Two-dimensional Dirac semimetals near a quantum critical point can be described by Gross–Neveu–Yukawa models. In view of recent experimental advances exhibiting a transition from Dirac semimetal to insulator in highly-tunable van-der-Waals heterostructures, a better understanding of finite-temperature effects is mandatory. Here, we study the Gross–Neveu–Yukawa phase diagram of the chiral Ising model with a non-perturbative field-theory approach at zero and finite temperature, both in the semimetallic phase and in the insulating phase with spontaneously broken ℤ_2 symmetry. At zero temperature, we find a quantum critical point with critical exponents that are close to the ones of the chiral Ising universality class, and show that relativistic symmetry is emergent close to the quantum critical point. At finite temperature, the ordered phase survives up to a finite critical temperature, at which we observe a classical phase transition into the disordered phase. We confirm that this transition lies in the two-dimensional Ising universality class. Finally, we determine the extent and scaling properties of the quantum critical fan, and the behavior of the quasiparticle weight, therein. In summary, we present a unified field-theoretical framework for the phase diagram of the chiral Ising model in the surroundings of its quantum critical point.

We study thermal order in conformal field theories (CFTs) in d=4-ε and d=3 dimensions where several scalars are coupled to an O(N) vector field. In (4-ε) dimensions, we consider models coupling a cubic model of M scalars with ℤ_2^M⋊ S_M symmetry (corresponding to sign flips and permutations of the scalars) to an O(N) vector model. For any M, we find a window of N in which two Wilson-Fisher-like fixed points exist. We show that the ℤ_2^M⋊ S_M symmetry is spontaneously broken at arbitrary nonzero temperatures for M=2 and sufficiently large N within this window, but it remains unbroken for M>2. Using the functional renormalisation group to continue these fixed points towards three dimensions, we find that they collide and move into the complex plane well before reaching d=3, suggesting that their continuation to d=3 does not yield unitary CFTs. We then work directly in three dimensions at large but finite N, with M≪ N, initially without assuming any permutation symmetry among the M scalars. By studying the RG flow, we show that these models possess a conformal manifold up to the leading nontrivial order in the 1/N expansion of the beta functions (which is 𝒪(1/N)). At each point on this manifold, the scalars split into two classes that are distinguished by how they couple to the O(N) vector field. We then restrict to a subspace of the conformal manifold where there is an additional symmetry under permutations of the scalars within each class. We prove that in a domain of this subspace, all M scalars acquire thermal expectation values such that the symmetry under sign flips of the scalars is spontaneously broken at all nonzero temperatures. This provides a novel class of large N CFTs that exhibit a rich pattern of persistent symmetry breaking at nonzero temperatures.

Experiments on twisted double bilayer tungsten diselenide have demonstrated that moir & eacute; semiconductors can realize a relativistic Mott transition, i.e., a quantum phase transition from a Dirac semimetal to a correlated insulating state, by twist-angle tuning. In addition, signatures of van Hove singularities were observed in the material's moir & eacute; valence bands, suggesting further potential for the emergence of strongly correlated states. Based on a continuum model, we provide a detailed analysis of the twist-angle dependence of the system's band structure, focusing on the evolution of the Dirac excitations and the Fermi-surface structure with its Lifshitz transitions across the van Hove fillings. We exhibit that the twist angle can be used to band engineer a high-order van Hove singularity, which can be accessed by gate tuning. We then study the magnetic phase diagram of an effective Hubbard model for twisted double bilayer tungsten diselenide on the effective honeycomb superlattice with tight-binding parameters fitted to the two topmost bands of the continuum model. To that end, we employ a self-consistent Hartree-Fock mean-field approach in real space. Fixing the angle-dependent Hubbard interaction based on the experimental findings, we explore a broad parameter range of twist angle, filling, and temperature. We find a rich variety of magnetic states that we expect to be accessible in future experiments, including, e.g., a noncoplanar spin-density wave with nonzero spin chirality and a half-metallic uniaxial spin-density wave. Finally, we employ a functional renormalization group approach to also study the competition between densitywave and superconducting instabilities. For twist angles of 0 = 2.0 degrees, 2.5 degrees, as well as 0 approximate to 3.5 degrees-where the highorder van Hove-singularity is found-we find clear indications for unconventional superconductivity.

