New exact results on density matrix for XXX spin chain

Using the fermionic basis we obtain the expectation values of all \slt-invariant and $C$-invariant local operators on 10 sites for the anisotropic six-vertex model on a cylinder with generic Matsubara data. This is equivalent to the generalised Gibbs ensemble for the XXX spin chain. In the case when the \slt and $C$ symmetries are not broken this computation is equivalent to finding the entire density matrix up to 10 sites. As application, we compute the entanglement entropy without and with temperature, and compare the results with CFT predictions.Comment: 20 pages, 4 figure

Finite type modules and Bethe Ansatz for quantum toroidal gl(1)

We study highest weight representations of the Borel subalgebra of the quantum toroidal gl(1) algebra with finite-dimensional weight spaces. In particular, we develop the q-character theory for such modules. We introduce and study the subcategory of `finite type' modules. By definition, a module over the Borel subalgebra is finite type if the Cartan like current \psi^+(z) has a finite number of eigenvalues, even though the module itself can be infinite dimensional. We use our results to diagonalize the transfer matrix T_{V,W}(u;p) analogous to those of the six vertex model. In our setting T_{V,W}(u;p) acts in a tensor product W of Fock spaces and V is a highest weight module over the Borel subalgebra of quantum toroidal gl(1) with finite-dimensional weight spaces. Namely we show that for a special choice of finite type modules $V$ the corresponding transfer matrices, Q(u;p) and T(u;p), are polynomials in u and satisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatz equation for the zeroes of the eigenvalues of Q(u;p). Then we show that the eigenvalues of T_{V,W}(u;p) are given by an appropriate substitution of eigenvalues of Q(u;p) into the q-character of V.Comment: Latex 42 page

Form factors and action of U_{\sqrt{-1}}(sl_2~) on infinite-cycles

Let ${\bf p}=\{P_{n,l}\}_{n,l\in\Z_{\ge 0}\atop n-2l=m}$ be a sequence of skew-symmetric polynomials in $X_1,...,X_l$ satisfying $\deg_{X_j}P_{n,l}\le n-1$, whose coefficients are symmetric Laurent polynomials in $z_1,...,z_n$. We call ${\bf p}$ an $\infty$-cycle if $P_{n+2,l+1}\bigl|_{X_{l+1}=z^{-1},z_{n-1}=z,z_n=-z} =z^{-n-1}\prod_{a=1}^l(1-X_a^2z^2)\cdot P_{n,l}$ holds for all $n,l$. These objects arise in integral representations for form factors of massive integrable field theory, i.e., the SU(2)-invariant Thirring model and the sine-Gordon model. The variables $\alpha_a=-\log X_a$ are the integration variables and $\beta_j=\log z_j$ are the rapidity variables. To each $\infty$-cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the $\infty$-cycles. In this paper, we define an action of $U_{\sqrt{-1}}(\widetilde{\mathfrak{sl}}_2)$ on the space of $\infty$-cycles. There are two sectors of $\infty$-cycles depending on whether $n$ is even or odd. Using this action, we show that the character of the space of even (resp. odd) $\infty$-cycles which are polynomials in $z_1,...,z_n$ is equal to the level $(-1)$ irreducible character of $\hat{\mathfrak{sl}}_2$ with lowest weight $-\Lambda_0$ (resp. $-\Lambda_1$). We also suggest a possible tensor product structure of the full space of $\infty$-cycles.Comment: 27 pages, abstract and section 3.1 revise