Non-diagonal boundary conditions for gl(1|1) super spin chains
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t Non-diagonal boundary conditions for gl (1 | super spin chains Andr´e M. Grabinski and Holger Frahm
Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover,Appelstraße 2, 30167 Hannover, Germany (Dated: November 18, 2018)
Abstract
We study a one-dimensional model of free fermions with gl (1 |
1) supersymmetry and demonstratehow non-diagonal boundary conditions can be incorporated into the framework of the graded Quan-tum Inverse Scattering Method (gQISM) by means of super matrices with entries from a superal-gebra. For super hermitian twists and open boundary conditions subject to a certain constraint,we solve the eigenvalue problem for the super transfermatrix by means of the graded algebraicBethe ansatz technique (gABA) starting from a fermionic coherent state. For generic boundaryconditions the algebraic Bethe ansatz can not be applied. In this case the spectrum of the supertransfer matrix is obtained from a functional relation. . INTRODUCTION For a long time studies of quantum integrable models in one spatial dimension have ledto important insights into the properties of many body systems and provided a sound basisfor the understanding of the non perturbative phenomena which arise due to the interplayof interactions and strong quantum fluctuations in low dimensional systems (see e.g. [1]). Aspecial way to introduce free parameters into these systems is by variation of their boundaryconditions. Considering all possible classes compatible with the integrability allows for acomplete classification of their low-energy quantum critical behaviour on one hand butalso to study in detail the effect of embedded impurities and contacts to an environment.Recently, there has been increased interest in twisted or non-diagonal boundary conditionswhich break certain bulk symmetries of integrable quantum spin chains [2, 3, 4, 5, 6, 7, 8]:although their hamiltonian is a member of a commuting family of operators the establishedalgebraic schemes for the computation of the spectrum fail unless additional constraints tothe boundary conditions are in place. For spin 1 / Z grading or higher rank symmetry. Although integrable non-diagonalopen boundary conditions have been constructed [9, 10, 11, 12] the solution of the spectralproblem is restricted to diagonal ones so far.In this paper we study this problem for the simplest possible case of spin chains with gl (1 |
1) supersymmetry. Since the corresponding bulk system describes free spinless fermionson a lattice this should provide a toy model to investigate in particular the applicability offunctional methods to the solution of the spectral problem. We begin with a short reviewof the graded Quantum Inverse Scattering Method [13, 14, 15]. Using a Grassmann valuedsuper matrix representation of the Yang Baxter algebra, spin chains subject to twisted pe-riodic boundary conditions can be embedded into this framework and are solved exactly. InSection III we construct the gl (1 |
1) super spin chain with generic open boundary conditionsbased on Sklyanin’s reflection algebra [16]. We study the spectrum of these super spin chainsfor certain classes of reflection matrices using the algebraic Bethe ansatz and finally extendthis solution to generic boundaries using functional methods.2
I. GRADED QUANTUM INVERSE SCATTERING METHOD
