A One Dimensional Ideal Gas of Spinons,
or
Some Exact Results on the XXX Spin Chain with Long Range Interaction.
D. Bernard, V. Pasquier and D. Serban,
Service de Physique Théorique de Saclay ^{1}^{1}1Laboratoire de la direction
des sciences de la matière du commissariat à l’énergie atomique.,
F91191, GifsurYvette, France.
Abstract: We describe a few properties of the XXX spin chain
with long range interaction. The plan of these notes is :
1 — The Hamiltonian.
2 — Symmetry of the model.
3 — The irreducible multiplets.
4 — The spectrum.
5 — Wave functions and statistics.
6 — The spinon description.
7 — The thermodynamics.
Introduction. The XXX spin chain with long range interaction is a variant of the spin half Heisenberg chain, with exchange inversely proportional to the square distance between the spins. It possesses the remarkable properties that its spectrum is additive and that the elementary excitations are spin half objects obeying a halffractional statistics intermediate between bosons and fermions. In this sense, it gives a model for an ideal gas of particles with fractional statistics. The model is gapless; its low energy properties belong to the same universality class as the Heisenberg model, and are described by the level one WZW conformal field theory.
Acknowledgements: It is a pleasure to thank Olivier Babelon, Michel Gaudin and Duncan Haldane for stimulating discussions.
1 The Hamiltonian.
The Hamiltonian of the trigonometric isotropic spin chain with long range interaction is given by [1, 2] :
(1) 
where is the operator which exchange the spins at the sites and . We restrict ourselves to the case, in which case the spin variables can only take two values: . The sum is over all the distinct pairs of sites labeled by integers ranging from to .
The spectrum of (1), which has been conjectured by Haldane [3], possesses a remarkable additivity property as well as a rich degeneracy. It can be described as follow. To each eigenstate multiplet is associated a set of rapidities which are nonconsecutive integers ranging from to . The energy of an eigenstate with rapidities is:
(2) 
The degeneracy of the multiplet with rapidities is described by its representation content as follows. Encode the rapidities in a sequence of labels or in which the ’s indicate the position of the rapidities; add two ’s at both extremities of the sequence which now has length . Since the rapidities are never equal nor differ by a unit, two labels cannot be adjacent. A sequence can be decomposed into the product of elementary motifs. A motif is a series of consecutives ’s, and it corresponds to a spin representation of . The representation content of a sequence is then the tensor product of its motifs.
The degeneracy of the spectrum can also be described in terms of path, in a way surprisingly similar to the path description of the sixvertex corner transfer matrix [4].
2 Symmetry of the model.
The symmetry algebra responsible for the degeneracy of the model was identified as the Yangian [5]. A Yangian is an infinite dimensional associative algebra generated by elements , with a positive integer, and in the case. These generators satisfy quadratic relations which can be arranged into an YangBaxter equation by introducing the transfer matrix , with matrix elements, . The commutation relations then take the following form [6, 7] :
(3) 
The matrix is the solution of the YangBaxter equation given by: , where is the permutation operator which exchanges the two auxiliary spaces. The transfer matrix was constructed in [8] . Its expression is :
(4) 
with , with , and is the canonical matrix acting on the spin only. The transfer matrix (4) form a representation of the exchange algebra (3) for any values of the complex numbers . The center of the Yangian algebra (3) is generated by the socalled quantum determinant defined by [9]:
(5) 
In the representation (4) , the quantum determinant is a pure number for any values of the ’s given by :
(6) 
with the characteristic polynomial of the matrix with entries : .
3 The irreducible multiplets.
Solving the model consists in finding all the irreducible components of the Yangian symmetry algebra and computing the energy in each of these blocks. For the values of the ’s induced by the spin chain, , the representation (4) is reducible. It is completely reducible since the transfer matrix is hermitic: . Each irreducible subrepresentation possesses a unique highest weight (h.w.) vector which is annihilated by and which is an eigenvector of the diagonal components of the transfer matrix :
Here, are rational functions in , but not operators. Since the quantum determinant (4) take the same value in any of the irreducible block, these two functions are related by :
Hence, only one of them, say , is independent. It uniquely characterizes the Yangian representation. We therefore have to compute all the functions arising from the decomposition of the Yangian representation induced by the spin chain, but also to identify the h.w. vectors in order to be able to compute the energy spectrum.
Obviously, the ferromagnetic vacuum is a h.w. vector: the corresponding is one, and the energy is zero. The h.w. vectors in the onemagnon sector are , with, : the corresponding eigenvalue is , with , and the onemagnon energy is .
In order to determine all the highest weight vectors, we decompose the hilbert space into subspaces of fixed magnon number. A magnon state has spin reversed :
(8) 
where denote the Pauli matrices acting on the spin located on the site . By construction, the coefficients of the magnon wave functions are symmetric in their indices. The wave function coefficients are unspecified for two coincident indices . By convention, we choose these coefficients to be zero.
