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arXiv:2008.00830 · Physical Review Research

Fermionic criticality of anisotropic nodal point semimetals away from the upper critical dimension: Exact exponents to leading order in 1Nf

Mikolaj D. Uryszek, Frank Krüger, Elliot Christou

Published Nov 20, 2020• 12 citations
arXivHTMLPDF

Abstract

We consider the fermionic quantum criticality of anisotropic nodal point semimetals in d = d L + d Q spatial dimensions that disperse linearly in d L dimensions, and quadratically in the remaining d Q dimensions.When subject to strong interactions, these systems are susceptible to semimetal-insulator transitions concurrent with spontaneous symmetry breaking.Such quantum critical points are described by effective field theories of anisotropic nodal fermions coupled to dynamical order parameter fields.We analyze the universal scaling in the physically relevant spatial dimensions, generalizing to a large number N f of fermion flavors for analytic control.Landau damping by gapless fermionic excitations gives rise to nonanalytic self-energy corrections to the bosonic order-parameter propagator that dominate the long-wavelength behavior.We show that perturbative momentum shell RG leads to nonuniversal, cutof-dependent results, as it does not correctly account for this nonanalytic structure.In turn, using a completely general soft cutoff formulation, we demonstrate that the correct IR scaling of the dressed bosonic propagator can be deduced by enforcing that results are independent of the cutoff scheme.Using the soft cutoff RG with the dressed dynamical RPA boson propagator, we compute the exact critical exponents for anisotropic semi-Dirac fermions (d L = 1, d Q = 1) to leading order in 1/N f and to all loop orders.Applying the same method to relativistic Dirac fermions, we reproduce the critical exponents obtained by other methods, such as conformal bootstrap.Unlike in the relativistic case, where the UV-IR connection is reestablished at the upper critical dimension, nonanalytic IR contributions persist near the upper critical line 2d L + d Q = 4 of anisotropic nodal fermions.We present expansions in both the number of linear and quadratic dimensions.The corrections to critical exponents are nonanalytic in , with a functional form that depends on the starting point on the upper critical line.

DOI
10.1103/physrevresearch.2.043265

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This Thesis explores the formation of novel phases in Dirac systems that are driven by strong many-body interactions and collective quantum fluctuations. The central focus is the half-filled honeycomb lattice, which is the prototypical Dirac semimetal with low-energy excitations described by Dirac fermions. Of particular interest are exotic phases of matter that are not favoured at mean-field but are stabilised by fluctuations. Self-consistent path integral techniques are used to demonstrate that extended repulsive interactions favour charge order that breaks translational and other lattice symmetries. This charge order outcompetes interaction induced topological insulating phases, which are dominant at the mean-field level. The interacting quantum critical point was found to be near the critical point of a topological Lifshitz transition. Interacting Dirac fermions exhibit perhaps the simplest example of fermionic quantum criticality. Gross-Neveu-Yukawa field theories describe the universality class of semimetal-insulator transitions, which are driven by on-site and nearest neighbour repulsive interactions on the honeycomb lattice. The Dirac fermions couple to the dynamical order parameter fields and play a crucial role in determining the universal behaviour. Naturally, this variety of criticality is outside of the Ginzburg-Landau-Wilson paradigm. This idea is extended to include the translational symmetry breaking charge order. Here it is found that the breaking of crystal symmetries corresponds to Lorentz violating effective field theories. The criticality is analysed with the renormalisation group. The low energy excitations at the topological transition are gapless semi-Dirac fermionic quasiparticles that disperse linearly in one direction and quadratically in the other. The instabilities induced by generic short-range repulsions are investigated. The criticality of this anisotropic system is analysed with the renormalisation group in two spatial dimensions, and it is found to exhibit anisotropic order parameter correlations.
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