We investigate the interplay between spin-density-wave (SDW) order and superconductivity in the bilayer nickelate La_3Ni_2O_7 using the functional renormalization group (fRG) applied to multiorbital weak-coupling models of both the ambient- and high-pressure crystal structures. As Hund's coupling increases, the leading instability evolves from superconductivity to an SDW state with ordering vector 𝐐_1 ≈ (π/2,π/2) (equivalently 𝐐_Y ≈ (0,π) in the orthorhombic Amam unit cell), in agreement with experimental observations. Surprisingly, the ambient- and high-pressure structures exhibit nearly identical non-interacting susceptibilities and leading fRG instabilities, indicating that the emergence of superconductivity under pressure cannot be explained solely by changes in the low-energy electronic structure. Instead, our results identify the suppression of orthorhombicity as a key ingredient for superconductivity. As the system approaches the tetragonal limit, symmetry-related SDW fluctuations become nearly degenerate, frustrating long-range magnetic order while enhancing pairing interactions. These findings highlight lattice symmetry as a central tuning parameter of the competing ordered states in bilayer nickelates and suggest that reducing orthorhombicity through uniaxial strain could stabilize bulk superconductivity already at ambient pressure.

We construct an ultraviolet-complete, local, and unitary quantum field theory in 2+1 dimensions that exhibits spontaneous breaking of space-time parity, persisting to arbitrarily high temperatures. The theory is defined by a renormalization group trajectory, triggered by a relevant deformation of a conformal field theory, consisting of a critical biconical vector model and a free massless Dirac fermion. This deformation couples the fermion to the scalar sector, generating a renormalization group flow that terminates at a nontrivial infrared fixed point described by a conformal Gross-Neveu-Yukawa model and a decoupled critical vector model. By construction, the quantum field theory is parity invariant at zero temperature. However, we show that at sufficiently high temperatures, parity symmetry is spontaneously broken and remains so even in the infinite-temperature limit. Our analysis relies on both, perturbative renormalization group techniques in 4-ε dimensions and functional renormalization group techniques directly in 2+1 dimensions.

Gross-Neveu-Yukawa-type models such as the chiral Ising, chiral XY, and chiral Heisenberg models, serve as effective descriptions of two-dimensional Dirac semi-metals undergoing quantum phase transitions into various symmetry-broken ordered states. Their relativistic quantum critical points govern the systems' physical behavior in the vicinity of the transition also at finite temperatures, which is strongly influenced by critical order-parameter and chiral fermion fluctuations. Here, we explore the effect of these fluctuations at zero and finite temperature, both in the Dirac phase and in the Mott phases with spontaneously broken symmetry. To that end, we set up a functional renormalization group approach, which allows us to systematically calculate the quantum phase diagrams and scaling behavior at and near quantum criticality. We explicitly estimate quantum critical exponents, calculate the quasiparticle weight of the chiral Dirac excitations, and determine the extent of the quantum critical fan. Furthermore, we expose a semi-metallic precondensation regime where order-parameter fluctuations destroy order at finite temperature and we show the related manifestation of the Coleman-Hohenberg-Mermin-Wagner theorem. For the chiral XY model, we also expose signatures of Berezinskii-Kosterlitz-Thouless physics in a system that includes strong fermion fluctuations. In view of recent experimental developments on correlated phases in highly tunable two-dimensional Dirac materials, our work aims at a more comprehensive theoretical description of relativistic quantum criticality in semi-metals, including non-Dirac-liquid behavior.

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.

Nontrivial quantum geometry of electronic bands has been argued to facilitate superconductivity even for the case of flat dispersions where the conventional contribution to the superfluid weight is suppressed by the large effective mass. However, most previous work focused on the case of conventional superconductivity while many contemporary superconducting quantum materials are expected to host unconventional pairing. Here, we derive a generalized expression for the superfluid weight employing mean-field BCS theory for systems with time-reversal symmetry in the normal state and arbitrary unconventional superconducting order with zero-momentum intraband pairing. Our derivation reveals the necessity of incorporating functional derivatives of the grand potential with respect to the superconducting gap function. Through perturbative analysis in the isolated narrow-bands limit, we demonstrate that this contribution arises from quantum geometrical effects, specifically due to a nontrivial Wilczek-Zee connection. Utilizing the newly obtained expressions for the superfluid weight, we apply our framework to an extended Kane-Mele model, contrasting conventional s-wave superconductivity with chiral d-wave superconductivity.

