Constraints of kinematic bosonization in two and higher dimensions
Arkadiusz Bochniak, Blazej Ruba, Jacek Wosiek, Adam Wyrzykowski
aa r X i v : . [ h e p - l a t ] A p r Constraints of kinematic bosonization in two and higherdimensions
Arkadiusz Bochniak ∗ , Blazej Ruba † , Jacek Wosiek ‡ , Adam Wyrzykowski § Institute of Theoretical Physics, Jagiellonian UniversityCracow, Poland
April 3, 2020
Abstract
Contrary to the common wisdom, local bosonizations of fermionic systemsexist in higher dimensions. Interestingly, resulting bosonic variables mustsatisfy local constraints of a gauge type. They effectively replace long distanceexchange interactions. In this work we study in detail the properties of such asystem which was proposed a long time ago. In particular, dependence of theconstraints on lattice geometry and fermion multiplicity is further elaboratedand is now classified for all two dimensional, rectangular lattices with arbitrarysizes. For few small systems the constraints are solved analytically and thecomplete spectra of reduced spin hamiltonias are shown to agree with theoriginal fermionic ones. The equivalence is also extended to fermions in anexternal Wegner Z field. It is also illustrated by an explicit calculation for aparticular configuration of Wegner variables. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] Introduction
Relations between fermionic and spin degrees of freedom is an old subject [1, 2], howeverit still attracts a fair amount of interest. There is a variety of motivations for suchstudies. Eliminating Grassmannian variables from the classical (and quantum) descriptionof fermionic field theories is only one example [3, 4]. Another one is provided by therelatively recent discovery of intriguing gauge structures in the equivalent spin systems [5,6]. Finally, intensive studies of quantum computers and ”quantum algorithms” stimulatesome progress also in the hamiltonian formulation [7].The connection is well understood, and exploited, in one space dimension. Howeverits extension to higher dimensions leads to complicated, non-local interactions and seemsto be not practical.In this paper we revisit the old proposal [4, 8] where the equivalent spins interact locallyand satisfy local constraints. Effectvely these constraints take care of the non-locality ofthe fermionic description in arbitrary space dimensions.Let us begin with a simple fermionic Hamiltonian on a one dimensional lattice H = i X n (cid:0) φ ( n ) † φ ( n + 1) − φ ( n + 1) † φ ( n ) (cid:1) , { φ ( m ) † , φ ( n ) } = δ mn . (1)Its equivalent in terms of spin variables reads H = 12 X n (cid:0) σ ( n ) σ ( n + 1) − σ ( n ) σ ( n + 1) (cid:1) , (2)where Pauli matrices σ k ( n ) commute between different sites labelled by n . The standardway to prove the above equivalence is via the Jordan-Wigner transformation [1]. Howeverin higher dimensions this leads to non-local spin-spin interactions. We therefore adoptanother route, which applies also to multidimensional systems.To this end, introduce the following Clifford variables X ( n ) = φ ( n ) † + φ ( n ) , Y ( n ) = i ( φ ( n ) † − φ ( n )) , (3)and rewrite the fermionic Hamiltonian (1) in terms of link (or hopping) operators H = 12 X n (cid:16) S ( n ) + ˜ S ( n ) (cid:17) , (4) S ( n ) = iX ( n ) X ( n + 1) , ˜ S ( n ) = iY ( n ) Y ( n + 1) . Link operators satisfy the following algebra[ S ( m ) , S ( n )] = 0 , m = n − , n + 1 , { S ( m ) , S ( n ) } = 0 , m = n − , n + 1 , (5)2 S ( m ) , ˜ S ( n )] = 0 . 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 thefollowing spin representation S ( n ) = σ ( n ) σ ( n + 1) , ˜ S ( n ) = − σ ( n ) σ ( 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 construc-tion exists in higher dimensions.Before concluding this Section we note that at the heart of the equivalence claim is anexpectation that if the two representations lead to the same algebra, the systems are infact equivalent. This is the common basis of many studies in this area [6], and has beenrecently 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 emergingconstraints, 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 discussconditions of equivalence for various lattice sizes. In Sect.3 the necessary and interestingreduction of Hilbert spaces is explicitly demonstrated and the spectra of both hamiltoniansare compared