Renyi Entropy and Parity Oscillations of the Anisotropic Spin-s Heisenberg Chains in a Magnetic Field

Using the density matrix renormalization group, we investigate the Renyi entropy of the anisotropic spin-s Heisenberg chains in a z-magnetic field. We considered the half-odd integer spin-s chains, with s=1/2,3/2 and 5/2, and periodic and open boundary conditions. In the case of the spin-1/2 chain we were able to obtain accurate estimates of the new parity exponents $p_{\alpha}^{(p)}$ and $p_{\alpha}^{(o)}$ that gives the power-law decay of the oscillations of the $\alpha-$Renyi entropy for periodic and open boundary conditions, respectively. We confirm the relations of these exponents with the Luttinger parameter $K$, as proposed by Calabrese et al. [Phys. Rev. Lett. 104, 095701 (2010)]. Moreover, the predicted periodicity of the oscillating term was also observed for some non-zero values of the magnetization $m$. We show that for $s>1/2$ the amplitudes of the oscillations are quite small, and get accurate estimates of $p_{\alpha}^{(p)}$ and $p_{\alpha}^{(o)}$ become a challenge. Although our estimates of the new universal exponents $p_{\alpha}^{(p)}$ and $p_{\alpha}^{(o)}$ for the spin-3/2 chain are not so accurate, they are consistent with the theoretical predictions.Comment: revised version, accepted to PRB. 9 pages, 3 Figures, 4 Table

Finite-size corrections of the Entanglement Entropy of critical quantum chains

Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\alpha$-Renyi entropy of a single interval for several critical quantum chains. We considered models with U(1) symmetry like the spin-1/2 XXZ and spin-1 Fateev-Zamolodchikov models, as well models with discrete symmetries such as the Ising, the Blume-Capel and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, that depend on the value of $\alpha$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\alpha$-Renyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\alpha$-Renyi entropies is $p_{\alpha}=2X_{\epsilon}/\alpha$ for $\alpha>1$ and $p_{1}=\nu$, where $X_{\epsilon}$ is the dimensions of the energy operator of the model and $\nu=2$ for all the models.Comment: final version, 9 Pages, 3 figures, 4 table

Conformal invariance studies of the Baxter-Wu model and a related site-colouring problem

The partition function of the Baxter-Wu model is exactly related to the generating function of a site-colouring problem on a hexagonal lattice. We extend the original Bethe ansatz solution of these models in order to obtain the eigenspectra of their transfer matrices in finite geometries and general toroidal boundary conditions. The operator content of these models are studied by solving numerically the Bethe-ansatz equations and by exploring conformal invariance. Since the eigenspectra are calculated for large lattices, the corrections to finite-size scaling are also calculated.Comment: 12 pages, latex, to appear in J. Phys. A: Gen. Mat

The Critical Behaviour of Potts models with symmetry breaking fields

The $Q$-state Potts model in two dimensions in the presence of external magnetic fields is studied. For general $Q\geq3$ special choices of these magnetic fields produce effective models with smaller $Z(Q')$ symmetry $(Q'< Q)$. The phase diagram of these models and their critical behaviour are explored by conventional finite-size scaling and conformal invariance. The possibility of multicritical behavior, for finite values of the symmetry breaking fields, in the cases where $Q>4$ is also analysed. Our results indicate that for effective models with $Z(Q')$ symmetry $(Q'\leq4)$ the multicritical point occurs at zero field. This last result is also corroborated by Monte Carlo simulations.Comment: 15 pages (standart LaTex), 2 figure (PostScript) available by request to [email protected]

The Critical Behaviour of the Spin-3/2 Blume-Capel Model in Two Dimensions

The phase diagram of the spin-3/2 Blume-Capel model in two dimensions is explored by conventional finite-size scaling, conformal invariance and Monte Carlo simulations. The model in its $\tau$-continuum Hamiltonian version is also considered and compared with others spin-3/2 quantum chains. Our results indicate that differently from the standard spin-1 Blume-Capel model there is no multicritical point along the order-disorder transition line. This is in qualitative agreement with mean field prediction but in disagreement with previous approximate renormalization group calculations. We also presented new results for the spin-1 Blume-Capel model.Comment: latex 18 pages, 4 figure