Since these indices range from to , to any magnon state is associated a symmetric polynomial in variables of degree less than such that . In the following, we restrict ourselves to the class of magnon states deriving from polynomials of the following form :
(9) 
with a symmetric polynomial of degree less than . This class of states does not include all the states of the spin chain but, as we will see, all the highest weight vectors are in this class.
As explained in the Appendix, the operators and act on this class of states. Therefore, the action of these operators on these magnon states induces an action on the polynomials. As shown in the Appendix, we find :
(10) 
and
(11) 
Here we have introduced differential operators which have recently been proved useful in the CalogeroSutherland model [10, 11, 12, 8]. For the following, we also need another set of differential operators . Both are defined by :
(12) 
Here, and, the operator exchanges the positions: . The differential are covariant under permutation of the positions, , whereas the ’s are not. On the other hand, the differentials commute, , but the differentials ’s does not: . The sum of the powers of both differentials form two sets of commuting operators. However, they are not independent thanks to the following relation:
(13) 
In Eq. (13) it is understood that the operators are acting on functions symmetric in their arguments.
From eq. (11), we learn that the highest weight vectors correspond to polynomials with given by :
(14) 
with a symmetric polynomial of degree less than . Thus, in the magnon sector, there are independent highest weight vectors.
From eq. (10), we learn that the eigenfunctions of are the eigenvectors of the commuting hamiltonians of the CalogeroSutherland model, or equivalently of the commuting operators . The symmetric eigenfunctions with,
are polynomials with degree between and if . Hence, the Mmagnon highest weight vectors , with wave function given by , are labeled by sets of M integers . Due to the Vandermond prefactor in eq.(9), these integers never coincide nor differ by a unit. Using the factorisation relation (13), one find that the value of for these highest weight vectors are :
(15) 
The dimension of the irreducible multiplets are encoded in the transfer matrix eigenvalues. The eigenvalues are given by eq. (15). The remaining eigenvalues are computed from the relation (4). They can also be written in a product form. The result is :
(16) 
with given in eq.(15), and and factorize :
(17) 
One can check that the factorization equation (17) admits solutions only if the roots of are not adjacent. This provides one way to recover the rapidities selection rules.
Let us decompose the sequence of rapidities in elementary motifs as explained in Section 1. To each motif of length , we associate a canonical transfer matrix defined by :
(18) 
where are the matrices forming the spin representation of and is the position of the most left label of the motif. It is easy to check that the matrix (18) satisfy the commutation relations (3) . The representation induced by the transfer matrix (16) is then seen to be equivalent to the irreducible tensor product of the transfer matrices associated to each motifs:
(19) 
This is proved by comparing the eigenvalues of the diagonal transfer matrix elements on the h.w. vector. Therefore, we find that the multiplet of a rapidity sequence is the tensor product of each of its motifs.
We have found one (and only one) irreducible representation for each rapidity sequence. Their direct sum is a vector space of dimension . Therefore, it fills the Hilbert space of the spin chain, and there is no other irreducible multiplet.
4 The spectrum.
Since all the irreducible multiplets are now identified, finding the spectrum consists in computing the action of the Hamiltonian on the highest weight vectors. The hamiltonian (1) is invariant, hence it acts on the magnon subspace. In the magnon basis, this action is :
The Hamiltonian act on the state of the form (9). Using eq. (43) given in the Appendix, one realizes that the action induced on the polynomials is :
(20)  
In Eq. (20) one recognizes the CalogeroSutherland Hamiltonian at a special value of the coupling constant. In other words, the spin chain in the magnon sector has been mapped on the body Calogero problem. The last equality in (20), gives the energy of a multiplet :
This completes the proof of the spectrum.
5 Wave functions and statistics.
Only the wave functions of the h.w. vectors are relevent since those of their descendents are obtained by recursive action of the transfer matrix. We now show how recent results on the Calogero models can be used to find explicit expressions for these wave functions. The latter are based on the construction of operators, which we denote by in the magnon sector, intertwining the Calogero Hamiltonian and the free Hamiltonian:
(21) 
These intertwiners were defined in [10, 13]. One of their definitions is the following Vandermond determinant of the operators :
In this formula, it is understood that is acting on antisymmetric functions. For example, for two magnons: .
The operators are antisymmetric. Therefore, the symmetric wave functions are obtained by acting first with the antisymmetrizer on the plane waves , and then with :
(22) 
It is easy to check that the wave functions (22) are symmetric polynomials vanishing at coincident points. If the rapidities are such that , these polynomials have degree less than and satisfy the condition (14). They thus are the wave functions of the h.w. vectors. In other words, since the plane waves are the wave functions of the onemagnon h.w. vectors, the operator map tensor products of onemagnon states into magnon states:
(23) 
This map is a generalization of the Slater determinant in the sense that it implements the rapidity selection rules: if two of the rapidities and are either equal or differ by a unit, then the resulting wave function vanish identically. The fact that the result vanish if two of the rapidities coincide is obvious from the definition, while the fact that it vanishes if they differ by a unit results from an explicit check on the twomagnon case (which has all generality thanks to the symmetry of the wave functions and the recursive definition of given in [13]).