The fundamental objects considered within the framework of the graded Quantum InverseScattering Method (gQISM) are representations T ( v ) of the graded Yang-Baxter algebra (gYBA) R ( u − v ) T ( u ) T ( v ) = T ( v ) T ( u ) R ( u − v ) . (2.1)The indices 1 and 2 label the linear spaces V , into which the respective operators areembedded by means of the super tensor product ⊗ s , defined through( A ⊗ s B )( C ⊗ s D ) ≡ ( − p ( B ) p ( C ) AC ⊗ s BD , (2.2)where p ( X ) refers to the parity function defined in the appendix. That is, to be precise T ( u ) ≡ T ( u ) ⊗ s , T ( u ) ≡ ⊗ s T ( u ) ,R ( u ) ≡ R ( u ) ⊗ s , R ≡ ⊗ s R ( u ) and R ( u ) = P R ( u ) P . (2.3)Here P ij is the graded permutation operator that interchanges two spaces V i and V j accordingto P ( x ⊗ s y ) ≡ ( − p ( x ) p ( y ) ( y ⊗ s x ). The R -matrix is subject to the consistency condition R ( u − v ) R ( u ) R ( v ) = R ( v ) R ( u ) R ( u − v ) , (2.4)known as Yang-Baxter equation (YBE). As a consequence one obtains local representations L j ( u ) ≡ R j ( u ) of the gYBA by a graded embedding of the R -matrix. These Lax-operators L j ( u ) act on an auxiliary space V , whereas their entries act on the j -th quantum space V j . Due to its comultiplication property, the gYBA allows for the construction of globalrepresentations as products of Lax-operators. This results in a particular representation onthe auxiliary space and the tensor product of the quantum spaces V q = V ⊗ s V ⊗ s · · · ⊗ s V N ,the monodromy matrix T ( u ) ≡ L N ( u ) L ,N − ( u ) . . . L ( u ) . (2.5)Taking the supertrace (A11) of this monodromy matrix, we obtain the super transfermatrix τ ( u ) = str { T ( u ) } which generates a set of commuting operators on V q . In particular,it is related to an integrable hamiltonian with periodic boundary conditions defined by H = ∂ u ln τ ( u ) | u =0 .For the gl (1 |
1) supersymmetric representations of the gYBA considered here, this con-struction leads to a model of free spinless fermions on a one-dimensional lattice with N sites.3n the case of periodic boundary conditions the hamiltonian reads H = N X j =1 H j,j +1 , H j,j +1 ≡ (cid:16) c † j c j +1 + c † j +1 c j (cid:17) − n j − n j +1 + 1 . (2.6)The corresponding R -matrix (cf. [15]) is R ( u ) = u + 1 u u u − y ˇ R ( u ) ≡ P R ( u ) = u uu − u (2.7)and a graded embedding yields L j ( u ) ≡ R j ( u ) = u + e j , e j , e j , u − e j , = u + ¯ n j c † j c j u − n j . (2.8)Generally we will define the hamiltonian density in terms of the checked R -matrix via H ij ≡ ∂ u ˇ R ij ( u ) | u =0 . A. Super hermitian twists
The simplest generalization of periodic boundary conditions are twists. They can easilybe incorporated into the above scheme by making use of the comultiplication property again.Let the twist matrix K be a representation of the gYBA on the auxiliary space. Then K · T ( u )is another global representation producing the super transfermatrix τ ( u ) = str { K · T ( u ) } = str { K L N ( u ) L ,N − ( u ) . . . L ( u ) } , (2.9)which results in a modified hamiltonian on V q which contains a boundary term H twist = N − X j =1 H j,j +1 + K − N H N K N . (2.10)As a specific twist matrix we choose K = a d E ( d E ) ♯ b a, b ∈ R , d ∈ C , (2.11)where E is the sole generator of C G (see Appendix A 2). Notice that this is the mostgeneral C G super matrix being hermitian with respect to the operation (A13). Taking into4ccount the properties of Grassmann numbers, K can be diagonalized by a super unitary transformation U = 1 a − b i( a − b ) d E ♯ d ∗ E i( a − b ) y U † U = U U † = , (2.12)such that e K ≡ U † KU = a b . (2.13)The Lax-operators (2.8) are super matrices over the algebra described in Appendix A 1,hence the comultiplication (2.9) will lead to products between fermionic operators (A3) andGrassmann numbers. For homogeneous elements C ∈ F and G ∈ C G N we define[ C, G ] ± = 0 and p ( CG ) = p ( GC ) ≡ p ( G ) + p ( C ) mod 2 . (2.14)In the periodic case (2.6), the spectrum can be obtained by means of the graded algebraicBethe ansatz (gABA) with the Fock-vacuum as a reference state. For diagonal (or uppertriangular) twist matrix K the Fock-vacuum would still provide a suitable reference statefor the gABA. For more general twists a different pseudo vacuum has to be used.Using the cyclicity of the supertrace we rewrite the super transfermatrix (2.9) as τ ( u ) = str n e K e L N ( u ) e L ,N − ( u ) . . . e L ( u ) o , (2.15)with