We apply the Linear Logarithmic Relaxation (LLR) method, which generalizes the Wang-Landau algorithm to quantum systems with continuous degrees of freedom, to the fermionic Hubbard model with repulsive interactions on the honeycomb lattice. We comput
e the generalized density of states of the average Hubbard field and divise two reconstruction schemes to extract physical observables from this result. By computing the particle density as a function of chemical potential we assess the utility of LLR in dealing with the sign problem of this model, which arises away from half filling. We show that the relative advantage over brute-force reweighting grows as the interaction strength is increased and discuss possible future improvements.
Hubbard-type models on the hexagonal lattice are of great interest, as they provide realistic descriptions of graphene and other related materials. Hybrid Monte Carlo simulations offer a first-principles approach to study their phase structure. Here,
we review the present status of our work in this direction.
We carry out lattice simulations of two-color QCD and spectroscopy at finite density with two flavors of rooted-staggered quarks and a diquark source term. As in a previous four-flavor study, for small values of the inverse gauge coupling we observe
a Goldstone spectrum which reflects the symmetry-breaking pattern of a Gaussian symplectic chiral random-matrix ensemble (GSE) with Dyson index $beta_D=4$, which corresponds to any-color QCD with adjoint quarks in the continuum instead of QC$_2$D wih fundamental quarks. We show that this unphysical behavior occurs only inside of the bulk phase of $SU(2)$ gauge theory, where the density of $Z_2$ monopoles is high. Using an improved gauge action and a somewhat larger inverse coupling to suppress these monopoles, we demonstrate that the continuum Goldstone spectrum of two-color QCD, corresponding to a Gaussian orthogonal ensemble (GOE) with Dyson index $beta_D=1$, is recovered also with rooted-staggered quarks once simulations are performed away from the bulk phase. We further demonstrate how this change of random-matrix ensemble is reflected in the distribution of eigenvalues of the Dirac operator. By computing the unfolded level spacings inside and outside of the bulk phase, we demonstrate that, starting with the low-lying eigenmodes which determine the infrared physics, the distribution of eigenmodes continuously changes from the GSE to the GOE one as monopoles are suppressed.
We calculate spectral functions of the relativistic $O(4)$ model from real-time lattice simulations in classical-statistical field theory. While in the low and high temperature phase of the model, the spectral functions of longitudinal $(sigma)$ and
transverse $(pi)$ modes are well described by relativistic quasi-particle peaks, we find a highly non-trivial behavior of the spectral functions in the cross over region, where additional structures appear. Similarly, we observe a significant broadening of the quasi-particle peaks, when the amount explicit $O(4)$ symmetry breaking is reduced. We further demonstrate that in the vicinity of the $O(4)$ critical point, the spectral functions develop an infrared power law associated with the critical dynamics, and comment on the extraction of the dynamical critical exponent $z$ from our simulations.
Using state of the art Hybrid-Monte-Carlo (HMC) simulations we carry out an unbiased study of the competition between spin-density wave (SDW) and charge-density wave (CDW) order in suspended graphene. We determine that the realistic inter-electron po
tential of graphene must be scaled up by a factor of roughly 1.6 to induce a semimetal-SDW phase transition and find no evidence for CDW order. A study of critical properties suggests that the universality class of the three-dimensional chiral Heisenberg Gross-Neveu model with two fermion flavors, predicted by renormalization group studies and strong-coupling expansion, is unlikely to apply to this transition. We propose that our results instead favor an interpretation in terms of a conformal phase transition. In addition, we describe a variant of the HMC algorithm which uses exact fermionic forces during molecular dynamics trajectories and avoids the use of pseudofermions. Compared to standard HMC this allows for a substantial increase of the integrator stepsize while achieving comparable Metropolis acceptance rates and leads to a sizable performance improvement.
Using first-principle Hybrid-Monte-Carlo (HMC) simulations, we carry out an unbiased study of the competition between spin-density wave (SDW) and charge-density wave (CDW) order in the extended Hubbard model on the two dimensional hexagonal lattice a
t half filling. We determine the phase diagram in the space of on-site and nearest-neighbor couplings $U$ and $V$ in the region $V<U/3$, which can be simulated without a fermion sign problem, and find that a transition from semimetal to a SDW phase occurs at sufficiently large $U$ for basically all $V$. Tracing the corresponding phase boundary from $V=0$ to the $V=U/3$ line, we find evidence for critical scaling in the Gross-Neveu universality class for the entire boundary. With rather high confidence we rule out the existence of the CDW ordered phase anywhere in the range of parameters considered. We also discuss several improvements of the HMC algorithm which are crucial to reach these conclusions, in particular the improved fermion action with exact sublattice symmetry and the complexification of the Hubbard-Stratonovich field to ensure the ergodicity of the algorithm.
We study the phase diagram of QCD at finite isospin density using two flavors of staggered quarks. We investigate the low temperature region of the phase diagram where we find a pion condensation phase at high chemical potential. We started a basic a
nalysis of the spectrum at finite isospin density. In particular, we measured pion, rho and nucleon masses inside and outside of the pion condensation phase. In agreement with previous studies in two-color QCD at finite baryon density we find that the Polyakov loop does not depend on the density in the staggered formulation.
We report on Hybrid-Monte-Carlo simulations at finite spin density of the $pi$-band electrons in monolayer graphene with realistic inter-electron interactions. Unlike simulations at finite charge-carrier density, these are not affected by a fermion-s
ign problem. Our results are in qualitative agreement with an interaction-induced warping of the Fermi contours, and a reduction of the bandwidth as observed in angle resolved photoemission spectroscopy experiments on charge-doped graphene systems. Furthermore, we find evidence that the neck-disrupting Lifshitz transition, which occurs when the Fermi level traverses the van Hove singularity (VHS), becomes a true quantum phase transition due to interactions. This is in-line with an instability of the VHS towards the formation of electronic ordered phases, which has been predicted by a variety of different theoretical approaches.
We use first-principles Quantum Monte-Carlo simulations to study the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between hydrogen adatoms attached to a graphene sheet. We find that the pairwise RKKY interactions at distances of a few lattice spa
cings are strongly affected by inter-electron interactions, in particular, the potential barrier between widely separated adatoms and the dimer configuration becomes wider and thus harder to penetrate. We also point out that anti-ferrromagnetic and charge density wave orderings have very different effects on the RKKY interaction. Finally, we analyze the stability of several regular adatom superlattices with respect to small displacements of a single adatom, distinguishing the cases of adatoms which populate either both or only one sublattice of the graphene lattice.
In this contribution we revisit simulations of two-color QCD with rooted staggered quarks at finite density, where baryon-number spontaneously breaks and a diquark condensate forms. We thereby pay special attention to simulating outside the lattice-a
rtifact bulk phase, in which $Z_2$ monopoles condense, and investigate some of the consequences of this, e.g. on the chiral and the diquark condensate which were known to be well described by chiral effective field theory. Not surprisingly, on finer lattices outside the bulk phase the quark condensate now requires additive renormalization before it can be compared with effective field theory predictions. The subtraction must necessarily depend on the chemical potential, however. The diquark condensate is not affected by this problem and remains in good agreement with these predictions. We also compare staggered with Wilson quarks to demonstrate that the two fermion discretizations yield qualitatively different results well below half-filling already. We close with prelimiary results for the Goldstone spectrum to demonstrate that the continuum pattern is recovered also with staggered quarks outside the bulk phase.
