fermionic and bosonic variables respectively, with an interaction between fermions and bosons. A convenient replacement of the variablesā mn by c mn andb mn by −c mn in the final integral leads us to isolate the minimal local action for the pure Ising model [ 16,17] with one pair of Grassmann variables per site: and the interaction part with the quantities The Ising part is the same action that results from the integration over a mn , b mn from the original Ising case. The introduction of nilpotent variables was necessary to achieve this partial extraction of the Ising contribution. The physical interpretation of the previous mixed representation is that it can be possible to describe the BC model with fermionic variables for the states S = ±1 and bosonic ones for states S = 0. In the limit ∆ 0 → −∞, the system is completely described in terms of fermions (Ising sector), while when ∆ 0 is increasing fermions and bosons begin to interact. Beyond a critical value of ∆ 0 , fermions form bosonic pairs and in the limit ∆ 0 → +∞, all fermions condense into bosons, leading to a purely bosonic system.
This view should be supported by further analysis.

Corrections to the effective action in the continuum limit
The integration of the previous action (26) over variables (η mn , τ mn ) can be performed perturbatively, as part of an expansion in the low momentum limit. We will define formally the derivatives of Grassmann variables [ 18], ∂ x c mn = c mn − c m−1n and ∂ y c mn = c mn − c mn−1 in the limit of large L. In this limit and in the Fourier space, the high order derivatives account in the action for a small contribution in momenta k = 2π(m, n)/L, with m, n ≪ L positive integers. We would like to obtain in this limit the non trivial part of the non-quadratic interaction in term of variables c mn ,c mn only. The procedure is described in reference [ 11] and based partially on substitution rules such as η mn τ mn → c mncmn , η mn →q mn , τ mn → q mn .
There are unfortunately more complicate terms in the resulting action than by using the substitution rules alone, such as but they can be discarded in the approximation scheme above in the sense they correspond to corrective terms higher than quartic polynomials or quantities of the The Ising part of the action can be written as with m BC = 1 − 2t − t 2 + g 0 and the quartic term can be express as with the potential We notice that the bare mass of the theory is given by

Critical behavior of the BC model: diagrammatic expansion
In this section, we further analyze the influence of the interaction potential V k,k ′ on the renormalized mass, in particular the shift of the critical temperature which was in reference [ 11] assumed to be given by the point where the bare mass m BC vanishes. We would like in particular to apply diagrammatic expansion of the effective action (30). For this, it is useful to express the Ising part of the action in term of Nambu-Gorkov representation of the fermions [ 23,24], using the two-component Formally, the Green functions can be defined within this representation by 2 × 2 where τ 3 is the Pauli matrix are from Ref. [ 20], the magenta triangles from Ref. [ 21], and the green squares from Ref. [ 22] (see also Table 1 for other numerical values at ∆ 0 = 0).
The unperturbed part of the Green functionĜ 0 is evaluted using the elements of the non diagonal but quadratic Ising action (32): where the momentum-dependent mass is defined by m k = m BC + it(t + 1)(k x − k y ).
The inverse is given byĜ Figure 2. Representation of the interaction part of the potential. The blob represents the potential interaction with incoming vector k ′′ and outgoing k ′′ − q.
With this representation and the unperturbed Green function, we can write the Ising part as where the set S contains half the momenta of the Brillouin zone. It is defined by the rule that if k ∈ S, then −k does not belong to S. The interaction part can be put, after some algebra, into the following form where the sum is not restricted to the ensemble S. We define the potential matrix The two diagonal elements of this matrix are not equal since V k,k ′ is not symmetric by exchange of the two momenta k and k ′ except when k = k ′ .
We now perform a diagrammatic expansion with respect with g 0 of the perturbed Green functionĜ(k) which will allow us to compute the corrections to the mass,  Figure 3. The four diagrams appearing at the lowest order in g 0 . Only diagram (b) contributes to the mass in the low momentum limit k → 0.
The first terms contributing to the self-energy are given in figure 3: The renormalized mass m R is given in the limit when k is zero by the diagonal components of the inverse-propagator Γ 11 (0) = Γ 22 (0). In this limit, only one diagram is not vanishing, which corresponds to the diagram (b) of figure (3): The last sum over q can be evaluated in the continuous limit L → ∞. Setting q = 2π( m L , n L ), we define for large L the two following integrals