transformed Lax-operators e L j ( u ) ≡ U † L j ( u ) U = u + ¯ n j − d ∗ a − b E ♯ c † j + da − b E c j c † j − da − b E c j − d ∗ a − b E ♯ u − n j − d ∗ a − b E ♯ c † j + da − b E c j . (2.16)By means of a super unitary transformation on the quantum space V j the Lax-operator e L j ( u ) can be written in the form (2.8): setting ρ ≡ d ∗ a − b E ♯ y ρ ♯ = da − b E (2.17)we define unitary operators Q j ≡ + ρ ♯ c j + ρc † j = e ρc † j + ρ ♯ c j y Q † j = − ρc † j − ρ ♯ c j = e − ( ρc † j + ρ ♯ c j ) (2.18)5hat map the fermionic creation and annihilation operators according to˜ c j = Q † j c j Q j = c j − ρ and ˜ c † j = Q † j c † j Q j = c † j − ρ ♯ y ˜ n j ≡ ˜ c † j ˜ c j = n j + ρc † j − ρ ♯ c j , ˜¯ n j ≡ ˜ c j ˜ c † j = 1 − ˜ n j = 1 − n j − ρc † j + ρ ♯ c j . (2.19)In terms of these new fermionic creation and annihilation operators we obtain e L j ( u ) = u + ˜¯ n j ˜ c † j ˜ c j u − ˜ n j . (2.20)After this transformation the gABA can be applied with the new Fock vacuum | ˜0 i = e − ρ P Nj =1 c † j | i (2.21)as the reference state. Note that the local Fock vacua | ˜0 j i = Q † j | j i = | j i − ρ | j i (2.22)are fermionic coherent states , i.e. eigenstates of the annihilation operator c j | ˜0 j i = ρ | ˜0 j i . III. GRADED REFLECTION ALGEBRA
We will now extend Sklyanin’s formalism for the treatment of integrable systems withopen boundary conditions [16] in a way that makes it applicable to supersymmetric models.Following [17, 18], for a given R -matrix we introduce two associative superalgebras T − and T + , subject to the graded reflection equation R ( u − v ) T − ( u ) R ( u + v ) T − ( v )= T − ( v ) R ( u + v ) T − ( u ) R ( u − v ) (3.1)and to the dual graded reflection equation R st ist ( v − u ) T + st ( u ) e R ( − u − v ) T + ist ( v )= T + ist ( v ) ¯ R ( − u − v ) T + st ( u ) R st ist ( v − u ) (3.2)respectively, whereas the new matrices e R and ¯ R are related to the R -matrix via e R st ( − u − v ) R st ( u + v ) = and (3.3)¯ R ist ( − u − v ) R ist ( u + v ) = . (3.4)6oreover the R -matrix (2.7) satisfies the unitarity condition R ( u − v ) R ( v − u ) ∼ .Under these conditions it is possible to show that the super transfermatrices τ ( u ) ≡ str (cid:8) T + ( u ) T − ( u ) (cid:9) (3.5)provide a family of commuting operators, i.e. [ τ ( u ) , τ ( v )] = 0, ∀ u, v ∈ C .Now open boundary conditions can be described by two auxiliary space matrices K − ( u )and K + ( u ) satisfying the reflection equations (3.1) and (3.2). Up to normalization, therestriction to C G essentially yields solutions K ± ( u ) = + u a ± b ± E f ± E ♯ − a ± (3.6)with complex coefficients a ± , b ± and f ± .Let T ( u ) be a representation of the gYBA (2.1). Then T ( u ) K − ( u ) T − ( − u ) is a furtherrepresentation of the graded reflection algebra T − and we have τ ( u ) = str (cid:8) K + ( u ) T ( u ) K − ( u ) T − ( − u ) (cid:9) . (3.7)The R -matrix is regular, that is R (0) = P , and for convenience let us choose the normaliza-tion such that K − (0) = . Since K + (0) has a vanishing supertrace we compute the secondderivative of the super transfermatix (3.7) and – bearing in mind that the R -matrix (2.7)complies with the unitarity condition only up to normalization – find d du τ ( u ) (cid:12)(cid:12)(cid:12)(cid:12) u =0 = 8 [1 + a + ] H (3.8)with the open chain hamiltionian H = N − X j =1 H j,j +1 + 12 ddu K − ( u ) (cid:12)(cid:12)(cid:12)(cid:12) u =0 + 12(1 + a + ) ddu N K + ( u ) (cid:12)(cid:12)(cid:12)(cid:12) u =0 . (3.9)Now we may address the question of what type of boundary terms the matrices K − and K + do generate, i.e. in what way such boundary conditions affect the hamiltonian of the givenmodel. Using the expressions (3.6) explicitly, the hamiltonian (3.9) can be written as H = N − X j =1 H j,j +1 + 12 a − d − E f − E ♯ − a − + 12(1 + a + ) a + d + E f + E ♯ − a + N . (3.10) Constant matrices of the form K ± = ( K ± ( u ) − ) /u can be employed as well.
