It is expected that the Gross-Neveu-Yukawa (GNY) chiral Ising transition of Dirac fermions coupled with a scalar field in
(2+1) dimensions will be the first fermionic quantum critical point that various methods, such as conformal bootstrap, perturbative renormalization group, and quantum Monte Carlo (QMC) simulations, would yield converged critical exponents---serving the same role as the Ising and
O(N) models in the textbooks of statistical and quantum physics. However, such an expectation has not been fully realized from the lattice QMC simulations due to the obstacles introduced by the UV finite-size effect. In this Letter, by means of the elective-momentum ultrasize (EMUS)-QMC method, we compute the critical exponents of the $O{(N/2)}^{2}\ensuremath{\rtimes}{\mathbb{Z}}_{2}$ GNY
N=8 chiral Ising transition on a two-dimensional $\ensuremath{\pi}$-flux fermion lattice model between Dirac semimetal and quantum spin Hall insulator phases. With the matching of fermionic and bosonic momentum transfer and collective update in momentum space, our QMC results provide fully consistent exponents with those obtained from the bootstrap and perturbative approaches. In this way, the EMUS now live happily on the
N=8 island and could explore the chiral Gross-Neveu-Yukawa archipelago with ease.
There is no an exact solution to three-dimensional (3D) finite-size Ising model (referred to as the Ising model hereafter for simplicity) and even two-dimensional (2D) Ising model with non-zero external field to our knowledge. Here by using an elementary but rigorous method, we obtain an exact solution to the partition function of the Ising model with
N lattice sites. It is a sum of
2N exponential functions and holds for
D-dimensional (
D=1,2,3,...) Ising model with or without the external field. This solution provides a new insight into the problem of the Ising model and the related difficulties, and new understanding of the classic exact solutions for one-dimensional (1D) (Kramers and Wannier, 1941) or 2D Ising model (Onsager, 1944). With this solution, the specific heat and magnetisation of a simple 3D Ising model are calculated, which are consistent with the results from experiments and/or numerical simulations. Furthermore, the solution here and the related approaches, can also be available to other models like the percolation and/or the Potts model.
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.
We introduce a half-filled Hamiltonian of spin-half lattice fermions that can be studied with the efficient meron-cluster algorithm in any dimension. As with the usual bipartite half-filled Hubbard models, the na\"{\i}ve
U(2) symmetry is enhanced to
SO(4). On the other hand, our model has a novel spin-charge flip
Z2C symmetry which is an important ingredient of free massless fermions. In this work we focus on one spatial dimension and show that our model can be viewed as a lattice-regularized two-flavor chiral-mass Gross-Neveu model. Our model remains solvable in the presence of the Hubbard coupling
U, which maps to a combination of Gross-Neveu and Thirring couplings in one dimension. Using the meron-cluster algorithm we find that the ground state of our model is a valence bond solid when
U=0. From our field theory analysis, we argue that the valence bond solid forms inevitably because of an interesting frustration between spin and charge sectors in the renormalization group flow enforced by the
Z2C symmetry. This state spontaneously breaks translation symmetry by one lattice unit, which can be identified with a ${\mathbb{Z}}_{2}^{\ensuremath{\chi}}$ chiral symmetry in the continuum. We show that increasing
U induces a quantum phase transition to a critical phase described by the
SU(2)1 Wess-Zumino-Witten theory. The quantum critical point between these two phases is known to exhibit a novel symmetry enhancement between spin and dimer. Here we verify the scaling relations of these correlation functions near the critical point numerically. Our study opens up the exciting possibility of numerical access to similar novel phase transitions in higher dimensions in fermionic lattice models using the meron-cluster algorithm.
Abstract We study the phase diagram of the (1 + 1)-dimensional Gross-Neveu model with both
gx2(ψψ)2 g x 2 ψ ¯ ψ 2 and
gy2(ψiγ5ψ)2 g y 2 ψ ¯ i γ 5 ψ 2 interaction terms on a spatial lattice. The continuous chiral symmetry, which is present in the continuum model when
gx2=gy2 g x 2 = g y 2 , has a mixed ’t Hooft anomaly with the charge conservation symmetry, which guarantees the existence of a massless mode. However, the same ’t Hooft anomaly also implies that the continuous chiral symmetry is broken explicitly in our lattice model. Nevertheless, from numerical matrix product state simulations we find that for certain parameters of the lattice model, the continuous chiral symmetry reemerges in the infrared fixed point theory, even at strong coupling. We argue that, in order to understand this phenomenon, it is crucial to go beyond mean-field theory (or, equivalently, beyond the leading order term in a 1/N expansion). Interestingly, on the lattice, the chiral Gross-Neveu model appears at a Landau-forbidden second order phase transition separating two distinct and unrelated symmetry-breaking orders. We point out the crucial role of two different ’t Hooft anomalies or Lieb-Schultz-Mattis obstructions for this Landau-forbidden phase transition to occur.
Partition function zeros are powerful tools in understanding critical behavior. In this paper, we present results of the Fisher zeros of two-dimensional Ising models, in the framework of free-fermion eight-vertex model. First, we succeed in finding special boundary conditions for the free-fermion model, under which the partition function of a finite lattice can be expressed in a double product form. Using appropriate mappings, these boundary conditions are transformed into the corresponding versions of the square, triangular, and honeycomb lattice Ising models. Each Ising model is studied in the cases of a zero field and of an imaginary field i ( π / 2 ) k B T . For the square lattice model, we rediscover the famous Brascamp-Kunz (B-K) boundary conditions. For the triangular and honeycomb lattice models, we obtain the B-K-type boundary conditions, and the Fisher zeros are conveniently solved from the product form of partition function. The advantage of B-K-type boundary conditions is that the Fisher zeros of any finite lattice exactly lie on certain loci, and the accumulation points of zeros can be easily determined in the thermodynamic limit. Our finding and method would be very helpful in studying the partition function zeros of vertex and Ising models.