Exercises with the universal R-matrix

Using the formula for the universal $R$-matrix proposed by Khoroshkin and Tolstoy, we give a detailed derivation of $L$-operators for the quantum groups associated with the generalized Cartan matrices $A_1^{(1)}$ and $A_2^{(1)}$.Comment: 36 page

Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field

We present a conjecture for the density matrix of a finite segment of the XXZ chain coupled to a heat bath and to a constant longitudinal magnetic field. It states that the inhomogeneous density matrix, conceived as a map which associates with every local operator its thermal expectation value, can be written as the trace of the exponential of an operator constructed from weighted traces of the elements of certain monodromy matrices related to $U_q (\hat{\mathfrak{sl}}_2)$ and only two transcendental functions pertaining to the one-point function and the neighbour correlators, respectively. Our conjecture implies that all static correlation functions of the XXZ chain are polynomials in these two functions and their derivatives with coefficients of purely algebraic origin.Comment: 35 page

Short-distance thermal correlations in the XXZ chain

Recent studies have revealed much of the mathematical structure of the static correlation functions of the XXZ chain. Here we use the results of those studies in order to work out explicit examples of short-distance correlation functions in the infinite chain. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field for various anisotropies in the massless regime $- 1 < \Delta < 1$. It turns out that the new formulae are numerically efficient and allow us to obtain the correlations functions over the full parameter range with arbitrary precision.Comment: 25 pages, 5 colored figure

Computation of static Heisenberg-chain correlators: Control over length and temperature dependence

We communicate results on correlation functions for the spin-1/2 Heisenberg-chain in two particularly important cases: (a) for the infinite chain at arbitrary finite temperature $T$, and (b) for finite chains of arbitrary length $L$ in the ground-state. In both cases we present explicit formulas expressing the short-range correlators in a range of up to seven lattice sites in terms of a single function $\omega$ encoding the dependence of the correlators on $T$ ($L$). These formulas allow us to obtain accurate numerical values for the correlators and derived quantities like the entanglement entropy. By calculating the low $T$ (large $L$) asymptotics of $\omega$ we show that the asymptotics of the static correlation functions at any finite distance are $T^2$ ($1/L^2$) terms. We obtain exact and explicit formulas for the coefficients of the leading order terms for up to eight lattice sites.Comment: 5 pages, 3 figures, v2: text slightly shortened, typos in eqns. (16), (17) corrected, Fig. 1 replaced, v3: typo in eqn. (11) correcte

Tibet, the Himalaya, Asian monsoons and biodiversity - In what ways are they related?

Prevailing dogma asserts that the uplift of Tibet, the onset of the Asian monsoon system and high biodiversity in southern Asia are linked, and that all occurred after 23 million years ago in the Neogene. Here, spanning the last 60 million years of Earth history, the geological, climatological and palaeontological evidence for this linkage is reviewed. The principal conclusions are that: 1) A proto-Tibetan highland existed well before the Neogene and that an Andean type topography with surface elevations of at least 4.5 km existed at the start of the Eocene, before final closure of the Tethys Ocean that separated India from Eurasia. 2) The Himalaya were formed not at the start of the India-Eurasia collision, but after much of Tibet had achieved its present elevation. The Himalaya built against a pre-existing proto-Tibetan highland and only projected above the average height of the plateau after approximately 15 Ma. 3) Monsoon climates have existed across southern Asia for the whole of the Cenozoic, and probably for a lot longer, but that they were of the kind generated by seasonal migrations of the Inter-tropical Convergence Zone. 4) The projection of the High Himalaya above the Tibetan Plateau at about 15 Ma coincides with the development of the modern South Asia Monsoon. 5) The East Asia monsoon became established in its present form about the same time as a consequence of topographic changes in northern Tibet and elsewhere in Asia, the loss of moisture sources in the Asian interior and the development of a strong winter Siberian high as global temperatures declined. 6) New radiometric dates of palaeontological finds point to southern Asia's high biodiversity originating in the Paleogene, not the Neogene