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Interaction-induced charge orders with electronic origin occur as states of spontaneously broken symmetry in several materials platforms. An electronic mechanism for charge order requires an attractive component in the effective charge vertex. We put forward such a mechanism for the formation of unconventional charge density waves in a metal. These states result from the condensation of particle-hole pairs with finite wave vector and non-zero angular momentum and correspond to bond or loop current order on a lattice. The mechanism we describe can be viewed as Kohn Luttinger analysis in the particle-hole channel with finite transferred momentum. It incorporates one-loop spin and pairing correctionsn, which are then used as an input for a summation in the charge channel triggering an instability. We extend our analysis to a spin-fluctuation approach, where the effective charge interaction is dressed by the particle-hole ladder with exchanged momentum. We argue that this mechanism works for weakly-interacting metals with nested Fermi surface and a large number of fermion flavors. We apply the Kohn-Luttinger-like approach to square- and triangular-lattice Hubbard models with SU($N_f$) flavour symmetry and show that it leads to different types of $p$-wave charge density waves. We also study effects beyond weak coupling at and away from Van Hove filling in terms of a phenomenological model with additional exchange interaction. In the vicinity of Van Hove filling, we obtain $d$-wave charge density waves with wave vectors determined by nesting as leading instabilities. In addition, we find another charge density wave with wave vector $K/4$ on the triangular lattice on both sides of Van Hove filling. We demonstrate that this $K/4$ instability can win the competition against pairing for $N_f=4$ via an unbiased functional renormalisation group calculation., Comment: 12+5 pages, 8+1 figures

The understanding of phenomena falling outside the Ginzburg-Landau paradigm of phase transitions represents a key challenge in condensed matter physics. A famous class of examples is constituted by the putative deconfined quantum critical points between two symmetry-broken phases in layered quantum magnets, such as pressurised SrCu$_2$(BO$_3$)$_2$. Experiments find a weak first-order transition, which simulations of relevant microscopic models can reproduce. The origin of this behaviour has been a matter of considerable debate for several years. In this work, we demonstrate that the nature of the deconfined quantum critical point can be best understood in terms of a novel dynamical mechanism, termed Nordic walking. Nordic walking denotes a renormalisation group flow arising from a beta function that is flat over a range of couplings. This gives rise to a logarithmic flow that is faster than the well-known walking behaviour, associated with the annihilation and complexification of fixed points, but still significantly slower than the generic running of couplings. The Nordic-walking mechanism can thus explain weak first-order transitions, but may also play a role in high-energy physics, where it could solve hierarchy problems. We analyse the Wess-Zumino-Witten field theory pertinent to deconfined quantum critical points with a topological term in 2+1 dimensions. To this end, we construct an advanced functional renormalisation group approach based on higher-order regulators. We thereby calculate the beta function directly in 2+1 dimensions and provide evidence for Nordic walking., Comment: 12 + 11 pages (typeset with sn-jnl.cls), 4 + 1 figures; comments welcome

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 $N_f=2$ 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 $N_f$, which disappears for a broad range of flavor numbers around the physical case, $N_f=2$, 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 $4-\epsilon$ spacetime dimensions. We determine the $\epsilon$ correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case $N_f=2$. However, this only happens very slowly, such that an extrapolation to $\epsilon=1$ 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., Comment: 13 pages, 6 figures, 2 tables

Two-dimensional van der Waals materials have recently been established experimentally as a highly-tunable condensed matter platform, facilitating the controlled manipulation of band structures and interactions. In several of these materials, Dirac cones are present in the low-energy regime near the Fermi level. Thus, fermionic excitations emerging in these materials close to the Dirac cones have a linear dispersion relation near the Fermi surface as massless relativistic Dirac fermions. Here, we study low-energy fermionic excitations of such Dirac materials in the presence of a mass gap that may be generated by symmetry breaking. Introducing a dynamical Fermi velocity and/or time-dependent mass gap for the Dirac quasiparticles, we exhibit the emergence of an analog of cosmological fermion pair production in terms of observables such as the expected occupation number or two-point correlation functions. We find that it is necessary and sufficient for quasiparticle production that only the ratio between the mass gap and the Fermi velocity is time-dependent. In this way, we establish that highly-tunable Dirac materials can serve as analog models for cosmological spacetime geometries, in particular, for Friedmann-Lema\^itre-Robertson-Walker expanding cosmologies. We briefly discuss possibilities for experimental realization, Comment: 15 pages, 4 figures