7n using standard representations of (A3) and by exploiting (A14) we can express the twomatrices from the latter equation by elements of the combined superalgebra. The first matrixyields a − d − E f − E ♯ − a − = a − a − + d − E + f − E ♯ (3.11)= a − − + d − E − E + f − E ♯ − E ♯ − (3.12)= a − ( − n ) + d − E c − f − E ♯ c † (3.13)and after repeating this procedure for the second matrix, the entire hamiltonian reads H = N − X j =1 H j,j +1 + 12 h a − − a − n + d − E c − f − E ♯ c † i + 12(1 + a + ) h a + − a + n N + d + E c N − f + E ♯ c † N i . (3.14)We point out that the non-diagonal boundary terms, which do not preserve the particlenumber, are Grassmann valued (i.e. ∼ E ). Such terms may arise, e.g., in the descriptionof the system coupled to a fermionic environment after integrating out the bath degrees offreedom. IV. GRADED ALGEBRAIC BETHE ANSATZ
In this section we show how the spectral problem for the hamiltonian (3.14) can besolved by means of a graded algebraic Bethe ansatz. For notational convenience we set T ( u ) ≡ T ( u ) K − ( u ) T − ( − u ) and consider T ( u ) as a 2 × T ( u ) ≡ A ( u ) B ( u ) C ( u ) D ( u ) (4.1)on the auxiliary space. The reflection equation (3.1) gives commutation relations betweenthe quantum space operators A ( u ) , B ( u ) , C ( u ) and D ( u ) of which the following three are of8articular interest B ( u ) B ( v ) = 1 − u + v u − v B ( v ) B ( u ) , (4.2a) A ( u ) B ( v ) = (1 − u + v )( v + u )(1 + u + v )( v − u ) B ( v ) A ( u ) + 11 + u + v B ( u ) (cid:26) u + vu − v A ( v ) − D ( v ) (cid:27) , (4.2b) D ( u ) B ( v ) = (1 − u + v )( v + u )(1 + u + v )( v − u ) B ( v ) D ( u ) + 11 + u + v B ( u ) (cid:26) u + vu − v D ( v ) − A ( v ) (cid:27) . (4.2c)Let | i be a pseudo-vacuum upon which T ( u ) acts as an upper triangular matrix, i.e. T ( u ) | i = A ( u ) | i B ( u ) | iC ( u ) | i D ( u ) | i = α ( u ) | i ∗ 6 = 00 δ ( u ) | i . (4.3)Here α ( u ) and δ ( u ) are scalar functions, called parameters , that are to be determined lateron. They are eigenvalues to A ( u ) and D ( u ) for the eigenstate | i . A. Diagonal boundary conditions
We begin by considering diagonal boundary matrices K − and K + , i.e. K − ( u ) = ua − − ua − and K + ( u ) = ua + − ua + . (4.4)This yields the super transfermatrix τ ( u ) = str { K + ( u ) T ( u ) } = (1 + ua + ) A ( u ) − (1 − ua + ) D ( u ) . (4.5)Using the commutation relations (4.2a) to (4.2c) we find B ( v ) . . . B ( v M ) | i to be an eigen-state of τ ( u ) with eigenvalueΛ( u ) = " M Y ℓ =1 (1 − u + v ℓ )( v ℓ + u )(1 + u + v ℓ )( v ℓ − u ) (1 + ua + ) α ( u ) − (1 − ua + ) δ ( u ) (cid:19) , (4.6)provided that the Bethe ansatz equations α ( v j ) δ ( v j ) = 1 − a + v j a + v j (4.7)are satisfied. Here the functions α ( u ) and δ ( u ) are obtained from the action of T ( u ) on theFock vacuum | i T ( u ) | i = T ( u ) K − ( u ) T − ( − u ) | i = α ( u ) B ( u )0 δ ( u ) | i . (4.8)9sing (2.5) and (2.8) we find α ( u ) = (cid:18) − u (cid:19) N (1 + ua − )[ u + 1] N δ ( u ) = (cid:18) − u (cid:19) N ( (1 − ua − ) u N + (1 + ua − ) u N u (cid:20) u + 1 u (cid:21) N − !) . (4.9)Therefore the Bethe ansatz equations (cid:18) v j + 1 v j (cid:19) N = 1 − a + v j a + ( v j + 1) 1 − a − ( v j + 1)1 + a − v j (4.10)determine the quantization of single particle momenta of the free fermions due to the bound-ary conditions.Finally, we find an explicit expression for the operators B ( u ), that generate eigenstatesof the super transfermatrix: B ( u ) = (cid:18) u − u (cid:19) N u + 1 N X ℓ =1 ( [1 + ua − ] (cid:18) u + 1 u (cid:19) j − + (cid:20) uu + 1 − a − (cid:21) (cid:18) uu + 1 (cid:19) j − ) c † ℓ . (4.11) B. Quasi-diagonal boundary conditions
