Constraints of kinematic bosonization in two and higher dimensions

Contrary to the common wisdom, local bosonizations of fermionic systems exist in higher dimensions. Interestingly, resulting bosonic variables must satisfy local constraints of a gauge type. They effectively replace long distance exchange interactions. In this work we study in detail the properties of such a system which was proposed a long time ago. In particular, dependence of the constraints on lattice geometry and fermion multiplicity is further elaborated and is now classified for all two dimensional, rectangular lattices with arbitrary sizes. For few small systems the constraints are solved analytically and the complete spectra of reduced spin hamiltonias are shown to agree with the original fermionic ones. The equivalence is also extended to fermions in an external Wegner $Z_2$ field. It is also illustrated by an explicit calculation for a particular configuration of Wegner variables.


Introduction
Relations between fermionic and spin degrees of freedom is an old subject [1,2], however it still attracts a fair amount of interest. There is a variety of motivations for such studies. Eliminating Grassmannian variables from the classical (and quantum) description of fermionic field theories is only one example [3,4]. Another one is provided by the relatively recent discovery of intriguing gauge structures in the equivalent spin systems [5,6]. Finally, intensive studies of quantum computers and "quantum algorithms" stimulate some progress also in the hamiltonian formulation [7].
The connection is well understood, and exploited, in one space dimension. However its extension to higher dimensions leads to complicated, non-local interactions and seems to be not practical.
In this paper we revisit the old proposal [4,8] where the equivalent spins interact locally and satisfy local constraints. Effectvely these constraints take care of the non-locality of the fermionic description in arbitrary space dimensions.
Let us begin with a simple fermionic Hamiltonian on a one dimensional lattice Its equivalent in terms of spin variables reads H = 1 2 n σ 1 (n)σ 2 (n + 1) − σ 2 (n)σ 1 (n + 1) , where Pauli matrices σ k (n) commute between different sites labelled by n. The standard way to prove the above equivalence is via the Jordan-Wigner transformation [1]. However in higher dimensions this leads to non-local spin-spin interactions. We therefore adopt another route, which applies also to multidimensional systems.
To this end, introduce the following Clifford variables and rewrite the fermionic Hamiltonian (1) in terms of link (or hopping) operators S(n) = iX(n)X(n + 1),S(n) = iY (n)Y (n + 1).
That is, they basically commute unless the two links share a common vertex. Now, the crucial point is that the same algebra is obeyed by link operators in the following spin representation S(n) = σ 1 (n)σ 2 (n + 1),S(n) = −σ 2 (n)σ 1 (n + 1), which gives immediately (2).
In this way we have changed fermionic and spin variables without invoking the Jordan-Wigner transformation. This lends itself an interesting possibility that similar construction exists in higher dimensions.
Before concluding this Section we note that at the heart of the equivalence claim is an expectation that if the two representations lead to the same algebra, the systems are in fact equivalent. This is the common basis of many studies in this area [6], and has been recently carefully reconsidered in detail in Ref. [9].
Second, above arguments work also for finite systems upon suitable modification. In fact the full discussion of finite size lattices, various boundary conditions and emerging constraints, reveals an interesting structure and is one of the goals of the present paper.
In the next Section we remind the equivalent spin model in two dimensions and discuss conditions of equivalence for various lattice sizes. In Sect.3 the necessary and interesting reduction of Hilbert spaces is explicitly demonstrated and the spectra of both hamiltonians are compared for few small systems. Finally in Sect.4 a simple and physical interpretation of all, possible in our construction, constraints is proposed and tested.

The equivalent spin model in two dimensions
Generalization of the above idea to two and higher space dimensions is known for a long time [4]. In two dimensions the fermionic Hamiltonian can be again rewritten in terms of two types of hopping operators labelled by links of a two dimensional lattice. Their definitions and algebra are a straightforward generalization from the one dimensional case. In short: the hopping operators commute unless corresponding links have one common site. The difference is that now four, instead of two anticommuting link operators, are attached to each lattice site. Consequently, one needs bigger matrices to satisfy the corresponding algebra in higher dimensions.
