Fermion masses can be generated through four-fermion condensates when symmetries prevent fermion bilinear condensates from forming. This less explored mechanism of fermion mass generation is responsible for making four reduced staggered lattice fermi
ons massive at strong couplings in a lattice model with a local four-fermion coupling. The model has a massless fermion phase at weak couplings and a massive fermion phase at strong couplings. In particular there is no spontaneous symmetry breaking of any lattice symmetries in both these phases. Recently it was discovered that in three space-time dimensions there is a direct second order phase transition between the two phases. Here we study the same model in four space-time dimensions and find results consistent with the existence of a narrow intermediate phase with fermion bilinear condensates, that separates the two asymptotic phases by continuous phase transitions.
The fermion bag is a powerful idea that helps to solve fermion lattice field theories using Monte Carlo methods. Some sign problems that had remained unsolved earlier can be solved within this framework. In this work we argue that the fermion bag als
o gives insight into a new mechanism of fermion mass generation, especially at strong couplings where fermion masses are related to the fermion bag size. On the other hand, chiral condensates arise due to zero modes in the Dirac operator within a fermion bag. Although in traditional four-fermion models the two quantities seem to be related, we show that they can be decoupled. While fermion bags become small at strong couplings, the ability of zero modes of the Dirac operator within fermion bags to produce a chiral condensate, can be suppressed by the presence of additional zero modes from other fermions. Thus, fermions can become massive even without a chiral condensate. This new mechanism of mass generation was discovered long ago in lattice field theory, but has remained unappreciated. Recent work suggests that it may be of interest even in continuum quantum field theory.
We study a lattice field theory model containing two flavors of massless staggered fermions with an onsite four-fermion interaction. The model contains a $SU(4)$ symmetry which forbids non-zero fermion bilinear mass terms, due to which there is a mas
sless fermion phase at weak couplings. However, even at strong couplings fermion bilinear condensates do not appear in our model, although fermions do become massive. While the existence of this exotic strongly coupled massive fermion phase was established long ago, the nature of the transition between the massless and the massive phase has remained unclear. Using Monte Carlo calculations in three space-time dimensions, we find evidence for a direct second order transition between the two phases suggesting that the exotic lattice phase may have a continuum limit at least in three dimensions. A similar exotic second order critical point was found recently in a bilayer system on a honeycomb lattice.
We study quantum critical behavior in three dimensional lattice Gross-Neveu models containing two massless Dirac fermions. We focus on two models with SU(2) flavor symmetry and either a $Z_2$ or a U(1) chiral symmetry. Both models could not be studie
d earlier due to sign problems. We use the fermion bag approach which is free of sign problems and compute critical exponents at the phase transitions. We estimate $ u = 0.83(1)$, $eta = 0.62(1)$, $eta_psi = 0.38(1)$ in the $Z_2$ and $ u = 0.849(8)$, $eta = 0.633(8)$, $eta_psi = 0.373(3)$ in the U(1) model.
We prove that sign problems in the traditional approach to some lattice Yukawa models can be completely solved when the fermions are formulated using fermion bags and the bosons are formulated in the worldline representation. We prove this within the
context of two examples of three dimensional models, symmetric under $U_L(1) times U_R(1) times Z_2 ({Parity})$ transformations, one involving staggered fermions and the other involving Wilson fermions. We argue that these models have interesting quantum phase transitions that can now be studied using Monte Carlo methods.
We introduce a new mathematical object, the fermionant ${mathrm{Ferm}}_N(G)$, of type $N$ of an $n times n$ matrix $G$. It represents certain $n$-point functions involving $N$ species of free fermions. When N=1, the fermionant reduces to the determin
ant. The partition function of the repulsive Hubbard model, of geometrically frustrated quantum antiferromagnets, and of Kondo lattice models can be expressed as fermionants of type N=2, which naturally incorporates infinite on-site repulsion. A computation of the fermionant in polynomial time would solve many interesting fermion sign problems.
We propose a new approach to the fermion sign problem in systems where there is a coupling $U$ such that when it is infinite the fermions are paired into bosons and there is no fermion permutation sign to worry about. We argue that as $U$ becomes fin
ite fermions are liberated but are naturally confined to regions which we refer to as {em fermion bags}. The fermion sign problem is then confined to these bags and may be solved using the determinantal trick. In the parameter regime where the fermion bags are small and their typical size does not grow with the system size, construction of Monte Carlo methods that are far more efficient than conventional algorithms should be possible. In the region where the fermion bags grow with system size, the fermion bag approach continues to provide an alternative approach to the problem but may lose its main advantage in terms of efficiency. The fermion bag approach also provides new insights and solutions to sign problems. A natural solution to the silver blaze problem also emerges. Using the three dimensional massless lattice Thirring model as an example we introduce the fermion bag approach and demonstrate some of these features. We compute the critical exponents at the quantum phase transition and find $ u=0.87(2)$ and $eta=0.62(2)$.
The dimensionless parameter $xi = M^2/(16 pi^2 F^2)$, where $F$ is the pion decay constant in the chiral limit and $M$ is the pion mass at leading order in the quark mass, is expected to control the convergence of chiral perturbation theory applicabl
e to QCD. Here we demonstrate that a strongly coupled lattice gauge theory model with the same symmetries as two-flavor QCD but with a much lighter $sigma$-resonance is different. Our model allows us to study efficiently the convergence of chiral perturbation theory as a function of $xi$. We first confirm that the leading low energy constants appearing in the chiral Lagrangian are the same when calculated from the $epsilon$-regime and the $p$-regime. However, $xi lesssim 0.002$ is necessary before 1-loop chiral perturbation theory predicts the data within 1%. However, for $xi > 0.0035$ the data begin to deviate qualitatively from 1-loop chiral perturbation theory predictions. We argue that this qualitative change is due to the presence of a light $sigma$-resonance in our model. Our findings may be useful for lattice QCD studies.