6 The spinon description.
The magnons are the excitations over the ferromagnetic vacuum; the excitations over the antiferromagnetic vacuum are conveniently described in terms of spinons.
For even, the antiferromagnetic vacuum corresponds to the alternating sequence of symbols . Its rapidities sequence is . The excitations are obtained by flipping and moving the symbols and . We classify the sequence by their number of rapidities. The spinon number of a sequence is then defined by . Since is an integer, is always even. The spin of the Yangian highest weight vector of the sequence is .
A sequence of rapidities , in the sector, can be decomposed into elementary motifs. As we defined them in Sect.1, an elementary motif is a series of consecutive . We will think about the elementary motifs as the possible orbitals for spin half objects, which are called spinons. At fixed , there are orbitals. To a spinon in the orbital, with , we assign a momemtum . Thus, the spinon momemta vary from zero to corresponds to the filling of the orbitals with respective occupation numbers , with and . The length of the elementary motif is then . By construction, the total occupation number is the spinon number : . By convention, a sequence of rapidities
(24) 
Since an elementary motif of length corresponds to a spin representation of , the full degeneracy of the sequences is then recovored by assuming that, at fixed spinon number, the spinon behaves as bosons. Notice that this property is specific to the spin chain.
The spinons are not bosons but “semions” since the number of available orbitals varies with the total occupation number [14]. In particular, the spinons are always created by pairs.
The energy of a collection of spinons is :
with .
The low energy, low temperature, behavior is classified [15] as the level one WZW conformal field theory. In the spinon description the states consist of semiinfinite sequences of symbols and . The two primary states, which correspond to the two integrable representations of the current algebra at level one, are the vacuum, with sequence and the spin half primary, with sequence . The excited states are given by finite rearrangement of the primary sequence. The Virasoro generator acts as , where are the primary sequences. Its spinon representation is :
(26) 
with , the total spinon number. The excitations over the vacuum possess an even number of spinons, wheras those over the spin half primary contain an odd number of spinons.
7 The thermodynamics.
Following ref.[3], the thermodynamics can be derived from the spinon description using methods similar to the thermodynamic Bethe ansatz [16].
First, we consider the system with a fixed spinon density . In the limit, the spinon orbitals are labeled by momenta continuously varying from zero to be the spinon occupation numbers of the orbital. By definition, they satisfy : . Let
(27) 
In the continuum limit, the energy per unit of length is :
(28)  
with and . In each orbital the spinons behave as bosons, therefore the entropy of a configuration of spinons is : ,
(29) 
The free energy per unit of length is :
(30) 
with the exterior magnetic field and the magnetization.
At fixed spinon density, the thermodynamic equilibrium is determined by minimizing (30) with respect to the variation of the spinon occupation numbers subject to the constraint (27). This gives :
(31) 
where is the Lagrange multiplier and is the dressed energy defined by :
(32) 
At fixed density, the Lagrange multiplier is determined from the constraint (27). This completely specifies the thermodynamics of the spinon gas.
The spin chain corresponds to a spinon gas of arbitrary density; i.e. the spinon chimical potential is zero. Minimazing the free energy with respect to the density fixes to be . The constraint (27) then gives the mean density .
The coupled eqs. (31,32) can be solved. Deriving twice eq.(32) with respect to gives :
(33) 
with the boundary conditions: , and . Eq.(33) is integrated by introducing the dressed momenta . It varies from to , and it satisfies
(34) 
with
(35) 
In the limit , the dressed energy is .
The free energy is found by integrating the thermodynamical relations :
(36) 
The result is the following simple answer :
(37) 
Notice that this is the free energy of a gas of noninteracting particles which, in the limit , have energies given by .
8 Appendix.
In this Appendix we prove the eqs.(10,11). First we compute the action of on these magnon states. We recall that can be written as , where is the projector on the spin acting on the spin only. The projector on the magnon states (8) gives states with one spin down marked of the form :
(38) 
They corresponds to polynomials symmetric in with the point distinguished. To evaluate the action of , we remark that on this class of states, acts as follows :
(39) 
with,
(40) 
where , and permutes the indices in position and . Therefore, can be recursively computed. Then,
(41) 
is obtained by symmetrizing the wave function coefficients of in all its indices. I.e. its wave function coefficients, denoted are :
(42) 
We now translate this action on the wave function coefficients into an action on the polynomials. We recall that in the basis of polynomials in one variable of degree specified by , the matrix elements of the derivatives are:
(43)  
The differentials , introduced in eq.(12), acts on polynomials of the form ; i.e. they preserve the form of these polynomials. Moreover, by comparing the formula, we recognize in eq.(40) the operator . Since the operators are covariant by permutation of the coordinates, the polynomial of is given by acting with on . Resuming all the contributions, we obtain :
The action of on these subclass of magnon states can be computed using the same methods. Its action is given by eq.(11). It is remarkable that the differentials appear naturally in the study of the spin transfer matrix.
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