Over the past two decades, the enigma of the deconfined quantum critical point (DQCP) has attracted broad attention across the condensed matter, quantum field theory, and high-energy physics communities, as it is expected to offer a new paradigm in theory, experiment, and numerical simulations that goes beyond the Landau-Ginzburg-Wilson framework of symmetry breaking and phase transitions. However, the nature of DQCP has been controversial. For instance, in the square-lattice spin-1/2 $J$-$Q$ model, believed to realize the DQCP between N\'eel and valence bond solid states, conflicting results, such as first-order versus continuous transition, and critical exponents incompatible with conformal bootstrap bounds, have been reported. The enigma of DQCP is exemplified in its anomalous logarithmic subleading contribution in its entanglement entropy (EE), which was discussed in recent studies. In the current work, we demonstrate that similar anomalous logarithmic behavior persists in a class of models analogous to the DQCP. We systematically study the quantum EE of square-lattice SU($N$) DQCP spin models. Based on large-scale quantum Monte Carlo computation of the EE, we show that for a series of $N$ smaller than a critical value, the anomalous logarithmic behavior always exists in the EE, which implies that the previously determined DQCPs in these models do not belong to conformal fixed points. In contrast, when $N\ge N_c$ with a finite $N_c$ that we evaluate to lie between $7$ and $8$, the DQCPs are consistent with conformal fixed points that can be understood within the Abelian Higgs field theory with $N$ complex components., Comment: The revised version focuses on the anomalous logarithmic correction to the EE arising from the smooth boundary, rather than corners. And the critical $N_c$ is determined based on the anomalous EE signal from smooth boundary

The discovery of correlated insulating states in moir\'e heterostructures has renewed the interest in strongly-coupled electron systems where spin and valley (or layer) degrees of freedom are intertwined. In the strong-coupling limit, such systems can be effectively described by SU(4) spin-valley models akin to Kugel-Khomskii models long studied in the context of spin-orbit coupled materials. However, typical moir\'e heterostructures also exhibit interactions that break the SU(4) symmetry down to SU(2)${}_{\mathrm{spin}}\otimes$U(1)${}_{\mathrm{valley}}$. Here we investigate the impact of such symmetry-breaking couplings on the magnetic phase diagram for triangular superlattices considering a filling of two electrons (or holes) per moir\'e unit cell. We explore a broad regime of couplings -- including XXZ anisotropies, Dzyaloshinskii-Moriya exchange and on-site Hund's couplings -- using semi-classical Monte Carlo simulations. We find a multitude of classically ordered phases, including (anti-)ferromagnetic, incommensurate, and stripe order, manifesting in different sectors of the spin-valley model's parameter space. Zooming in on the regimes where quantum fluctuations are likely to have an effect, we employ pseudo-fermion functional renormalization group (pf-FRG) calculations to resolve quantum disordered ground states such as spin-valley liquids, which we indeed find for certain parameter regimes. As a concrete example, we discuss the case of trilayer graphene aligned with hexagonal boron nitride using material-specific parameters., Comment: 20 pages, 16 figures

