Algebraic Bethe ansatz for the XXZ Heisenberg spin chain
Algebraic Bethe ansatz for the XXZ Heisenberg spin chain Igor Salom (Institute of Physics, Belgrade) and Nenad Manojlovic (University of Algarve, Portugal) Talk outline
Introduction to algebraic Bethe ansatz A few words on motivation Define XXX chain problem Introduce standard notions (R matrices, Lax operator, monodromy, transfer matrix, Bethe equations) Solving ABA Contemporary research generalizations
XXZ chain interaction anisotropy Nontrivial boundary conditions How do we solve this? Announce another integrable system Gaudin model - that Prof. Manojlovic is going to discuss in the next talk Motivation?
So few problems that we can exactly solve Any such is precious Some models have more or less natural potential application Explains dominant magnetic behavior Related to conformal field theories In 2 dimensions Expected to provide insights to realistic problems
Basic problem spin chain Heisenberg spin chain Hamiltonian (nearest neighbor interaction): Solve = find energies, wave functions, other conserved quantities
Coordinate Bethe ansatz 90 years old approach: Does solve the main problem, but not allowing easy generalizations We will not follow this approach Algebraic Bethe Ansatz
(ABA) Introduce additional Lax operator: auxiliary space and a
Relation to permutation operator Lax operator can be written as: Where P is permutation operator: How Lax operators commute?
Consider: ? where: This RLL relation or fundamental commutation relation can here be confirmed by using: R matrices
These R matrices satisfy the quantum Yang-Baxter equation: Which can be seen to follow from RLL relation, since: while also: Monodromy Now we define Monodromy, that acts on the
entire chain: That can be seen as a 2x2 matrix whose elements are operators on the basic Hilbert space: From RLL we find how T-s commute, i.e. RTT: Transfer matrix The trace: due to
commutes for different values of the parameter: Why is this important? We have families of mutually commuting operators, e.g. obtained from power series expansion. And one of such operators is the Hamiltonian! From: one obtains:
which is proportional to the Hamiltonian: also: Solving the transfer matrix eigenproblem If we know eigenvalues of the transfer matrix t(u), we know values of all conserved quantities!
First we show that is an eigenstate: thus: also we note: The rest of eigenvectors? We show that these are obtained by action of B
operators. First we neeed commutation relations: From the commutation relations If we define vectors: we obtain off-shell action:
unwanted terms =0 eigenvalue Bethe equations But the approach is far
more general! Each R matrix satisfying Yang-Baxter equation defines an integrable spin chain! If and It is proved following the same procedure:
Yet, it is not easy to solve the t eigenproblem Research generalizations Interaction does not have to be isotropic XXZ chain XYZ chain
Nonperiodic, more general boundary conditions? Extension to long range interaction (Gaudin model)? Find the Bethe vectors and the off-shell action XXZ chain R matrix Arbitrary spin:
Uq(sl(2)) generators A few usual steps&relations Now:
Nontrivial boundary conditions Introduce 2x2 matrices K+ and K that parameterize boundary conditions Following Sklyanin, these must satisfy reflection equations: New monodromy matrix
It incorporates boundary conditions: where: satisfying Commutation relations Transfer matrix The transfer matrix is now constructed as:
and corresponds to Hamiltonian: XXZ anisotropy nontrivial boundary terms
Finding Bethe vectors vacuum Start with vacuum: where and we set =0 First order excitation One magnon excitation vector:
e u l a nv e eig
unwanted terms Bethe equations However, there is a freedom Due to the following relation:
There is actually a whole class of solutions, satisfying identical off-shell equation: This is a generic feature, we dont know that was discussed before! Second order excitation
One of intermediary steps It satisfies off-shell equation: But also does:
which is a consequence of identities: General solution Off shell action:
Conclusion We illustrated the ABA method Following that approach, we have solved a spinchain model with anisotropy and non-periodic boundary condition We pointed out the degrees of freedom that appear in off-shell solutions We set up the stage to derive another model, of
long range interactions Gaudin model Thank you. Connection with physical entities Rapidities ui are related to momenta of magnons
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