In two dimensions we choose the Euclidean Dirac matrices and set (c.f. Fig.1) It is a straightforward exercise to show that the two dimensional extension of the algebra (5) remains intact. Hence our hamiltonian in the spin representation reads Generalization to higher dimensions is simple. One just needs representations of higher Clifford algebra, e.g. by larger Dirac matrices. In d dimensions they admit 2d anticommuting ones which corresponds to the 2d links meeting at one lattice site. Consequently, we have a viable candidate for a local bosonic system equivalent to free fermions in arbitrary dimensions.
The story is not over however, since the representation (7) is redundant with respect to the fermionic one. In fact, in two space dimensions, it doubles the number of degrees of freedom per lattice site compared to the original fermionic system. Evidently one needs additional constraints for above spins to render the exact correspondence.
Such constraints are provided by the plaquette operators P n (from now on n is a two dimensional index n = (n x , n y )). If we denote by C n an elementary plaquette labelled by its lower-left corner, say, then P n = l∈Cn S(l).
These operators are identically 1 in the fermionic representations, while only P 2 n = 1 in the spin representation. Hence imposing all constraints should provide necessary reduction of the Hilbert space. Details of how it works depend on sizes of lattices, boundary conditions and other specifications. It was shown for few simple observables, that such reduction indeed works in two and three dimensions [4] . Later this problem has been revisited in [SZ, BR].
The advent of symbolic computations allows for further, also analytic, understanding of this and related questions. This is the aim of present work as continued in the next sections.

The constraints
The precise form of constraints required to satisfy the above fermion-spin equivalence depends on a geometry of a lattice. Consider two dimensional, cubic lattices, possibly with different sizes (e.g. L x and L y in each direction). Periodic or antiperiodic boundary conditions are used. Different periodicity conditions for fermions and equivalent spins are allowed.
and similarly for the other direction.
We seek to impose N = L x L y , L x , L y 3, constraints (10) to eliminate abundant degrees of freedom. However not all of them are independent. For example, in the spin representation plaquette operators satisfy the identity n P n = 1, which leaves only N − 1 independent constraints.
In addition, on finite periodic lattices, one can also construct "Polyakov line" operators In fermionic representation they are just pure numbers sensitive to the boundary conditions, while in spin representation their squares are unity, similarly to the plaquette operators. Hence again they provide additional projectors. In principle there are L x + L y line operators, but in fact they can be shifted perpendicularly by multiplication with appropriate rows/columns of plaquette operators. Therefore, altogether there are only two more candidates for independent projectors.
It turns out that even this set of N − 1 plaquettes and two line projectors is overcomplete. The additional structure is revealed once we consider the operator of fermion number at each site (i.e. the fermion density).
Since Hamiltonian (6) is moving fermions between neighbouring sites only, the total number of fermions, N = n N(n), is conserved, but obviously their density N(n) is not.
In the spin representation the number operator is related to the Γ 5 matrix where η = ±1 represents the freedom of defining a fermion-empty and a fermion-occupied state in the spin representation. As in the fermionic representation N is conserved, while the number densities N(n) are not. On the other hand, the plaquette and line operators do commute with the local densities. This will be exploited below when we diagonalize constraints.
Given these definitions, it is easy to show the equality which implies the additional relation between Polyakov line projectors.
Summarizing, the complete set of independent projectors on a two dimensional, finite lattice consists of ones associated with N − 1 plaquettes and one Polyakov line. Hence the reduced Hilbert space is indeed 2 N dimensional corresponding to N fermionic degrees of freedom.