We study quantum multicritical behavior in a (2+1)-dimensional Gross-Neveu-Yukawa field theory with eight-component Dirac fermions coupled to two triplets of order parameters that act as Dirac masses, and transform as $(1,0) + (0,1)$ representation under the SO(4)$\simeq$SO(3)$\times$SO(3) symmetry group. This field theory is relevant to spin-1/2 fermions on honeycomb or $\pi$-flux lattices, for example, near the transition points between an $s$-wave superconductor and a charge-density wave, on one side, and N\'eel order, on the other. Two triplets of such order parameters always allow for a common pair of two other order parameters that would complete them to the maximal set of compatible (anticommuting) orders of five. We first derive a unitary transformation in the Nambu (particle-hole) space which maps any two such triplets, possibly containing some superconducting orders, onto purely insulating order parameters. This allows one to consider a universal SO(4) Gross-Neveu-Yukawa description of the multicriticality without any Nambu doubling. We then proceed to derive the renormalization-group flow of the coupling constants at one-loop order in $4-\epsilon$ space-time dimensions, allowing also a more general set of order parameters transforming under SO($n_a$)$\times$SO($n_b$). While for $n_a=n_b > 2 $ in the bosonic sector and with fermions decoupled there is a stable fixed point of the flow, the Yukawa coupling to fermions quickly leads to its elimination by a generic fixed-point collision in the relevant range of fermion flavor numbers $N_f$. This suggests the replacement of the critical behavior by a runaway flow in the physical case $n_a=n_b=3$. The structure of the RG flow at $n_a\neq n_b$ is also discussed, and some non-perturbative arguments in favor of the stability of the decoupled critical point when $n_a=3$ and $n_b=1$ in $D=2+1$ are provided., Comment: 13 pages, 4 figures

We study competing orders of spinless fermions in the triangular-lattice Hubbard model with nearest-neighbor interaction. We calculate the effective, momentum-resolved two-particle vertex in an unbiased way in terms of the functional renormalization group method and compare two different schemes for the momentum discretization, one based on dividing the Fermi surface into patches and one based on a channel decomposition. We study attractive and repulsive nearest-neighbor interaction and find a competition of pairing and charge instabilities. In the attractive case, a Pomeranchuk instability occurs at Van Hove filling and $f$-wave and $p$-wave pairing emerge when the filling is reduced. In the repulsive case, we obtain a charge density wave at Van Hove filling and extended $p$-wave pairing with reduced filling. The $p$-wave pairing solution is doubly degenerate and can realize chiral $p+ip$ superconductivity with different Chern numbers in the ground state. We discuss implications for strongly correlated spin-orbit coupled hexagonal electron systems such as moir\'e heterostructures.

Evidence for correlated insulating and superconducting phases around regions of high density of states was reported in the strongly spin-orbit coupled van-der Waals material twisted tungsten diselenide (tWSe$_2$). We investigate their origin and interplay by using a functional renormalization group approach that allows to describe superconducting and spin/charge instabilities in an unbiased way. We map out the phase diagram as function of filling and perpendicular electric field, and find that the moir\'e Hubbard model for tWSe$_2$ features mixed-parity superconducting order parameters with $s/f$-wave and topological $d/p$-wave symmetry next to (incommensurate) density wave states. Our work systematically characterizes competing interaction-driven phases in tWSe$_2$ beyond mean-field approximations and provides guidance for experimental measurements by outlining the fingerprint of correlated states in interacting susceptibilities., Comment: 7 pages, 3 figures, supplemental material

A simple yet paradigmatic model for the interplay of strong electronic correlations and geometric frustration is the triangular lattice Hubbard model. Recently it was proposed that moir\'e structures of transition metal dichalcogenides can be used to simulate extended versions that include non-local density-density interactions. We study competing instabilities of interacting electrons in such an extended Hubbard model on the triangular lattice near a filling where the density of states has a Van Hove singularity. We employ a truncated-unity functional renormalization group approach to investigate two cases: a paradigmatic minimally extended Hubbard model and a specific model with parameters that are applicable to hetero-bilayers of transition metal dichalcogenides. We unravel rich phase diagrams, including tendencies to spin-density-wave order and unconventional pairing, which can give rise to topological superconductivity. We classify the symmetry of the superconducting instabilities according to their irreducible representations and show that higher lattice harmonics are dominant when the nearest-neighbor interaction is sizable indicating pair formation between second-nearest neighbors. The phenomenological consequences can be enhanced spin and thermal quantum Hall responses in a topological superconductor., Comment: 11+7 pages, 11+2 figures

