We perform large scale quantum Monte Carlo simulations of the Hubbard model at half filling with a single Dirac cone close to the critical point, which separates a Dirac semi-metal from an antiferromagnetically ordered phase where SU(2) spin rotational symmetry is spontaneously broken. We discuss the implementation of a single Dirac cone in the SLAC formulation for eight Dirac components and the influence of dynamically induced long-range super-exchange interactions. The finite size behavior of dimensionless ratios and the finite size scaling properties of the Hubbard model with a single Dirac cone are shown to be superior compared to the honeycomb lattice. We extract the critical exponent believed to belong to the chiral Heisenberg Gross-Neveu-Yukawa universality class: The critical exponent ${\nu = 1.02(3)}$ coincides for the two lattice types once honeycomb lattices of linear dimension ${L\ge 15}$ are considered. In contrast to the SLAC formulation, where the anomalous dimensions are estimated to be ${\eta_{\phi}=0.73(1)}$ and ${\eta_{\psi}=0.09(1)}$, they remain less stable on honeycomb lattices, but tend towards the estimates from the SLAC formulation.

Thomas C. Lang Andreas M. Läuchli

Given a finite Coxeter system $(W,S)$ and a Coxeter element $c$, we construct a simple polytope whose outer normal fan is N. Reading's Cambrian fan $F_c$, settling a conjecture of Reading that this is possible. We call this polytope the $c$-generalized associahedron. Our approach generalizes Loday's realization of the associahedron (a type $A$ $c$-generalized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the $c$-singleton cones, the cones in the $c$-Cambrian fan which consist of a single maximal cone from the Coxeter fan. Moreover, if $W$ is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to $W$, then we show that our realizations have integer coordinates in this lattice.

Christophe Hohlweg Carsten Lange Hugh Thomas

We study the orbits of a polynomial f in C[X], namely the sets {e,f(e),f(f(e)),...} with e in C. We prove that if nonlinear complex polynomials f and g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C^d with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both this result and the Mordell--Lang conjecture.

Dragos Ghioca Thomas J. Tucker Michael E. Zieve

Let $(W,S)$ be a finite Coxeter system acting by reflections on an $\mathbb R$-Euclidean space with simple roots $\Delta=\{\a_s | s\in S\}$ of the same length and fundamental weights $\Delta^*=\{v_s | s\in S\}$. We set $M(e)=\sum_{s\in S}\kappa_s v_s$, $\kappa_s>0$, and for $w\in W$ we set $M(w)=w(M(e))$. The permutahedron $Perm(W)$ is the convex hull of the set $\{M(w) | w\in W\}$. Given a Coxeter element $c\in W$, we have defined in a previous work a generalized associahedron $Asso_c(W)$ whose normal fan is the corresponding $c$-Cambrian fan $F_c$ defined by N. Reading. By construction, $Asso_c(W)$ is obtained from $Perm(W)$ by removing some halfspaces according to a rule prescribed by $c$. In this work, we classify the isometry classes of these realizations. More precisely, for $(W,S)$ an irreducible finite Coxeter system and $c,c'$ two Coxeter elements in $W$, we have that $Asso_{c}(W)$ and $Asso_{c'}(W)$ are isometric if and only if $\mu(c') = c$ or $\mu(c')=w_0c^{-1}w_0$ for $\mu$ an automorphism of the Coxeter graph of $W$ such that $\kappa_s=\kappa_{\mu(s)}$ for all $s\in S$. As a byproduct, we classify the isometric Cambrian fans of $W$.

Nantel Bergeron Christophe Hohlweg Carsten Lange Hugh Thomas

In this contribution we present results from quenched QCD simulations with the parameterized fixed-point (FP) and the chirally improved (CI) Dirac operator. Both these operators are approximate solutions of the Ginsparg-Wilson equation and have good chiral properties. We focus our discussion on observables sensitive to chirality. In particular we explore pion masses down to 210 MeV in light hadron spectroscopy, quenched chiral logs, the pion decay constant and the pion scattering length. We discuss finite volume effects, scaling properties of the FP and CI operators and performance issues in their numerical implementation.

