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arXiv:2112.15511 · International Symposium on Lattice Field Theory (LATTICE)

Critical exponents for a spin-charge flip symmetric fixed point in 2+1d with massless Dirac fermions

Emilie Huffman

Published May 16, 2022• 0 citations
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Abstract

In the Hamiltonian picture, free spin-$1/2$ Dirac fermions on a bipartite lattice have an $O(4)$ (spin-charge) symmetry. Here we construct an interacting lattice model with an interaction $V$, which is similar to the Hubbard interaction but preserves the spin-charge flip symmetry. By tuning the coupling $V$, we show that we can study the phase transition between the massless fermion phase at small-$V$ and a massive fermion phase at large-$V$. We construct a fermion bag algorithm to study this phase transition and find evidence for it to be second order. Numerical study shows that the universality class of the transition is different from the one studied earlier involving the Hubbard coupling $U$. Here we obtain some critical exponents using lattices up to $L=48$.

DOI
10.22323/1.396.0208

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In the Hamiltonian picture, free spin-1/21/21/2 Dirac fermions on a bipartite lattice have an O(4)O(4)O(4) (spin-charge) symmetry. Here we construct an interacting lattice model with an interaction VVV, which is similar to the Hubbard interaction but preserves the spin-charge flip symmetry. By tuning the coupling VVV, we show that we can study the phase transition between the massless fermion phase at small-VVV and a massive fermion phase at large-VVV. We construct a fermion bag algorithm to study this phase transition and find evidence for it to be second order. Numerical study shows that the universality class of the transition is different from the one studied earlier involving the Hubbard coupling UUU. Here we obtain some critical exponents using lattices up to L=48L=48L=48.
We study a quantum phase transition of electrons on a two-dimensional square lattice. Our lattice model preserves the full O(4)\mathrm{O}(4)O(4) symmetry of free spin-12\frac{1}{2}21​ Dirac fermions on a bipartite lattice. In particular, it not only preserves the usual SO(4)\mathrm{SO}(4)SO(4) (spin-charge) symmetry like in the half-filling Hubbard model, but also preserves a Z2\mathbb{Z}_2Z2​ spin-charge flip symmetry. Using sign-problem-free Monte Carlo simulation, we find a second order quantum phase transition from a massless Dirac phase to a massive phase with spontaneously chosen spin order or charge order, which become simultaneously critical at the critical point. We analyze all the possible 4-fermion couplings in the continuum respecting the lattice symmetry, and identify the terms whose effective potential in the broken phase is consistent with the numerical results. Using renormalization group calculations in the continuum, we show the existence of the new spin-charge flip symmetric fixed point and calculate its critical exponents.
We study a quantum phase transition from a massless to massive Dirac fermion phase in a new two-dimensional bipartite lattice model of electrons that is amenable to sign-free quantum Monte Carlo simulations. Importantly, interactions in our model are not only invariant under SU(2) symmetries of spin and charge like the Hubbard model, but they also preserve an Ising-like electron spin-charge flip symmetry. From unbiased fermion bag Monte Carlo simulations with up to 2304 sites, we show that the massive fermion phase spontaneously breaks this Ising symmetry, picking either antiferromagnetism or superconductivity, and that the transition at which both orders are simultaneously quantum critical belongs to a new "chiral spin-charge symmetric" universality class. We explain our observations using effective potential and renormalization group calculations within the framework of a continuum field theory.
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)U(2)U(2) symmetry is enhanced to SO(4)SO(4)SO(4). On the other hand, our model has a novel spin-charge flip Z2C{\mathbb{Z}}_{2}^{C}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 UUU, 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=0U=0U=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{\mathbb{Z}}_{2}^{C}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 UUU induces a quantum phase transition to a critical phase described by the SU(2)1SU(2{)}_{1}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.
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.
Quantum Critical Phenomena in an O(4)O(4)O(4) Fermion Chain
We construct a fermionic lattice model containing interacting spin-12\frac{1}{2}21​ fermions with an O(4)O(4)O(4) symmetry. In addition the model contains a Z2\mathbb{Z}_2Z2​ chiral symmetry which prevents a fermion mass term. Our model is motivated by the ability to study its physics using the meron-cluster algorithm. By adding a strong repulsive Hubbard interaction UUU, we can transform it into the regular Heisenberg anti-ferromagnet. While we can study our model in any dimension, as a first project we study it in one spatial dimension. We discover that our model at U=0U=0U=0 can be described as a lattice-regularized 2-flavor Gross-Neveu model, where fermions become massive since the Z2\mathbb{Z}_2Z2​ chiral symmetry of the model is spontaneously broken. We show numerically that the theory remains massive when UUU is small. At large values of UUU the model is equivalent to the isotropic spin-half anti-ferromagnetic chain, which is massless for topological reasons. This implies that our model has a quantum phase transition from a Z2\mathbb{Z}_2Z2​ broken massive phase to a topologically massless phase as we increase UUU. We present results obtained from our quantum Monte Carlo method near this phase transition.
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.