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arXiv:1912.11776

Fermion Bag Approach for Hamiltonian Lattice Field Theories

Emilie Huffman

Published Dec 26, 2019• 2 citations
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Abstract

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 $64^2$ 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 $χ^2/DOF=.90$, showing strong evidence for a second-order critical phase transition, and hence a non-perturbative QFT can be defined at the critical point.

DOI
10.48550/arxiv.1912.11776

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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 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 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 3d3d3d 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=2304N=2304N=2304 sites. We argue that even sizes up to N=10,000N=10,000N=10,000 sites should be accessible with supercomputers available today.
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.
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 ϵϵϵ, when $ϵ\ensuremath{\rightarrow}0$, conventional continuous-time Hamiltonian lattice field theories are recovered. The fermion-bag algorithms run relatively faster when ϵ=1ϵ=1ϵ=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 ϵϵϵ. As an example of this new approach, here we study the Nf=1{N}_{f}=1Nf​=1 Gross-Neveu chiral-Ising universality class in 2+12+12+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{100}^{2}1002 sites when ϵ=1ϵ=1ϵ=1, while with comparable resources we can reach lattice sizes of only up to 642{64}^{2}642 when $ϵ\ensuremath{\rightarrow}0$. The critical exponents obtained in both these studies match within errors.
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.
According to the Landau criterion, a phase transition should be first order when cubic terms of order parameters are allowed in its effective Ginzburg-Landau free energy. Recently, it was shown by renormalization-group (RG) analysis that continuous transition can happen at putatively first-order Z3{Z}_{3}Z3​ transitions in two-dimensional (2D) Dirac semimetals and such non-Landau phase transitions were dubbed ``fermion-induced quantum critical points'' (FIQCPs) [Z.-X. Li, Y.-F. Jiang, S.-K. Jian, and H. Yao, Nat. Commun. 8, 314 (2017)]. The RG analysis, controlled by the 1/N1/N1/N expansion with NNN the number of flavors of four-component Dirac fermions, shows that the FIQCP occurs for $N\ensuremath{\ge}{N}_{c}$. Previous quantum Monte Carlo simulations of a microscopic model of SU(N\mathrm{SU}(NSU(N) fermions on the honeycomb lattice showed that the FIQCP occurs at the transition between Dirac semimetals and Kekule valence bond solids for $N\ensuremath{\ge}2$. However, the precise value of the lower bound Nc{N}_{c}Nc​ has not been established. Especially, the case of N=1N=1N=1 has not been explored by studying microscopic models so far. Here, by introducing a generalized SU(N\mathrm{SU}(NSU(N) fermion model with N=1N=1N=1 (namely, spinless fermions on the honeycomb lattice), we perform large-scale sign-problem-free Majorana quantum Monte Carlo simulations and find convincing evidence of the FIQCP for N=1N=1N=1. Consequently, our results suggest that the FIQCP can occur in 2D Dirac semimetals for all positive integers $N\ensuremath{\ge}1$.