for few small systems. Finally in Sect.4 a simple and physical interpretationof all, possible in our construction, constraints is proposed and tested. Generalization of the above idea to two and higher space dimensions is known for a longtime [4]. In two dimensions the fermionic Hamiltonian H f = i X ~n,~e (cid:0) φ ( ~n ) † φ ( ~n + ~e ) − φ ( ~n + ~e ) † φ ( ~n ) (cid:1) = 12 X l (cid:16) S ( l ) + ˜ S ( l ) (cid:17) , l = ( ~n, ~e ) (6)can be again rewritten in terms of two types of hopping operators labelled by links ofa two dimensional lattice. Their definitions and algebra are a straightforward general-ization from the one dimensional case. In short: the hopping operators commute unlesscorresponding links have one common site. The difference is that now four, instead of twoanticommuting link operators, are attached to each lattice site. Consequently, one needsbigger matrices to satisfy the corresponding algebra in higher dimensions.In two dimensions we choose the Euclidean Dirac matrices and set (c.f. Fig.1) S ( ~n, ˆ x ) = Γ ( ~n )Γ ( ~n + ˆ x ) , S ( ~n, ˆ y ) = Γ ( ~n )Γ ( ~n + ˆ y ) , ˜ S ( ~n, ˆ x ) = ˜Γ ( ~n )˜Γ ( ~n + ˆ x ) , ˜ S ( ~n, ˆ y ) = ˜Γ ( ~n )˜Γ ( ~n + ˆ y ) , (7)3Γ k = i Π j = k Γ j . • • • • • • • • • Figure 1: Assignment of the Dirac matrices to lattice links (7).It is a straightforward exercise to show that the two dimensional extension of thealgebra (5) remains intact. Hence our hamiltonian in the spin representation reads H s = 12 X l (cid:16) S ( l ) + ˜ S ( l ) (cid:17) . (8)Generalization to higher dimensions is simple. One just needs representations of higherClifford algebra, e.g. by larger Dirac matrices. In d dimensions they admit 2 d anticom-muting 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 arbi-trary dimensions.The story is not over however, since the representation (7) is redundant with respectto the fermionic one. In fact, in two space dimensions, it doubles the number of degrees offreedom per lattice site compared to the original fermionic system. Evidently one needsadditional 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 twodimensional index n = ( n x , n y )). If we denote by C n an elementary plaquette labelled byits lower-left corner, say, then P n = Y l ∈ C n S ( l ) . (9)These operators are identically 1 in the fermionic representations, while only P n = 1 inthe spin representation. Hence imposing all constraints P n = 1 , (10)4hould provide necessary reduction of the Hilbert space. Details of how it works dependon sizes of lattices, boundary conditions and other specifications. It was shown for fewsimple 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, understandingof this and related questions. This is the aim of present work as continued in the nextsections. The precise form of constraints required to satisfy the above fermion-spin equivalencedepends on a geometry of a lattice. Consider two dimensional, cubic lattices, possiblywith different sizes (e.g. L x and L y in each direction). Periodic or antiperiodic boundaryconditions are used. Different periodicity conditions for fermions and equivalent spins areallowed. φ ( n + L x ˆ x ) = ǫ x φ ( n ) , Γ k ( n + L x ˆ x ) = ǫ ′ x Γ k ( n ) , ǫ x , ǫ ′ x = ± , (11)and similarly for the other direction.We seek to impose N = L x L y , L x , L y >
3, constraints (10) to eliminate abundantdegrees of freedom. However not all of them are independent. For example, in the spinrepresentation plaquette operators satisfy the identity Y n P n = 1 , (12)which leaves only N − L x ( n y ) = L x Y n x =1 S ( n x , n y , ˆ x ) , L y ( n x ) = L y Y n y =1 S ( n x , n y , ˆ y ) . (13)In fermionic representation they are just pure numbers sensitive to the boundary con-ditions, while in spin representation their squares are unity, similarly to the plaquetteoperators. 