MC-JYF
These two quantities are finite when m BC vanishes. To see why, we can consider polar coordinates q x = q cos θ and q y = q sin θ, so that, near the origin q = 0 the second integral for example behaves like . (49) When m BC = 0, this integral is finite since there is no singularity in the denominator. Here the renormalization only concerns the total coefficient of m BC and this does not affect the critical line location: can be generalized for any value of the spin S, in particular for higher values of S.
The equations obtained in section 2, equations (16) and (17), are general for they represent the fermionization of any spins-S model.
The construction of the fermionic action is however not an easy task, unlike the BC model which is a simpler case, but we expect to be able to extract a bare mass associated to non kinetic terms, or term involving derivatives with respect with space variables. At first approximation, we assume that the partition function and the free energy are singular in the low momentum limit when this bare mass vanishes. In the continuum limit, the c's coefficients defined by relations (13)  Also a term a α mnā α mn can be combined with b α mn (x αb α mn ) to give the same contribution. Since the q (2k) are ordered, there are also signs to take into account and coming from the fact the variables c α mn have to be moved in the correct order before integration. We obtain after some algebra the general relation where we have define the following quantities with initial condition R 0 = u 2 0 , and σ(k, l) = 1 if k and l are both even, and σ(k, l) = −1 otherwise. We can apply this result to different cases to check the validity of this relation. For the Ising model (S = 1/2, q = 1) u 0 = ch(K/4) and u 1 = 4sh(K/4), we obtain m 1/2 = 2 cosh(∆/4)(u 2 0 − u 0 u 1 /2 − u 2 1 /16), or which vanishes at the Ising critical temperature T c ≃ 0.567 296 or with the normalization t c ≡ T c /S 2 = 2.269 185, which is independent, as expected, of ∆ 0 . For the Blume-Capel model (S = 1, q = 2) we have m 1 = 1 + 2 exp(∆)(1 − 2u 1 + 2u 2 − u 2 1 − 2u 1 u 2 + u 2 2 ), or more explicitly  [ 25,29] This mass is directly proportionnal to the mass m BC found in the previous section.
Indeed, we have the relation and therefore both masses vanish on the same line of critical points. The coefficient g 0 comes from a global rescaling of the Grassmann variables in the original weights W mn which leads to the coefficient g −L 2 0 in the BC function partition (19) and (22), instead of the coefficient (u 0 ) 2L 2 = 1 in front of (17). For ∆ 0 = 0 we find in particular that t c = 2/arcsinh(3/2) ≃ 1.673 971 (see table 1).
The other masses are deduced by iteration of formula (51) For general spin S, we can extend the previous results to the formula which is bounded by t c = 2/9 log(1 + and in particular for ∆ 0 = 0, we have the following expansion for large S m S (t, ∆ 0 = 0) ≃ a(t)S − 1 + 4 3tS + 8 21t 3 S 6 + · · · (60) with a(t) = 2 − 2 2/t √ 2/t 0 sh(x 2 ) dx. We observe that the rescaled mass m S /S vanishes in this case when t c = 0.925 148, in good agreement with numerical works for this model [ 25,29], and it is worth noting that equation (59) also possesses a non trivial solution at t = 0 which is simply given by ∆ 0 = 4/ √ 3 = 2.309 401. This value is different from the value 2 expected for all finite S models [ 8]. It can be suggested that there also exists a tricritical point before this non physical value is reached.

Conclusion
In this review paper, we have presented a method which tries to operate a correspondence between classical spin models and fermionic systems. We have extended Plecho's method [ 6,10] based on Ising model to generalized spin-S systems. The method is based on the projection onto q polynomial components, q depending specifically on the value of spin S, of Boltzmann local weights given by equation (2).
Then the introduction of 2q pairs of Grassmann variables per site and the use of special symmetries such as mirror and associative symmetries in 2D for Grassmannian objects allows us to perform exactly the sum over the spin variables. This gives a representation of spin-S models in term of fermionic multiple integrals (17). Effective actions can in principle be deduced from this representation. We have shown that such action can be built exactly for the Blume Capel model S = 1 (30) and the bare mass (35) gives accurate description of the second-order critical line. We have seen that there is no shift of this mass due to the effect of quartic potential of the effective theory at the lowest order expansion in the coupling parameter g 0 , implying that corrections to the critical temperature may be indeed small. This quartic potential is however responsible for the presence of a tricritical point, rendering the second order line instable by changing the sign of the stiffness coefficient or making the free-fermion spectrum itself instable. For general spin-S model, the bare mass can also be generalized and calculated directly in the low momentum limit (51) without knowing the full effective action, and still gives accurate description of second-order critical points even in the limit of the continuous Ising model.