Application of the graded Bethe ansatz for non -diagonal boundary matrices is only pos-sible when a suitable reference state can be found. Here we consider a super hermitian leftboundary matrix K + K + ( u ) = + u a + d + E d ∗ + E ♯ − a + with a + ∈ R und d + ∈ C , (4.12)which is diagonalized by the super unitary transformation U = 12 a + a + d + E ♯ d ∗ + E a + , e K + ( u ) = U † K + ( u ) U = ua + − ua + . (4.13)Now we proceed as in Section II A: the transformation U leaves the Lax-operators shape-invariant, and we find e L j ( u ) = U † L j ( u ) U = u + ˜¯ n j ˜ c † j ˜ c j u − ˜ n j , (4.14)10here ˜ c j = c j − ρ and ˜ c † j = c † j − ρ ♯ ; but now we have ρ ≡ d ∗ + a + E ♯ y ρ ♯ = d + a + E . (4.15)Due to the cyclicity of the supertrace the super transfermatrix can be written as τ ( u ) = str n K + ( u ) T ( u ) o = str ( e K + ( u ) e T ( u ) ) , (4.16)where e T ( u ) = e T ( u ) e K − ( u ) e T − ( − u ) ≡ e A ( u ) e B ( u ) e C ( u ) e D ( u ) . (4.17)Here we have introduced e T ( u ) = e L N ( u ) . . . e L ( u ) and e K − is the transformed right boundarymatrix (3.6) e K − ( u ) = U † K − ( u ) U = 1 a + a + (1 + ua − ) ( a + d − − d + a − ) u E ( a + f − − d ∗ + a − ) u E ♯ a + (1 − ua − ) . (4.18)Now, choosing the parameters in (4.18) to satisfy the constraint a + f − = d ∗ + a − , (4.19)the transformed boundary matrix e K − is upper triangular and the graded algebraic Betheansatz can be performed again with a pseudo vacuum constructed from the fermionic coher-ent state (2.21) by using the definition (4.15) for ρ (see Ref. 2 for a similar approach in the un-graded case). Furthermore, since the transformed quantum space operators e A ( u ) , e B ( u ) , e C ( u )and e D ( u ) obey the same fundamental commutation relations (4.2a) to (4.2c) as their originalcounterparts, the Bethe ansatz equations (4.10) remain unchanged.Compared to the diagonal case we find that the addition of non-diagonal boundary pa-rameters subject to the constraint (4.19) does not affect the eigenvalues of the super transfer-matrix: the energy spectrum of the chain is determined by the diagonal parameters a ± ofthe boundary matrices alone. The Bethe states are generated by the action of the operator e B on the new pseudo vacuum. Due to the unitary transformation it contains a Grassmannvalued shift e B ( u ) = (cid:18) u − u (cid:19) N u + 1 N X ℓ =1 ( [1 + ua − ] (cid:18) u + 1 u (cid:19) j − + (cid:20) uu + 1 − a − (cid:21) (cid:18) uu + 1 (cid:19) j − ) ˜ c † ℓ + (cid:18) u − u (cid:19) N (cid:18) d − − d + a + a − (cid:19) u E . (4.20)11herefore, the Bethe states e B ( v ) . . . e B ( v M ) | e i are linear combinations of states with up to M particles added to the coherent state Fock vacuum (2.21). C. Generic boundary conditions: functional relations
Finally, we want to address the question to what extent the spectral problem of the gl (1 | / k , ± be the eigenvalues of the boundary matrices K ± ( u ), then (4.6) can be rewrittenas a functional relation for an unknown function q ( u )Λ( u ) = q ( u − q ( u ) f ( u ) (4.21)where f ( u ) is a known function: f ( u ) ≡ k α ( u ) − k δ ( u )= k (cid:18) − u (cid:19) N k − [ u + 1] N − k (cid:18) u − u (cid:19) N ( k − + k − u (cid:20) u + 1 u (cid:21) N − !) . (4.22)By construction Λ( u ) is a polynomial in u . Therefore Eq. (4.21) has to be complementedwith the condition that its RHS is analytic. In particular the residues at the zeroes of theunknown function q ( u ) have to vanish. With a polynomial ansatz q ( u ) ≡ M Y ℓ =1 ( − u − − v ℓ )( u − v ℓ ) , (4.23)this leads immediately to the Bethe equations (4.10).12or spin 1 / and in the generic off-diagonalboundary conditions: there, only the eigenvalues of the boundary matrices enter the equationexplicitly while the deviation from constraints such as (4.19) in the non-diagonal case changesthe asymptotic