Quantum critical phenomena may be qualitatively different when massless Dirac fermions are present at criticality. Using our recently-discovered fermion-sign-free Majorana quantum Monte Carlo method introduced by us in (Li et al 2015 Phys. Rev. B 91 241117 ), we investigate the quantum critical phenomena of spinless Dirac fermions at their charge-density-wave phase transitions on the honeycomb lattice having sites with largest L = 24. By finite-size scaling, we accurately obtain critical exponents of this so-called Gross–Neveu chiral-Ising universality class of two (two-component) Dirac fermions in 2+1D: , , and , which are qualitatively different from the mean-field results but are reasonably close to the ones obtained from renormalization group calculations.
A second-order phase transition for the three-dimensional Gross-Neveu model is established for one fermion species,
N=1. This transition breaks a paritylike discrete symmetry. It constitutes its peculiar universality class with critical exponent $\ensuremath{\nu}=0.63$ and scalar and fermionic anomalous dimensions ${\ensuremath{\eta}}_{\ensuremath{\sigma}}=0.31$ and ${\ensuremath{\eta}}_{\ensuremath{\psi}}=0.11,$ respectively. We also compute critical exponents for other N. Our results are based on exact renormalization-group equations.
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.
We study the critical properties of the
QED3-Gross-Neveu model with
2N flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For
N=1, this theory has recently been suggested to be dual to the SU(2) noncompact
CP1 model that describes the deconfined phase transition between the N\'eel antiferromagnet and the valence bond solid on the square lattice. For
N=2, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-
1/2 systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the
QED3-Gross-Neveu model for all values of
N. This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain
QED3 (without scalar-field coupling) unstable at low values of
N. The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter $\ensuremath{\epsilon}=4\ensuremath{-}D$. We compute the scalar-field anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, the correlation-length exponent $\ensuremath{\nu}$, as well as the scaling dimension of the flavor-symmetry-breaking bilinear $\overline{\ensuremath{\psi}}{\ensuremath{\sigma}}^{z}\ensuremath{\psi}$ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.
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
403 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.