Entanglement entropy of two disjoint intervals in c=1 theories

We study the scaling of the Renyi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge c=1. We provide the analytic conformal field theory result for the second order Renyi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor network techniques that allowed to obtain the reduced density matrices of disjoint blocks of the spin-chain and to check the correctness of the predictions for Renyi and entanglement entropies from conformal field theory. In order to match these predictions, we have extrapolated the numerical results by properly taking into account the corrections induced by the finite length of the blocks to the leading scaling behavior.Comment: 37 pages, 23 figure

Entanglement entropy of two disjoint blocks in XY chains

We study the Renyi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking properly into account the Jordan-Wigner string between the two blocks. From this we can evaluate any Renyi entropy of finite integer order. We study in details critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results in the gapped phase and after a quantum quench.Comment: 34 pages, 11 figure

A mechanically active heterotypic E-cadherin/N-cadherin adhesion enables fibroblasts to drive cancer cell invasion

Cancer-associated fibroblasts (CAFs) promote tumour invasion and metastasis. We show that CAFs exert a physical force on cancer cells that enables their collective invasion. Force transmission is mediated by a heterophilic adhesion involving N-cadherin at the CAF membrane and E-cadherin at the cancer cell membrane. This adhesion is mechanically active; when subjected to force it triggers β-catenin recruitment and adhesion reinforcement dependent on α-catenin/vinculin interaction. Impairment of E-cadherin/N-cadherin adhesion abrogates the ability of CAFs to guide collective cell migration and blocks cancer cell invasion. N-cadherin also mediates repolarization of the CAFs away from the cancer cells. In parallel, nectins and afadin are recruited to the cancer cell/CAF interface and CAF repolarization is afadin dependent. Heterotypic junctions between CAFs and cancer cells are observed in patient-derived material. Together, our findings show that a mechanically active heterophilic adhesion between CAFs and cancer cells enables cooperative tumour invasion

Fine-mapping of prostate cancer susceptibility loci in a large meta-analysis identifies candidate causal variants

Prostate cancer is a polygenic disease with a large heritable component. A number of common, low-penetrance prostate cancer risk loci have been identified through GWAS. Here we apply the Bayesian multivariate variable selection algorithm JAM to fine-map 84 prostate cancer susceptibility loci, using summary data from a large European ancestry meta-analysis. We observe evidence for multiple independent signals at 12 regions and 99 risk signals overall. Only 15 original GWAS tag SNPs remain among the catalogue of candidate variants identified; the remainder are replaced by more likely candidates. Biological annotation of our credible set of variants indicates significant enrichment within promoter and enhancer elements, and transcription factor-binding sites, including AR, ERG and FOXA1. In 40 regions at least one variant is colocalised with an eQTL in prostate cancer tissue. The refined set of candidate variants substantially increase the proportion of familial relative risk explained by these known susceptibility regions, which highlights the importance of fine-mapping studies and has implications for clinical risk profiling. © 2018 The Author(s).Prostate cancer is a polygenic disease with a large heritable component. A number of common, low-penetrance prostate cancer risk loci have been identified through GWAS. Here we apply the Bayesian multivariate variable selection algorithm JAM to fine-map 84 prostate cancer susceptibility loci, using summary data from a large European ancestry meta-analysis. We observe evidence for multiple independent signals at 12 regions and 99 risk signals overall. Only 15 original GWAS tag SNPs remain among the catalogue of candidate variants identified; the remainder are replaced by more likely candidates. Biological annotation of our credible set of variants indicates significant enrichment within promoter and enhancer elements, and transcription factor-binding sites, including AR, ERG and FOXA1. In 40 regions at least one variant is colocalised with an eQTL in prostate cancer tissue. The refined set of candidate variants substantially increase the proportion of familial relative risk explained by these known susceptibility regions, which highlights the importance of fine-mapping studies and has implications for clinical risk profiling. © 2018 The Author(s).Peer reviewe