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 ap-propriate rows/columns of plaquette operators. Therefore, altogether there are only twomore candidates for independent projectors.It turns out that even this set of N − N ( n ) = φ † ( n ) φ ( n ) . (14)5ince Hamiltonian (6) is moving fermions between neighbouring sites only, the total num-ber of fermions, N = P 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 matrixΓ ( n ) = η ( − N ( n ) = η (1 − N ( n )) . (15)where η = ± N is conserved, whilethe number densities N ( n ) are not. On the other hand, the plaquette and line operatorsdo commute with the local densities. This will be exploited below when we diagonalizeconstraints.Given these definitions, it is easy to show the equalityΠ ≡ L y Y n y =1 L x ( n y ) L x Y n x =1 L y ( n x ) = ( − ǫ ′ x ) L y ( − ǫ ′ y ) L x ( − η ) L x L y ( − N , (16)which implies the additional relation between Polyakov line projectors.Summarizing, the complete set of independent projectors on a two dimensional, finitelattice consists of ones associated with N − N dimensional corresponding to N fermionic degreesof freedom.However such a reduction occurs only in certain sectors, labelled by fermionic multi-plicity. The required condition follows immediately from (16) upon comparing RHS withthe correponding expression in the fermionic representation( − N = η L x L y (cid:18) − ǫ ′ x ǫ x (cid:19) L x (cid:18) − ǫ ′ y ǫ y (cid:19) L y . (17)Above discussion is valid quantitatively only for (odd)x(odd) lattices. Nevertheless,it illustrates generically the interplay between various constraints, lattice geometry andfermion multiplicity. For other lattice sizes the explicit forms of constraints are slightlydifferent and will be discussed below in detail. In all four cases, however, the final numberof independent constraints turns out to be 2 N leading to the correct ”fermionic” dimensionof the restricted spin space.Of course consistency of dimensions of both spaces is only a necessary condition forthe equivalence. The next step is to actually solve the constraints and to show that thespin 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, sucha program can be carried through for a few small lattices thanks to the rapid growth ofcomputing power and symbolic calculations. It is shown below how this works in practice.6 .1 Some explicit examples The complete Hilbert space of our system of spins on L x x L y lattice has 4 N dimensions, N = L x L y . States are represented by configurations { i , i , . . . , i N } . (18)of N Dirac indices, i n = 1 , . . . , n = 1 , . . . , N labelling sites of a lattice. All oper-ators are constructed from tensor products of N -fold four dimensional gamma matricesand the unity . In principle they require (4 N ) elements of computer storage, howeverin general they are sparse matrices and take only O(4 N ) memory size. Still, the memoryrequirement is the main limitation for such a direct approach and restricts aviable sizesto ca. N ∼ Γ Γ Γ Γ − − − i i − i i − i
00 0 0 1 i − i − − Table 1: Explicit representation of euclidean Dirac matrices used in this Section.To reduce further the memory demand, we split the whole Hilbert space into N + 1sectors of the fixed fermion multiplicity p = 0 , , . . . , N . In the fermionic representa-tion the total number of fermions is obviously conserved. The same is true in our spinrepresentation. Namely, the corresponding number operator N = X n
12 (1 − η Γ ( n )) , (19)commutes with the hamiltonian (8). Moreover, it also commutes with all plaquette andline operators. This allows to carry out the analysis of constraints in the sectors of fixedN seperately for each p . Choosing the sector of fixed multiplicity amounts to restrictingthe 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 = − a = (2 ,
3) and b = (1 , N + 1 fixed multiplicitysectors of the full H seperately, the size of each sector being2 N (cid:18) N p (cid:19) −→ (cid:18) N p (cid:19) , (20)before and after imposing constraints in the spin representation. We use the specific representation of Γ k (cf. Table 1), any other equivalent choice is possible. N butalso with each of the individual densities N ( n ). This allows to further split the problem byperforming the reduction of Hilbert space in each sub-sector of fixed p and fixed positionsof p spin excitations r , r , . . . , r p , or just fermionic coordinates, in the configuration space.Now the reduction looks like 2 N −→ . (21)The last step not only saves the computer memory, but first of all can provide a clearinterpretation of the eventual solution of the constraints problem. The eigenvectors ofall constraints depend classically on space coordinates of p fermions. This is valid for alllattice sizes and might help to better understand the nature of the former. It should benoted, however, that (21) is