Two-dimensional semi-Dirac fermions are quasiparticles that disperse linearly in one direction and quadratically in the other. We investigate instabilities of semi-Dirac fermions toward charge and spin density wave and superconducting orders, driven by short-range interactions. We analyze the critical behavior of the Yukawa theories for the different order parameters using Wilson momentum shell renormalization group. We generalize to a large number
Nf of fermion flavors to achieve analytic control in
2+1 dimensions and calculate critical exponents at one-loop order, systematically including
1/Nf corrections. The latter depend on the specific form of the bosonic infrared propagator in
2+1 dimensions, which needs to be included to regularize divergencies. The
1/Nf corrections are surprisingly small, suggesting that the expansion is well controlled in the physical dimension. The order parameter correlations inherit the electronic anisotropy of the semi-Dirac fermions, leading to correlation lengths that diverge along the spatial directions with distinct exponents, even at the mean-field level. We conjecture that the proximity to the critical point may stabilize novel modulated order phases.
Quantum electrodynamics in 2+1 dimensions ( QED3 ) has been proposed as a critical field theory describing the low-energy effective theory of a putative algebraic Dirac spin liquid or of quantum phase transitions in two-dimensional frustrated magnets. We provide compelling evidence that the intricate spectrum of excitations of the elementary but strongly frustrated J1−J2 Heisenberg model on the triangular lattice is in one-to-one correspondence to a zoo of excitations from QED3 , in the quantum spin liquid regime. This evidence includes a large manifold of explicitly constructed monopole and bilinear excitations of QED3 , which is thus shown to serve as an organizing principle of phases of matter in triangular lattice antiferromagnets and their low-lying excitations. Moreover, we observe signatures of emergent valence-bond solid (VBS) correlations, which can be interpreted either as evidence of critical VBS fluctuations of an emergent Dirac spin liquid or as a transition from the 120° Néel order to a VBS whose quantum critical point is described by QED3 . Our results are obtained by comparing ansatz wave functions from a parton construction to exact eigenstates obtained using large-scale exact diagonalization up to N=48 sites. Published by the American Physical Society 2024
We discuss the renormalization of the quantum antiferromagnet in two dimensions. With an analysis of the spin-wave Hamiltonian of the quantum Heisenberg model or the quantum nonlinear \ensuremath{\sigma} model (NL\ensuremath{\sigma}M), we consider O(n) invariant observable 〈
Sl\ensuremath{\cdot}
Sj〉 and evaluate \ensuremath{\beta} functions in the dimensional regularization method up to two-loop order. We show that quantum and classical NL \ensuremath{\sigma} M are connected by a simple crossover function. It follows from this crossover phenomenon that the critical temperature
Tc for d>2 is reduced due to the quantum effects and that the thermal fluctuations reduce the critical coupling constant
gc. The disappearance of
gc indicates thermal restoration of the asymptotic freedom. Finally we present an exact expression of the correlation length.
Quantum electrodynamics in
2+1 dimensions is an effective gauge theory for the so-called algebraic quantum liquids. A new type of such a liquid, the algebraic charge liquid, has been proposed recently in the context of deconfined quantum critical points [R. K. Kaul et al., Nat. Phys. 4, 28 (2008)]. In this context, by using the renormalization group in $d=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions, we show that a deconfined quantum critical point occurs in a SU(2) system provided the number of Dirac fermion species ${N}_{f}\ensuremath{\ge}4$. The calculations are done in a representation where the Dirac fermions are given by four-component spinors. The critical exponents are calculated for several values of
Nf. In particular, for
Nf=4 and $\ensuremath{\epsilon}=1$
(d=2+1), the anomalous dimension of the N\'eel field is given by ${\ensuremath{\eta}}_{N}=1/3$, with a correlation length exponent $\ensuremath{\nu}=1/2$. These values considerably change for
Nf>4. For instance, for
Nf=6, we find ${\ensuremath{\eta}}_{N}\ensuremath{\approx}0.75\text{ }191$ and $\ensuremath{\nu}\ensuremath{\approx}0.660\text{ }09$. We also investigate the effect of chiral symmetry breaking and analyze the scaling behavior of the chiral holon susceptibility, ${G}_{\ensuremath{\chi}}(x)\ensuremath{\equiv}⟨\overline{\ensuremath{\psi}}(x)\ensuremath{\psi}(x)\overline{\ensuremath{\psi}}(0)\ensuremath{\psi}(0)⟩$.
Evidence for relativistic quantum criticality of antiferromagnetism and superconductivity in two-dimensional Dirac fermion systems has been found in large-scale quantum Monte Carlo simulations. However, the corresponding (
2+1)-dimensional Gross--Neveu--Yukawa field theory with
Nf=2 four-component Dirac fermions coupled to two triplets of order parameters does not exhibit a renormalization group fixed point at one-loop order. Instead, the theory only features a critical point for a large or very small fractional number of fermion flavors
Nf, which disappears for a broad range of flavor numbers around the physical case,
Nf=2, due to fixed-point annihilation. This raises the question on how to explain the observed scaling collapse in the quantum Monte Carlo data. Here, we extend previous renormalization-group analyses by studying a generalized model at two-loop order in $4\ensuremath{-}\ensuremath{\epsilon}$ spacetime dimensions. We determine the $\ensuremath{\epsilon}$ correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case
Nf=2. However, this only happens very slowly, such that an extrapolation to $\ensuremath{\epsilon}=1$ still suggests the absence of criticality in
2+1 dimensions. Thereby, we consolidate the finding that the continuum field theory does not feature a stable renormalization-group fixed point and no true quantum criticality would be expected for the considered system. We briefly discuss a possible reconciliation in terms of a complex conformal field theory. Further, we also explore the fixed-point structure in an enlarged theory space and identify a candidate stable fixed-point solution.
We study the universal critical properties of the
QED3-Gross-Neveu-Yukawa model with
N flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the $\ensuremath{\epsilon}$ expansion below four dimensions. For
N=1, the model is conjectured to be infrared dual to the
SU(2)-symmetric noncompact
CP1 model, which describes the deconfined quantum critical point of the N\'eel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For
N=2, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagome antiferromagnet. For general
N we determine the order-parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of
SU(N) singlet and adjoint fermion mass bilinears at the critical point. We use Pad\'e approximants to obtain estimates of critical properties in 2+1 dimensions.