Experimental demonstrations of tunable correlation effects in magic-angle twisted bilayer graphene have put two-dimensional moir\'e quantum materials at the forefront of condensed-matter research. Other twisted few-layer graphitic structures, boron-nitride, and homo- or hetero-stacks of transition metal dichalcogenides (TMDs) have further enriched the opportunities for analysis and utilization of correlations in these systems. Recently, within the latter material class, strong spin-orbit coupling or excitonic physics were experimentally explored. The observation of a Mott insulating state and other fascinating collective phenomena such as generalized Wigner crystals, stripe phases and quantum anomalous Hall insulators confirmed the relevance of many-body interactions, and demonstrated the importance of their extended range. Since the interaction, its range, and the filling can be tuned experimentally by twist angle, substrate engineering and gating, we here explore Fermi surface instabilities and resulting phases of matter of hetero-bilayer TMDs. Using an unbiased renormalization group approach, we establish in particular that hetero-bilayer TMDs are unique platforms to realize topological superconductivity with winding number $|\mathcal{N}|=4$. We show that this state reflects in pronounced experimental signatures, such as distinct quantum Hall features., Comment: 8 pages, 3 figures

Moir\'e materials provide a highly tunable environment for the realization of band structures with engineered physical properties. Specifically, moir\'e structures with Fermi surface flat bands - a synthetic environment for the realization of correlated phases - have moir\'e unit cells containing thousands of atoms and tantalizingly complex bands structures. In this paper we show that statistical principles go a long way in explaining universal physical properties of these systems. Our approach builds on three conceptual elements: the presence of quantum chaos caused by the effective irregularity of the atomic configurations on short length scales, Anderson localization in momentum space, and the presence of approximate crystalline symmetries. Which of these principles dominates depends on material parameters such as the extension of the Fermi surface or the strength of the moir\'e lattice potential. The phenomenological consequences of this competition are predictions for the characteristic group velocity of moir\'e bands, a primary indicator for their average flatness. In addition to these generic features, we identify structures outside the statistical context, notably almost flat bands close to the extrema of the unperturbed spectra, and the celebrated zero energy `magic angle' flat bands, where the latter require exceptionally fine tuned material parameters., Comment: 15 pages, 10 figures

Two-dimensional spin-orbital magnets with strong exchange frustration have recently been predicted to facilitate the realization of a quantum critical point in the Gross-Neveu-SO(3) universality class. In contrast to previously known Gross-Neveu-type universality classes, this quantum critical point separates a Dirac semimetal and a long-range-ordered phase, in which the fermion spectrum is only partially gapped out. Here, we characterize the quantum critical behavior of the Gross-Neveu-SO(3) universality class by employing three complementary field-theoretical techniques beyond their leading orders. We compute the correlation-length exponent $\nu$, the order-parameter anomalous dimension $\eta_\phi$, and the fermion anomalous dimension $\eta_\psi$ using a three-loop $\epsilon$ expansion around the upper critical space-time dimension of four, a second-order large-$N$ expansion (with the fermion anomalous dimension obtained even at the third order), as well as a functional renormalization group approach in the improved local potential approximation. For the physically relevant case of $N=3$ flavors of two-component Dirac fermions in 2+1 space-time dimensions, we obtain the estimates $1/\nu = 1.03(15)$, $\eta_\phi = 0.42(7)$, and $\eta_\psi = 0.180(10)$ from averaging over the results of the different techniques, with the displayed uncertainty representing the degree of consistency among the three methods., Comment: 18 pages, 7 figures, 3 tables, 1 ancillary file "GNSO3-exponents.m". v2: additional explanations, minor corrections