valid only for the purpose of studying the constraints. Thereduced spin hamiltonian has to be calculated in the bigger sectors of fixed p only. Onthe other hand, the basis of (cid:18) N p (cid:19) vectors obtained in (cid:18) N p (cid:19) steps (21) is an appropriatebasis of constraints-satisfying spin excitations in the larger sector (20).To proceed, we define the projection operators associated with all plaquettes and twoPolyakov lines Σ m,n = 12 (1 + P m,n ) Σ Z = 12 (1 + L Z ) , Z = x, y, (22)and calculate their matrix representations, at fixed total multiplicity, p . For illustrationwe explicitly display below traces of successive products of all relevant projectors on 3 × × p= 0 1 2 3 4 5 6 7 8 9 Tr Σ
256 2304 9216 21504 32256 32256 21504 9216 2304 256Tr Σ Σ
128 1152 4608 10752 16128 16128 10752 4608 1152 128Tr Σ Σ Σ
64 576 2304 5376 8064 8064 5376 2304 576 64Tr Σ Σ ... Σ
32 288 1152 2688 4032 4032 2688 1152 288 32Tr Σ Σ ... Σ
16 144 576 1344 2016 2016 1344 576 144 16Tr Σ Σ ... Σ Σ ... Σ Σ ... Σ Σ ... Σ Σ ... Σ x Σ ... Σ y × p -particle sectors. Periodicboundary conditions are assumed.For 3 × p and proceeds according to the scheme (20). Indeed, including successive8rojectors reduces dimensions by half, as expected. The last (here Σ ) plaquette projectordoes not change anything according to what is said above. Moreover, final result is non-trivial only for multiplicities which satisfy (17). Finally, the second Polyakov line isalso inactive (i.e. it depends on the other projectors) for allowed multiplicities, while it isincompatible with the rest for forbidden values of p . All this is in complete agreement withthe discussion of (odd)x(odd) lattices in Sect.3. Notice, that the final dimensionalitiesof the fully reduced spin spaces agree with the sizes of the corresponding sectors with p indistinguishable fermions (20), as it should be the case.Sector ( p ) even, 0 ≤ p ≤
16 odd, 0 < p < H il b e rt s p a ce r e du c t i o n Tr Σ Σ ... Σ ... Σ ... Σ ... Σ ... Σ ... Σ ... Σ ... Σ ... Σ ... Σ ... Σ ... Σ ... Σ x ... Σ y ... Σ ... Σ ≤ p ≤
16, and fixedcoordinates, on a 4 × × p fermionic coordinates(scheme (21)). All of them have the same size, independently of p . As in the previous caseadding subsequent plaquette projectors cuts the size by half until we reach the last twoplaquettes. 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 aredependent. This is easy to explain: for even-by-even lattices one can split all plaquettesinto even and odd ones (i.e. according to the parity of the lower-left corner, say). Then The parity of a vertex is defined as ( − n x + n y Y n P n = ( − N , n − even, n − odd, (23)independently. Consequently there are two dependent plaquettes on (even)x(even) lat-tices, which explains the above observation.On the other hand both Polyakov line projectors seem now to be independent. Thiscan be understood as follows. As explained above the horizontal/vertical line operatorscan be shifted perpendicularly by multiplying by a row/column of adjacent plaquettes.This allows to write the product (16) asΠ ∼ L x (1) L y L y (1) L x , (24)where the ∼ means that we have ignored all plaquette operators as they do not matterin the argument. It follows that fixing the value of the product (24) can determine thesign of a line operator only if the corresponding dimension of the lattice is odd. Foreven L x ( L y ) corresponding Polyakov line operator L y (1) ( L x (1)) is not restricted by theconstraint (16), i.e. it remains independent. This was readily seen in our 4 × N since for4 × N − L x L y plaquettes lines multiplicityodd odd N − L x or L y oddodd even N − L x oddeven odd N − L y oddeven even N − L x and L y evenTable 4: Number of independent projectors and consistent multiplicities for periodicboundary conditions in both representations, ǫ = ǫ ′ = 1.The final test of our hypothesis is to calculate the spectrum of the spin hamiltonian inthe eigenspace of all above constraints. One way to do it is to employ results of the scheme(21). Every of the (cid:18) N p (cid:19) single vectors obtained in each of (cid:18) N p (cid:19) reductions providedone eigenvector of all constraints. Upon repeating the procedure for all positions of p spin excitations one generates a complete eigenbasis in a bigger sector (20). For thesmall lattices considered in this example (see also the next Section) all eigenvectors areanalytically generated by Mathematica [10]. Given these, the reduced spin hamiltonianmatrix, and its spectrum can be readily, obtained. The exercise was repeated for fewmultiplicity sectors on above lattices. In all cases considered, the complete spectrum ofknown eigenenergies of p free fermions was analytically reproduced. Correponding to the eigenvalue 1, i.e. invariant under all constraints. Generalization to the whole family of constraints
