We present a new approach to some four-fermion lattice field theories which we call the generalized fermion bag approach.The basic idea is to identify unpaired fermionic degrees of freedom that cause sign problems and collect them in a bag.Paired fermions usually act like bosons and do not lead to sign problems.A resummation of all unpaired fermion degrees of freedom inside the bag is sufficient to solve the fermion sign problem in a variety of interesting cases.Using a concept of duality we then argue that the size of the fermion bags is small both at strong and weak couplings.This allows us to construct efficient algorithms in both these limits.Using the fermion bag approach, we study the quantum phase transition of the 3D massless lattice Thirrring model which is of interest in the context of Graphene.Using our method we are able to solve the model on lattices as large as 40 3 with moderate computational resources.We obtain the precise location of the quantum critical point and the values of the critical exponents through this study.
Two formidable bottlenecks to the applicability of QMC include: (1) the sign problem and (2) algorithmic update inefficiencies. In this thesis, I overcome both these difficulties for a class of problems by extending the fermion bag approach to the Hamiltonian formalism and demonstrating progress using the example of a specific quantum system known as the
t-
V model, which exhibits a transition from a semimetal to an insulator phase for a single flavor of four-component Dirac fermions. The success of this extension is demonstrated in two ways: first, through solutions to sign problems, and second, through the development of new efficient QMC algorithms. In addressing the first point, I present a solution to the sign problem for the aforementioned
t-
V model, which is then extended to many other Hamiltonian models within a class that involves fermions interacting with quantum spins. Some of these models contain an interesting quantum phase transition between a massless/semimetal phase to a massive/insulator phase in the Gross-Neveu universality class. The second point is addressed through the construction of a Hamiltonian fermion bag algorithm, which is then used to compute critical exponents for the second-order phase transition in the
t-
V model and is described in detail here. The largest lattice sizes of
642 at a comparably low temperature are reachable due to efficiency gains from this Hamiltonian fermion bag algorithm. The two independent critical exponents I find, which completely characterize the phase transition, are
η=.51(3) and
ν=.89(1). The finite size scaling fit is excellent with a
χ2/DOF=.90, showing strong evidence for a second-order critical phase transition, and hence a non-perturbative QFT can be defined at the critical point.
The fermion bag approach is a new method to tackle fermion sign problems in lattice field theories. Using this approach it is possible to solve a class of sign problems that seem unsolvable by traditional methods. The new solutions emerge when partition functions are written in terms of fermion bags and bosonic worldlines. In these new variables it is possible to identify hidden pairing mechanisms which lead to the solutions. The new solutions allow us for the first time to use Monte Carlo methods to solve a variety of interesting lattice field theories, thus creating new opportunities for understanding strongly correlated fermion systems.
Motivated by the fermion-bag approach, we construct a new class of Hamiltonian lattice field theories that can help us to study fermionic quantum critical points, particularly those with four-fermion interactions. Although these theories are constructed in discrete time with a finite temporal lattice spacing
ϵ, when $ϵ\ensuremath{\rightarrow}0$, conventional continuous-time Hamiltonian lattice field theories are recovered. The fermion-bag algorithms run relatively faster when
ϵ=1 as compared to $ϵ\ensuremath{\rightarrow}0$ but still allow us to compute universal quantities near the quantum critical point even at such a large value of
ϵ. As an example of this new approach, here we study the
Nf=1 Gross-Neveu chiral-Ising universality class in
2+1 dimensions by calculating the critical scaling of the staggered mass order parameter. We show that we are able to study lattice sizes up to
1002 sites when
ϵ=1, while with comparable resources we can reach lattice sizes of only up to
642 when $ϵ\ensuremath{\rightarrow}0$. The critical exponents obtained in both these studies match within errors.
We extend the idea of fermion bags to Hamiltonian lattice field theories in the continuous time formulation. Using a class of models we argue that the temperature is a parameter that splits the fermion dynamics into small spatial regions that can be used to identify fermion bags. Using this idea we construct a continuous time quantum Monte Carlo algorithm and compute critical exponents in the
3d Ising Gross-Neveu universality class using a single flavor of massless Hamiltonian staggered fermions. We find $\ensuremath{\eta}=0.54(6)$ and $\ensuremath{\nu}=0.88(2)$ using lattices up to
N=2304 sites. We argue that even sizes up to
N=10,000 sites should be accessible with supercomputers available today.
We report on studies of quantum critical behavior in three dimensional lattice Gross-Neveu models with one flavor of staggered fermions.We focus on two models, one with SU(2) × Z 2 symmetry and the other with an SU(2) × U(1) symmetry.Both these models could not be studied earlier with conventional Monte Carlo methods due to sign problems.However, the fermion bag approach is free of sign problems for these models and allows us to compute the critical exponents at the quantum phase transition that separates the massless fermion phase at small couplings and the massive fermion phase at large couplings.Our results help resolve some old puzzles in the field.