behaviour of its solution. This leads to non-polynomial solutions q ( u ) to thecorresponding difference equations and therefore Bethe like equations are not easily obtained.Based on this observation we propose that the eigenvalues of the super transfermatrix(3.7) satisfy Eq. (4.21) with f ( u ) parametrized by the eigenvalues of the generic boundarymatrices K ± ( u ) as in (4.22). We have verified this hypothesis for small system sizes wherewe are able to explicitly construct the super transfermatrix as a square even super matrix ofcorresponding finite dimension. Taking into account the peculiarities arising from gradingas well as the nilpotency of Grassmann generators, it is perfectly possible to perform anexact diagonalization by the use of computer algebra systems. For chains with up to N = 6sites we have computed the eigenvalues for the most general boundary matrices K − ( u )and K + ( u ) and found that the functional equation (4.21) is indeed satisfied. Unlike thesituation for spin 1 / q ( u ) are still polynomial as in (4.23)which allows to compute the eigenvalues by solving the Bethe equations (4.10) for genericboundary conditions!As an simple example we consider a system with just one site, i.e. N = 1: the exactdiagonalization of the corresponding super transfermatrix yields the two eigenvaluesΛ ± ( u ) = − uu − a + + u ( u ±
1) [ a + + a − (1 + a + )]) . (4.24)On the other hand, assuming that the eigenvalues satisfy (4.21) with polynomial q ( u ) (4.23)we can determine the values of the parameters v ℓ from the requirement, that Λ( u ) hasvanishing residues at the poles at u = v ℓ and u = − − v ℓ . For M = 0 we immediatelyobtain Λ + ( u ) while for M = 1 we find v = − ( ± s a − + a + ( a − − − a + + a − (1 + a + ) ) (4.25)and thereby recover the second eigenvalue Λ − ( u ).13 . SUMMARY AND CONCLUSION In this paper we have studied gl (1 | U (1)particle number conservation of the bulk system. The boundary conditions could be em-bedded into the reflection algebra formalism resulting in quantum integrable models. Forthe solution of the spectral problem we have applied the graded algebraic Bethe ansatzfor a class of boundary conditions satisfying a constraint (4.19). In these cases both theeigenvalues and the eigenstates of the super transfermatrix are obtained by the action ofcreation operators on a suitably chosen reference state. For generic boundary conditionssuch a vacuum state could not be constructed. Motivated by recent findings for spin chainswithout grading we have proposed the hypothesis that the eigenvalues can still be obtainedfrom Bethe equations and verified this conjecture for small system sizes using numericalmethods. In this case, however, it is not clear how the eigenstates are parametrized by theBethe roots.Although the case of gl (1 | q -deformation of the system presented here. Non-diagonal solutions to the reflection equations for the corresponding small-polaron modelhave been constructed in the past [20, 21]. Studies of the spectral problem for these chains,however, have been restricted to the diagonal case. Acknowledgments
We thank A. Seel for numerous discussions. This work has been supported by theDeutsche Forschungsgemeinschaft under grant no. Fr 737/6.14
PPENDIX A: SUPERALGEBRAS AND -MATRICES1. General linear Lie Superalgebras
Let
N, m, n ∈ N and (cid:8) e j , βα (cid:9) j =1 ,...,Nα,β =1 ,...,m + n be a homogeneous basis of an associative super-algebra, subject to the commutation relations (cid:2) e j , βα , e k , δγ (cid:3) ± = δ jk (cid:16) δ βγ e j , δα − ( − p ( e j , βα ) p ( e k , δγ ) δ δα e j , βγ (cid:17) , (A1)whereas [ X, Y ] ± ≡ XY − ( − p ( X ) p ( Y ) Y X denotes the so-called super commutator and p ( X )gives the parity of a homogeneous element X of the superalgebra, that is p ( X ) = X is an element of the even subspace, or1 if X is an element of the odd subspace . (A2)Considering the super commutator as a generalized Lie product, the generators e j , βα consti-tute the Lie superalgebra gl ( m | n ). We restrict ourselves to the special case m = n = 1. Byidentifying c j ≡ e j , , c † j ≡ e j , , n j ≡ c † j c j ≡ e j , and ¯ n j ≡ c j c † j = 1 − n j ≡ e j , (A3)we find gl (1 |
