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arXiv:1912.11237 · International Symposium on Lattice Field Theory (LATTICE)

Quantum Critical Phenomena in an $O(4)$ Fermion Chain

Hanqing Liu, Shailesh Chandrasekharan A, Ribhu K. Kaul

Published Jan 4, 2020• 2 citations
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Abstract

We construct a fermionic lattice model containing interacting spin-$\frac{1}{2}$ fermions with an $O(4)$ symmetry. In addition the model contains a $\mathbb{Z}_2$ 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 $U$, 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 $U=0$ can be described as a lattice-regularized 2-flavor Gross-Neveu model, where fermions become massive since the $\mathbb{Z}_2$ chiral symmetry of the model is spontaneously broken. We show numerically that the theory remains massive when $U$ is small. At large values of $U$ 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 $\mathbb{Z}_2$ broken massive phase to a topologically massless phase as we increase $U$. We present results obtained from our quantum Monte Carlo method near this phase transition.

DOI
10.22323/1.363.0222

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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.
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.
We present quantum Monte Carlo simulations for the chiral Heisenberg Gross-Neveu-Yukawa quantum phase transition of relativistic fermions with N=4 Dirac spinor components subject to a repulsive, local four fermion interaction in (2+1)D. Here we employ a two-dimensional lattice Hamiltonian with a single, spin-degenerate Dirac cone, which exactly reproduces a linear energy-momentum relation for all finite size lattice momenta in the absence of interactions. This allows us to significantly reduce finite size corrections compared to the widely studied honeycomb and π-flux lattices. A Hubbard term dynamically generates a mass beyond a critical coupling of U_{c}=6.76(1) as the system acquires antiferromagnetic order and SU(2) spin rotational symmetry is spontaneously broken. At the quantum phase transition, we extract a self-consistent set of critical exponents ν=0.98(1), η_{ϕ}=0.53(1), η_{ψ}=0.18(1), and β=0.75(1). We provide evidence for the continuous degradation of the quasiparticle weight of the fermionic excitations as the critical point is approached from the semimetallic phase. Finally, we study the effective "speed of light" of the low-energy relativistic description, which depends on the interaction U, but is expected to be regular across the quantum phase transition. We illustrate that the strongly coupled bosonic and fermionic excitations share a common velocity at the critical point.
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 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.
Motivated by the physics of spin-orbital liquids, we study a model of interacting Dirac fermions on a bilayer honeycomb lattice at half filling, featuring an explicit global SO(3)×\times×U(1) symmetry. Using large-scale auxiliary- field quantum Monte Carlo (QMC) simulations, we locate two zero-temperature phase transitions as function of increasing interaction strength. First, we observe a continuous transition from the weakly-interacting semimetal to a different semimetallic phase in which the SO(3) symmetry is spontaneously broken and where two out of three Dirac cones acquire a mass gap. The associated quantum critical point can be understood in terms of a Gross-Neveu-SO(3) theory. Second, we subsequently observe a transition towards an insulating phase in which the SO(3) symmetry is restored and the U(1) symmetry is spontaneously broken. While strongly first order at the mean-field level, the QMC data is consistent with a direct and continuous transition. It is thus a candidate for a new type of deconfined quantum critical point that features gapless fermionic degrees of freedom.
Motivated by the physics of spin-orbital liquids, we study a model of interacting Dirac fermions on a bilayer honeycomb lattice at half filling, featuring an explicit global SO(3)×U(1) symmetry. Using large-scale auxiliary-field quantum Monte Carlo (QMC) simulations, we locate two zero-temperature phase transitions as function of increasing interaction strength. First, we observe a continuous transition from the weakly interacting semimetal to a different semimetallic phase in which the SO(3) symmetry is spontaneously broken and where two out of three Dirac cones acquire a mass gap. The associated quantum critical point can be understood in terms of a Gross-Neveu-SO(3) theory. Second, we subsequently observe a transition toward an insulating phase in which the SO(3) symmetry is restored and the U(1) symmetry is spontaneously broken. While strongly first order at the mean-field level, the QMC data are consistent with a direct and continuous transition. It is thus a candidate for a new type of deconfined quantum critical point that features gapless fermionic degrees of freedom.