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

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

Biological treatment strategies for disc degeneration: potentials and shortcomings

Recent advances in molecular biology, cell biology and material sciences have opened a new emerging field of techniques for the treatment of musculoskeletal disorders. These new treatment modalities aim for biological repair of the affected tissues by introducing cell-based tissue replacements, genetic modifications of resident cells or a combination thereof. So far, these techniques have been successfully applied to various tissues such as bone and cartilage. However, application of these treatment modalities to cure intervertebral disc degeneration is in its very early stages and mostly limited to experimental studies in vitro or in animal studies. We will discuss the potential and possible shortcomings of current approaches to biologically cure disc degeneration by gene therapy or tissue engineering. Despite the increasing number of studies examining the therapeutic potential of biological treatment strategies, a practicable solution to routinely cure disc degeneration might not be available in the near future. However, knowledge gained from these attempts might be applied in a foreseeable future to cure the low back pain that often accompanies disc degeneration and therefore be beneficial for the patien

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

Thermodynamics and short-range correlations of the XXZ chain close to its triple point

The XXZ quantum spin chain has a triple point in its ground state $h$-$1/\Delta$ phase diagram. This first order critical point is located at the joint end point of the two second order phase transition lines marking the transition from the gapless phase to the fully polarized phase and to the N\'eel ordered phase, respectively. We explore the magnetization and the short-range correlation functions in its vicinity using the exact solution of the model. In the critical regime above the triple point we observe a strong variation of all physical quantities on a low energy scale of order $1/\Delta$ induced by the transversal quantum fluctuations. We interpret this phenomenon starting from a strong-coupling perturbation theory about the highly degenerate ground state of the Ising chain at the triple point. From the perturbation theory we identify the relevant scaling of the magnetic field and of the temperature. Applying the scaling to the exact solutions we obtain explicit formulae for the magnetization and short-range correlation functions at low temperatures.Comment: 18 pages, 7 figures, v2: figures rearranged, v3: a typo correcte

Emptiness formation probability at finite temperature for the isotropic Heisenberg chain

We present an integral formula for a special correlation function of the isotropic spin-1/2 antiferromagnetic Heisenberg chain. The correlation function describes the probability for the occurrence of a string of consecutive up-spins as a function of temperature, magnetic field and length of the string.Comment: 3 pages, 1 figure, submitted to SCES'0

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

The Arteriovenous Loop: Engineering of Axially Vascularized Tissue

Background: Most of the current treatment options for large-scale tissue defects represent a serious burden for the patients, are often not satisfying, and can be associated with significant side effects. Although major achievements have already been made in the field of tissue engineering, the clinical translation in case of extensive tissue defects is only in its early stages. The main challenge and reason for the failure of most tissue engineering approaches is the missing vascularization within large-scale transplants. Summary: The arteriovenous (AV) loop model is an in vivo tissue engineering strategy for generating axially vascularized tissues using the own body as a bioreactor. A superficial artery and vein are anastomosed to create an AV loop. This AV loop is placed into an implantation chamber for prevascularization of the chamber inside, e.g., a scaffold, cells, and growth factors. Subsequently, the generated tissue can be transplanted with its vascular axis into the defect site and anastomosed to the local vasculature. Since the blood supply of the growing tissue is based on the AV loop, it will be immediately perfused with blood in the recipient site leading to optimal healing conditions even in the case of poorly vascularized defects. Using this tissue engineering approach, a multitude of different axially vascularized tissues could be generated, such as bone, skeletal or heart muscle, or lymphatic tissues. Upscaling from the small animal AV loop model into a preclinical large animal model could pave the way for the first successful attempt in clinical application. Key Messages: The AV loop model is a powerful tool for the generation of different axially vascularized replacement tissues. Due to minimal donor site morbidity and the possibility to generate patient-specific tissues variable in type and size, this in vivo tissue engineering approach can be considered as a promising alternative therapy to current treatment options of large-scale defects

Matrix metalloproteinase expression levels suggest distinct enzyme roles during lumbar disc herniation and degeneration

The disruption of the extracellular disc matrix is a major hallmark of disc degeneration. This has previously been shown to be associated with an up-regulation of major matrix metalloproteinase (MMP) expression and activity. However, until now hardly any data are available for MMP/TIMP regulation and thereby no concept exists as to which MMP/TIMP plays a major role in disc degeneration. The objective of this study was, therefore, to identify and quantify the putative up-regulation of MMPs/TIMPs on the mRNA and protein level and their activity in disc material in relation to clinical data and histological evidence for disc degeneration. A quantitative molecular analysis of the mRNA expression levels for the MMPs (MMPs-1, -2, -3, -7, -8, -9, -13) and the MMP inhibitors (TIMPs-1 and -2) was performed on 37 disc specimens obtained from symptomatic disc herniation or degeneration. In addition, disc specimens from patients without disc degeneration/herniation (=controls) were analyzed. Expression of MMPs-1, -2, -3, -7, -8, -9, -13 and TIMPs-1, -2 was analyzed using quantitative RT-PCR, normalized to the expression level of a house keeping gene (GAPDH). Gene expression patterns were correlated with MMP activity (in situ zymography), protein expression patterns (immunohistochemistry), degeneration score (routine histology) and clinical data. MMP-3 mRNA levels were consistently and substantially up-regulated in samples with histological evidence for disc degeneration. A similar but less pronounced up-regulation was observed for MMP-8. This up-regulation was paralleled by the expression of TIMP-1 and to a lesser extent TIMP-2. In general, these findings could be confirmed with regard to protein expression and enzyme activity. This study provides data on the gene and protein level, which highlights the key role of MMP-3 in the degenerative cascade leading to symptomatic disc degeneration and herniation. Control of the proteolytic activity of MMP-3 may, therefore, come into the focus when aiming to develop new treatment options for early disc degeneratio