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arXiv:1411.7147 · International Symposium on Lattice Field Theory (LATTICE)

Solution to new sign problems with Hamiltonian Lattice Fermions

Emilie Huffman

Published May 21, 2015• 0 citations
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Abstract

We present a solution to the sign problem in a class of particle-hole symmetric Hamiltonian lattice fermion models on bipartite lattices using the idea of fermion bags. The solution remains valid when the particle-hole symmetry is broken through a staggered chemical potential term. This solution allows, for the first time, simulations of some massless four-fermion models with minimal fermion doubling and with an odd number of fermion flavors using ultra-local actions. One can thus study a variety of quantum phase transitions that have remained unexplored so far due to sign problems.

DOI
10.22323/1.214.0058

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We present a solution to the sign problem in a class of particle-hole symmetric Hamiltonian lattice fermion models on bipartite lattices using the idea of fermion bags. The solution remains valid when the particle-hole symmetry is broken through a staggered chemical potential term. This solution allows, for the first time, simulations of some massless four-fermion models with minimal fermion doubling and with an odd number of fermion flavors using ultra-local actions. One can thus study a variety of quantum phase transitions that have remained unexplored so far due to sign problems.
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 ttt-VVV 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 ttt-VVV 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 ttt-VVV model and is described in detail here. The largest lattice sizes of 64264^2642 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)η=.51(3)η=.51(3) and ν=.89(1)ν=.89(1)ν=.89(1). The finite size scaling fit is excellent with a χ2/DOF=.90χ^2/DOF=.90χ2/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.
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.
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.
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 40340^3403 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 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.