Above discussion addressed solely the case where all plaquette operators were constrainedto unity. In principle, however, one could consider the whole family of 2 N constraints P n = ± , n N . (25)Such sectors obviously exist in the bigger, unconstraint spin system what raises the ques-tion of their interpretation. The answer is simple and instructive, as discussed in thisSection.Consider the following modification of the original fermionic Hamiltonian (1) H f = i X ~n,~e (cid:0) U ( ~n, ~n + ~e ) φ ( ~n ) † φ ( ~n + ~e ) − U ( ~n, ~n + ~e ) φ ( ~n + ~e ) † φ ( ~n ) (cid:1) (26)= 12 X l (cid:16) U ( l ) S ( l ) + U ( l ) ˜ S ( l ) (cid:17) , (27)where U ( l ) is an additional Z field assigned to each link l . In the spin representationthis goes into H s = 12 X l (cid:16) U ( l ) S ( l ) + U ( l ) ˜ S ( l ) (cid:17) . (28)with the same variables U ( l ) and S ( l ) given by (7). Clearly these hamiltonians describefermions and/or corresponding spins in an external Z field. As in the free case theyshould be equivalent as long as we restrict the spin Hilbert space identically as discussedin the previous Section. This provides an extension of the fermion-spin equivalence tosystems coupled minimally to external fields .On the other hand, one can absorb the U ( l ) factors into the new link operators anddefine S ′ ( l ) = U ( l ) S ( l ); ˜ S ′ ( l ) = U ( l ) ˜ S ( l ) , (29)without any change of the commutation rules between link variables. Now the spinhamiltonian does not depend on the external field H ′ s = 12 X l (cid:16) S ′ ( l ) + ˜ S ′ ( l ) (cid:17) , (30)but the constraints on the new spin variables do. They readily follow from (9) P ′ n = Y l ∈ C n U ( l ) . (31)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 withthe external field: An early version was already considered in Ref. [8]
11) the standard one by putting explicitly link variables into the hamiltonian andimposing ”free” form of the constraints (10), and2) 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 inthe 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 variousboundary conditions are probing different gauge invariant classes of the Z variables [12].It is instructive to test the new hypothesis explicitly on a small lattice. This will bedone below. There exists a particular configuration of Wegner variables, namely U x ( x, y ) = ( − y , U y ( x, y ) = 1 , (32)for which the fermionic problem can be solved analytically. The spectrum of the fermionichamiltonian (27) reads E (1) magnetic ( k x , k y ) = ± s sin (cid:18) πk x L x (cid:19) + sin (cid:18) πk y L y (cid:19) k x L x , k y L y / , (33)to be contrasted with the free case E (1) free ( k x , k y ) = 2 sin (cid:18) πk x L x (cid:19) + 2 sin (cid:18) πk y L y (cid:19) , k z L z , z = x, y. (34)Configuration (32) can be realized only for an even L y and results in all plaquettes beingequal P n = − , n N , (35)hence it is a Wegner version of a constant magnetic field.We have therefore repeated the procedure of Sect 3.1 for the 3 × 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 , the results in the intermediate stagesare different and it is worthwhile to analyze them in detail. ”Routes” A and B differ onlyby the order of the two line projectors which are added at the end of the process. Beforethat, we employ all N = 12 plaquette projectors and, as discussed before, the last one is And again consistent with the condition (17). Tr 1 49152Tr Σ Σ . . . Σ . . . Σ . . . Σ . . . Σ . . . Σ . . . Σ . . . Σ . . . Σ
48 Tr . . . Σ
48 Tr . . . Σ x . . . Σ
24 Tr . . . Σ
24 Tr . . . Σ y . . . Σ
24 Tr . . . Σ
24 Tr . . . Σ . . . Σ x
12 Tr . . . Σ y
24 Tr . . . Σ . . . Σ y
12 Tr . . . Σ x
12 Tr . . . Σ A B C