Christof Gattringer Meinulf Göckeler Peter Hasenfratz Simon Hauswirth Kieran Holland Thomas Jörg K. J. Juge C. B. Lang Ferenc Niedermayer P. E. L. Rakow Stefan Schaefer Andreas Schäfer

We present quantum Monte Carlo simulations for the chiral Heisenberg Gross-Neveu-Yukawa quantum phase transition of relativistic fermions with $N=4$ Dirac spinor components subject to a repulsive, local four fermion interaction in 2+1$d$. Here we employ a two dimensional lattice Hamiltonian with a single, spin-degenerate Dirac cone, which exactly reproduces a linear energy-momentum relation for all finite size lattice momenta in the absence of interactions. This allows us to significantly reduce finite size corrections compared to the widely studied honeycomb and $\pi$-flux lattices. A Hubbard term dynamically generates a mass beyond a critical coupling of ${U_c = 6.76(1)}$ as the system acquires antiferromagnetic order and SU(2) spin rotational symmetry is spontaneously broken. At the quantum phase transition we extract a self-consistent set of critical exponents ${\nu = 0.98(1)}$, ${\eta_{\phi} = 0.53(1)}$, ${\eta_{\psi} = 0.18(1)}$, ${\beta = 0.75(1)}$. We provide evidence for the continuous degradation of the quasi-particle weight of the fermionic excitations as the critical point is approached from the semimetallic phase. Finally we study the effective "speed of light" of the low-energy relativistic description, which depends on the interaction $U$, but is expected to be regular across the quantum phase transition. We illustrate that the strongly coupled bosonic and fermionic excitations share a common velocity at the critical point.

Thomas C. Lang Andreas M. Läuchli

We examine a basic lattice model of interacting fermions that exhibits quadratic band crossing points (QBCPs) in the non-interacting limit. In particular, we consider spinless fermions on the honeycomb lattice with nearest neighbor hopping $t$ and third-nearest neighbor hopping $t''$, which exhibits fine-tuned QBCPs at the corners of the Brillouin zone for ${t'' = t/2}$. In this situation, the density of states remains finite at the Fermi level of the half-filled band and repulsive nearest-neighbor interactions $V$ lead to a charge-density-wave (CDW) instability at infinitesimally small $V$ in the random-phase approximation or mean-field theory. We examine the fragility of the QBCPs against dispersion renormalizations in the ${t\mbox{-}t''\mbox{-}V}$ model using perturbation theory, and find that the $t''$-value needed for the QBCPs increases with $V$ due to the hopping renormalization. However, the instability toward CDW formation always requires a nonzero threshold interaction strength, i.e., one cannot fine-tune $t''$ to recover the QBCPs in the interacting system. These perturbative arguments are supported by quantum Monte Carlo simulations for which we carefully compare the corresponding threshold scales at and beyond the QBCP fine-tuning point. From this analysis, we thus gain a quantitative microscopic understanding of the fragility of the QBCPs in this basic interacting fermion system.

Stephan Hesselmann Carsten Honerkamp Stefan Wessel Thomas C. Lang

We prove a dynamical version of the Mordell-Lang conjecture in the context of Drinfeld modules. We use analytic methods similar to the ones employed by Skolem, Chabauty, and Coleman for studying diophantine equations.

Dragos Ghioca Thomas J. Tucker

We prove a dynamical version of the Mordell-Lang conjecture for subvarieties of quasiprojective varieties X, endowed with the action of a morphism f:X --> X. We use an analytic method based on the technique of Skolem, Mahler, and Lech, along with results of Herman and Yoccoz from nonarchimedean dynamics.

Dragos Ghioca Thomas J. Tucker

We have implemented a Langevin approach for the transport of heavy quarks in the UrQMD hybrid model. The UrQMD hybrid approach provides a realistic description of the background medium for the evolution of relativistic heavy ion collisions. We have used two different sets of drag and diffusion coefficients, one based on a $T$-Matrix approach and one based on a resonance model for the elastic scattering of heavy quarks within the medium. In case of the resonance model we have investigated the effects of different decoupling temperatures of the heavy quarks from the medium, ranging between $130\,\text{MeV}$ and $180\,\text{MeV}$. We present calculations of the nuclear modification factor $R_{AA}$, as well as of the elliptic flow $v_2$ in Au+Au collisions at $\sqrt{s_{NN}}=200\,\text{GeV}$ and Pb+Pb collisions at $\sqrt{s_{NN}}=2.76\,\text{TeV}$. To make our results comparable to experimental data at RHIC and LHC we have implemented a Peterson fragmentation and a quark coalescence approach followed by the semileptonic decay of the D- and B-mesons to electrons. We find that our results strongly depend on the decoupling temperature and the hadronization mechanism. At a decoupling temperature of $130\,\text{MeV}$ we reach a good agreement with the measurements at both, RHIC and LHC energies, simultaneously for the elliptic flow $v_2$ and the nuclear modification factor $R_{AA}$.

Thomas Lang Hendrik van Hees Jan Steinheimer Marcus Bleicher