However such a reduction occurs only in certain sectors, labelled by fermionic multiplicity. The required condition follows immediately from (16) upon comparing RHS with the correponding expression in the fermionic representation Above discussion is valid quantitatively only for (odd)x(odd) lattices. Nevertheless, it illustrates generically the interplay between various constraints, lattice geometry and fermion multiplicity. For other lattice sizes the explicit forms of constraints are slightly different and will be discussed below in detail. In all four cases, however, the final number of independent constraints turns out to be 2 N leading to the correct "fermionic" dimension of the restricted spin space.
Of course consistency of dimensions of both spaces is only a necessary condition for the equivalence. The next step is to actually solve the constraints and to show that the spin hamiltonian (8) in the reduced Hilbert space is indeed equivalent to the fermionic one (6). Although the explicit solution for arbitrary lattice sizes still remains a challenge, such a program can be carried through for a few small lattices thanks to the rapid growth of computing power and symbolic calculations. It is shown below how this works in practice.

Some explicit examples
The complete Hilbert space of our system of spins on L x xL y lattice has 4 N dimensions, N = L x L y . States are represented by configurations of N Dirac indices, i n = 1, . . . , 4 with n = 1, . . . , N labelling sites of a lattice. All operators are constructed from tensor products of N -fold four dimensional gamma matrices and the unity 1 . In principle they require (4 N ) 2 elements of computer storage, however in general they are sparse matrices and take only O(4 N ) memory size. Still, the memory requirement is the main limitation for such a direct approach and restricts aviable sizes to ca. N ∼ 16. To reduce further the memory demand, we split the whole Hilbert space into N + 1 sectors of the fixed fermion multiplicity p = 0, 1, . . . , N . In the fermionic representation the total number of fermions is obviously conserved. The same is true in our spin representation. Namely, the corresponding number operator commutes with the hamiltonian (8). Moreover, it also commutes with all plaquette and line operators. This allows to carry out the analysis of constraints in the sectors of fixed N seperately for each p. Choosing the sector of fixed multiplicity amounts to restricting the full basis to states (18) with N − p indices in the "vacuum class" a, and p ones in the "one-excitation class" b. With our choice of gamma matrices and η = −1, a = (2, 3) and b = (1, 4). In practical terms we will now be dealing with the N + 1 fixed multiplicity sectors of the full H seperately, the size of each sector being before and after imposing constraints in the spin representation.
Moreover, constraints operators commute not only with the total multiplicity N but also with each of the individual densities N(n). This allows to further split the problem by performing the reduction of Hilbert space in each sub-sector of fixed p and fixed positions of p spin excitations r 1 , r 2 , . . . , r p , or just fermionic coordinates, in the configuration space. Now the reduction looks like 2 N −→ 1.
The To proceed, we define the projection operators associated with all plaquettes and two Polyakov lines and calculate their matrix representations, at fixed total multiplicity, p. For illustration we explicitly display below traces of successive products of all relevant projectors on 3 × 3 and 4 × 4 lattices.   (Table 2) the reduction was performed in sectors of fixed fermion multiplicity p and proceeds according to the scheme (20). Indeed, including successive projectors reduces dimensions by half, as expected. The last (here Σ 33 ) plaquette projector does not change anything according to what is said above. Moreover, final result is nontrivial only for multiplicities which satisfy (17). Finally, the second Polyakov line is also inactive (i.e. it depends on the other projectors) for allowed multiplicities, while it is incompatible with the rest for forbidden values of p. All this is in complete agreement with the discussion of (odd)x(odd) lattices in Sect. 3. Notice, that the final dimensionalities of the fully reduced spin spaces agree with the sizes of the corresponding sectors with p indistinguishable fermions (20), as it should be the case.   (3,1), (4,1) and (1,2).