1) to be the algebra F of operators c † j and c j creating and annihilating spinlessfermions on a one-dimensional lattice respectively, j being the site index. In this case theeven subspace is spanned by n j and ¯ n j while c † j and c j span the odd subspace.For a more detailed introduction to the construction of superalgebras on graded vectorspaces, we refer to [22], [15] and section 12.3 in [1].
2. Grassmann algebras
Grassmann numbers, being the elements of a Grassmann algebra, are one of the keyingredients in the formulation of non-diagonal boundary conditions for super spin chains.The
N ∈ N generators of a Grassmann algebra will be denoted by E , E , . . . , E N and inaccordance with [22] we define a product between them such that for all j, k, l = 1 , , . . . , N
1. the product is associative, ( E j E k ) E l = E j ( E k E l ) , (A4)15. any two generators mutually anticommute, E j E k = − E k E j , (A5)3. and each non-zero product E j E j . . . E j r , ≤ r ≤ N (A6)involving r generators is linearly independent of products involving less than r gener-ators. In particular, this means that Grassmann generators E j have no inverse .For consistency reasons it is customary to supplement the set of generators by an iden-tity 1 with the defining properties 1 · E j = E j E j . Using multi-indexnotation, each product of | µ | generators can be written as E µ ≡ E j E j . . . E j | µ | , whereas µ = (cid:8) j , j , . . . , j | µ | (cid:9) is an, without loss of generality, ascendingly ordered set of naturalnumbers 1 ≤ j n ≤ N . The identity may be incorporated by setting E ∅ ≡
1. Finally,this enables us to express every Grassmann number G as a linear combination of generatorproducts E µ with complex coefficients G µ , G = G µ E µ . (A7)Here the summation is to be carried out over all multi-indices µ . In the following textthis complex Grassmann algebra with N generators will be labeled C G N . We impose aconvenient grading, setting p ( E µ ) ≡ | µ | mod 2 . (A8)The complex conjugation of a Grassmann number G is given by the complex conjugation ofthe linear coefficients in (A7), i.e. G ∗ ≡ ( G µ ) ∗ E µ . Moreover we define the adjoint G ♯ of aGrassmann number G by G ♯ ≡ ( − i) p ( E µ ) ( G µ ) ∗ E µ . (A9)
3. Super matrices
Just like the elements of the above superalgebras, super matrices are graded objects. Herewe will only make use of square even invertable super matrices M , having the partitioning M = A BC D , (A10)16uch that all entries of the submatrices A and D are even elements of a superalgebra, whereasall entries of the submatrices B and C are odd elements of the same superalgebra. We defineconvenient analogs to the usual matrix operations. The supertrace is given bystr { M } ≡ tr { A } − tr { D } . (A11)In contrast to the ordinary matrix transposition, the super transposition ( ) st is not aninvolution. Therefore, we have an additional inverse super transposition ( ) ist , M st ≡ A T C T − B T D T , M ist ≡ A T − C T B T D T . (A12)If the underlying superalgebra is C G N there are two more important operations, namelythe adjoint operation M † ≡ ( A ♯ ) T ( C ♯ ) T ( B ♯ ) T ( D ♯ ) T , (A13)where A ♯ is defined by entrywise application of (A9), and the multiplication of a super matixby a Grassmann number G of definite parity, G · M ≡ G dim A
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