Table 5: Reduction of the spin Hilbert space for 3 × y is inactive, i.e.dependent, while Σ x is independent and reduces the remaining space, independently ofthe different order of insertion of Σ y and Σ y in routes A and B. The situation is differentin 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 inthe apparent disagreement with (24). One should remember however that on the routeC the line projectors are acting before the last three plaquettes. In this situation Σ y isindeed independent of the previous N − L x (1) = 1 , L y (1) = 1, and2) magnetic (35) and L x (1) = − , L y (1) = 1.In both cases the correct fermionic spectrum was reproduced from the effective spinhamiltonian providing nice check of the correspondence, as well as the confirmation of theinterpretation of the constraints. 13 Summary
The old proposal for local bosonization of fermionic degrees of freedom in higher di-mensions was revisited. Resulting spin systems are indeed local. However they have tosatisfy additional constraints which, even though local themselves, introduce effectivelylong range interactions. In particular, they are sensitive to the lattice size, its geometryand also to fermionic multiplicities.In this paper we have studied and classified this dependence in detail. The necessaryreduction of spin Hilbert space was demonstrated analytically for few small lattices. Thenumber of regularities was found which apply to larger systems as well. In particular, forgiven lattice sizes, the fermion-spin equivalence works only in specific sectors of fermionicmultiplicity. Only in this sectors the complete reduction to the correct fermionic Hilbertspace 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 Math-ematica. Consequently, complete eigenbases of spin states fulfilling the constraints areknown analytically. Their structure is tantalizingly simple, however the explicit general-ization to arbitrary sizes still remains a challenge.The second step was to calculate the spectra of effective spin hamiltonians reducedto the sectors which satisfy the constraints. In all studied above cases the well knownfermionic eigenenergies were readily reproduced.Then, the equivalence was generalized to fermions coupled minimally to the external Z gauge field. Apart from being interesting by itself, it provided a simple and intuitiveinterpretation of the constraints. Namely, the constraints of the spin system equivalentto fermions in a given external field are determined uniquely by this field. Moreover, allother constraints conceivable for this system are represented by other (nontrivial) gaugeinvariant classes of the external Z field. Apart from a simple and general proof, thisobservation was also checked for a particular configuration of Z variables - the Wegneranalog of a constant magnetic field. Indeed, the analytically obtained spectrum of thespin hamiltonian, reduced to the constraint-invariant sector, reproduced the fermioniceigenenergies in this field.For the simplicity, most of the discussion and our calculations concentrated on the twodimensional case. Nevertheless, extension to higher space dimensions does not presentany objective difficulties.This work is supported in part by the NCN grant: UMO-2016/21/B/ST2/01492. References [1] P. Jordan, E. Wigner, ¨Uber das Paulische ¨Aquivalenzverbot , Z. Phys (1928) 631.142] Y. Nambu, A Note on the Eigenvalue Problem in Crystal Statistisc. , Progr. Theor.Phys. (1950) 1.[3] J. B. Kogut, An Introduction to Lattice Gauge Theory and Spin Systems , Rev. Mod.Phys. (1979) 659.[4] J. Wosiek, A local representation for fermions on a lattice , Acta. Phys. Polon.
B13 (1982) 543.[5] A. Kitaev,
Anyons in an exactly solved model and beyond , Ann. Phys. (2006) 2.[6] Y-A Chen, A. Kapustin, D. Radicevic,
Exact bosonization in two spatial dimensionsand a new class of lattice gauge theories , Ann. Phys. (2018) 234.[7] E. Zohar, J. I. Cirac,
Eliminating fermionic matter fields in lattice gauge theories ,Phys. Rev.
B98 (2018) 075119.[8] A. Szczerba,
Spins and Fermions on Arbitrary Lattices , Comm. Math. Phys. (1985) 513.[9] A. Bochniak, B. Ruba, Model of bosonization by flux attachment on hamiltonian lat-tices of arbitrary dimension , arXiv:2003.06905v1.[10] https://support.wolfram.com/41360.[11] F. J. Wegner,
Duality in Generalized Ising Models and Phase Transitions WithoutLocal Order Parameters , J. Math. Phys. (1971) 2259.[12] E. Fradkin, S. H. Shenker, Phase diagrams of lattice gauge theory with Higgs fields ,Phys. Rev.19