In the 4 × 4 case the reduction was done in subsectors of fixed p fermionic coordinates (scheme (21)). All of them have the same size, independently of p. As in the previous case adding subsequent plaquette projectors cuts the size by half until we reach the last two plaquettes. Interestingly, both of them do not reduce further the remaining Hilbert space. This means that for 4 x 4 lattice (and generally for (even)x(even) ones) two plaquettes are dependent. This is easy to explain: for even-by-even lattices one can split all plaquettes into even and odd ones (i.e. according to the parity 2 of the lower-left corner, say). Then each of the two groups satisfies the condition n P n = (−1) N , n − even, n − odd, independently. Consequently there are two dependent plaquettes on (even)x(even) lattices, which explains the above observation.
On the other hand both Polyakov line projectors seem now to be independent. This can be understood as follows. As explained above the horizontal/vertical line operators can be shifted perpendicularly by multiplying by a row/column of adjacent plaquettes. This allows to write the product (16) as where the ∼ means that we have ignored all plaquette operators as they do not matter in the argument. It follows that fixing the value of the product (24) can determine the sign of a line operator only if the corresponding dimension of the lattice is odd. For even L x (L y ) corresponding Polyakov line operator L y (1) (L x (1)) is not restricted by the constraint (16), i.e. it remains independent. This was readily seen in our 4 × 4 example ( Table 3). Notice that the final number of independent projectors remains 2 N since for 4 × 4 lattice there are only N − 2 independent plaquette operators.
The whole discussion can be repeated for other situations as well. The results are summarized in Table 4 for all four cases. L x L y plaquettes lines multiplicity odd odd N − 1 L x or L y odd odd even N − 1 L x odd even odd N − 1 L y odd even even N − 2 L x and L y even Table 4: Number of independent projectors and consistent multiplicities for periodic boundary conditions in both representations, ǫ = ǫ ′ = 1.
The final test of our hypothesis is to calculate the spectrum of the spin hamiltonian in the eigenspace of all above constraints. One way to do it is to employ results of the scheme . For the small lattices considered in this example (see also the next Section) all eigenvectors are analytically generated by Mathematica [10]. Given these, the reduced spin hamiltonian matrix, and its spectrum can be readily, obtained. The exercise was repeated for few multiplicity sectors on above lattices. In all cases considered, the complete spectrum of known eigenenergies of p free fermions was analytically reproduced.

Generalization to the whole family of constraints
Above discussion addressed solely the case where all plaquette operators were constrained to unity. In principle, however, one could consider the whole family of 2 N constraints Such sectors obviously exist in the bigger, unconstraint spin system what raises the question of their interpretation. The answer is simple and instructive, as discussed in this Section.
Consider the following modification of the original fermionic Hamiltonian (1) U( n, n + e)φ( n) † φ( n + e) − U( n, n + e)φ( n + e) † φ( n) where U(l) is an additional Z 2 field assigned to each link l. In the spin representation this goes into with the same variables U(l) and S(l) given by (7). Clearly these hamiltonians describe fermions and/or corresponding spins in an external Z 2 field. As in the free case they should be equivalent as long as we restrict the spin Hilbert space identically as discussed in the previous Section. This provides an extension of the fermion-spin equivalence to systems coupled minimally to external fields 4 .
On the other hand, one can absorb the U(l) factors into the new link operators and define S ′ (l) = U(l)S(l);S ′ (l) = U(l)S(l), without any change of the commutation rules between link variables. Now the spin hamiltonian does not depend on the external field but the constraints on the new spin variables do. They readily follow from (9) That is, the system of new spins is not free, but remembers the interactions via constraints (31) only. In another words: there are two ways of introducing minimal interaction with the external field: 1) the standard one by putting explicitly link variables into the hamiltonian and imposing "free" form of the constraints (10), and 2) use the free spin hamiltonian (8), but impose the "interacting" constraints (31).
On the fermionic side, the hamiltonian (27) is that of two dimensional fermions in the fixed, external gauge field of the Wegner type [11]. The gauge field is not dynamical. On the other hand our spin system is also coupled to the same gauge filed and various boundary conditions are probing different gauge invariant classes of the Z 2 variables [12].
It is instructive to test the new hypothesis explicitly on a small lattice. This will be done below.

A soluble example
There exists a particular configuration of Wegner variables, namely for which the fermionic problem can be solved analytically. The spectrum of the fermionic hamiltonian (27) reads to be contrasted with the free case Configuration (32) can be realized only for an even L y and results in all plaquettes being equal P n = −1, 1 n N , hence it is a Wegner version of a constant magnetic field.
We have therefore repeated the procedure of Sect 3.1 for the 3 × 4 lattice, in order to check the above prediction. Table 5. shows, familar by now, pattern of the reduction of Hilbert spaces. All proceeds as before, the new element is the distinguished role of the line projector associated with L x , as predicted in Table 4. Table 5 displays results for the three different orderings (A, B, C) of all projectors. Although the final effect of the three is the same 5 , the results in the intermediate stages are different and it is worthwhile to analyze them in detail. "Routes" A and B differ only by the order of the two line projectors which are added at the end of the process. Before that, we employ all N = 12 plaquette projectors and, as discussed before, the last one is  evidently dependent on the rest. Then, among the two line projectors, Σ y is inactive, i.e. dependent, while Σ x is independent and reduces the remaining space, independently of the different order of insertion of Σ y and Σ y in routes A and B. The situation is different in the scheme C where the line projectors are inserted before the last three plaquettes.
Here the Σ y acts as an independent projector, contrary to the A and B schemes and in the apparent disagreement with (24). One should remember however that on the route C the line projectors are acting before the last three plaquettes. In this situation Σ y is indeed independent of the previous N − 3 ones and accordingly reduces the Hilbert space by a half. At the same time, among the three plaquettes following the line projectors along the route C, two are now inactive (i.e. dependent on the previously employed set) leading finally to the correct final size of the one particle sector.
In both cases the correct fermionic spectrum was reproduced from the effective spin hamiltonian providing nice check of the correspondence, as well as the confirmation of the interpretation of the constraints.

Summary
The old proposal for local bosonization of fermionic degrees of freedom in higher dimensions was revisited. Resulting spin systems are indeed local. However they have to satisfy additional constraints which, even though local themselves, introduce effectively long range interactions. In particular, they are sensitive to the lattice size, its geometry and also to fermionic multiplicities.
In this paper we have studied and classified this dependence in detail. The necessary reduction of spin Hilbert space was demonstrated analytically for few small lattices. The number of regularities was found which apply to larger systems as well. In particular, for given lattice sizes, the fermion-spin equivalence works only in specific sectors of fermionic multiplicity. Only in this sectors the complete reduction to the correct fermionic Hilbert space could be achieved. The general analytic conditions when this occurs were derived.
For the above small lattices all relevant constraints were solved with the aid of Mathematica. Consequently, complete eigenbases of spin states fulfilling the constraints are known analytically. Their structure is tantalizingly simple, however the explicit generalization to arbitrary sizes still remains a challenge.
The second step was to calculate the spectra of effective spin hamiltonians reduced to the sectors which satisfy the constraints. In all studied above cases the well known fermionic eigenenergies were readily reproduced.
Then, the equivalence was generalized to fermions coupled minimally to the external Z 2 gauge field. Apart from being interesting by itself, it provided a simple and intuitive interpretation of the constraints. Namely, the constraints of the spin system equivalent to fermions in a given external field are determined uniquely by this field. Moreover, all other constraints conceivable for this system are represented by other (nontrivial) gauge invariant classes of the external Z 2 field. Apart from a simple and general proof, this observation was also checked for a particular configuration of Z 2 variables -the Wegner analog of a constant magnetic field. Indeed, the analytically obtained spectrum of the spin hamiltonian, reduced to the constraint-invariant sector, reproduced the fermionic eigenenergies in this field.
For the simplicity, most of the discussion and our calculations concentrated on the two dimensional case. Nevertheless, extension to higher space dimensions does not present any objective difficulties.