Two-dimensional materials with interacting Dirac excitations can host quantum multicritical behavior near the phase boundaries of the semimetallic and two-ordered phases. We study such behavior in Gross-Neveu-Yukawa field theories where N-f flavors of Dirac fermions are coupled to two order-parameter fields with SO(N-A) and SO(N-B) symmetry, respectively. To that end, we employ the perturbative renormalization group up to three-loop order in 4-is an element of spacetime dimensions. We distinguish two key scenarios: (i) The two orders are compatible as characterized by anticommuting mass terms, and (ii) the orders are incompatible. For the first case, we explore the stability of a quantum multicritical point with emergent SO(N-A+N-B) symmetry. We find that the stability is controlled by increasing the number of Dirac fermion flavors. Moreover, we extract the series expansion of the leading critical exponents for the chiral SO(4) and SO(5) models up to third order in E. Notably, we find a tendency towards rapidly growing expansion coefficients at higher orders, rendering an extrapolation to is an element of = 1 difficult. For the second scenario, we study a model with SO(4)similar or equal to SO(3) x SO(3) symmetry, which was recently suggested to describe criticality of antiferromagnetism and superconductivity in Dirac systems. However, it was also argued that a physically admissible renormalization-group fixed point only exists for Nf above a critical number Nc>. We determine the corresponding series expansion at three-loop order as N-c degrees approximate to 16.83-7.14 is an element of-7.12 is an element of(2). This suggests that the physical choice of Nf = 2 may be a borderline case, where true criticality and pseudocriticality, as induced by fixed-point annihilation, are extremely challenging to distinguish.

Electron-electron and electron-phonon interactions are responsible for the formation of spin, charge, and superconducting correlations in layered quantum materials. A paradigmatic model for such materials that captures both kinds of interactions is the two-dimensional Hubbard-Holstein model with a dispersionless Einstein phonon. In this work, we provide a detailed analysis of the magnetic, density, and superconducting fluctuations at and away from half filling. To that end, we employ the functional renormalization group using the recently introduced extension of the single-boson exchange formulation. More precisely, we go beyond previous approaches to the model by resolving the full frequency dependence of the two-particle vertex and taking into account the feedback from the electronic self-energy. We perform broad parameter scans in the space of Hubbard repulsion, electron-phonon coupling strength, and phonon frequency to explore the leading magnetic, density, and superconducting susceptibilities from the adiabatic to the anti-adiabatic regime. Our numerical data reveal that self-energy effects lead to an enhancement of the d-wave superconducting susceptibility toward larger phonon frequencies, in contrast to earlier isotope-effect studies. At small phonon frequencies, large density contributions to the s-wave superconducting susceptibility change sign and eventually lead to a reduction of s-wave superconductivity with increasing electron-phonon coupling, signaling the breakdown of Migdal-Eliashberg theory. We analyze our findings systematically, employing detailed diagnostics of the intertwined fluctuations and pinning down the various positive and negative isotope effects of the physical susceptibilities. Our results pave the way for a more comprehensive understanding of the interplay of electron-electron and electron-phonon interactions in correlated quantum materials.

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 nonzero 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 corrections, 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 a nested Fermi surface and a large number of fermion flavors. We apply the Kohn-Luttinger-like approach to square-PLXINSERT-, and triangular-lattice Hubbard models with SU(Nf) 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 the Van Hove filling. We demonstrate that this K/4 instability can win the competition against pairing for Nf = 4 via an unbiased functional renormalization group calculation.

Two-dimensional van der Waals materials offer a highly tunable platform for engineering electronic band structures and interactions. By employing techniques such as twisting, gating, or applying pressure, these systems enable precise control over the electronic excitation spectrum. In moir & eacute; bilayer graphene, the tunability facilitates the transition from a symmetric Dirac semimetal phase through a quantum critical point into an interaction-induced long-range ordered phase with a finite band gap. At charge neutrality, the ordered state proposed to emerge from twist-angle tuning is the Kramers intervalley-coherent insulator. In this case, the transition falls into the quantum universality class of the relativistic Gross-Neveu-XY model in 2 + 1 dimensions. Here, we refine estimates for the critical exponents characterizing this universality class using an expansion around the lower critical space-time dimension of two. We compute the order-parameter anomalous dimension q phi and the correlation-length exponent v at one-loop order, and the fermion anomalous dimension q psi at two-loop order. Combining these results with previous findings from the expansion around the upper critical dimension, we obtain improved estimates for the universal exponents in 2 + 1 dimensions via Pad & eacute; interpolation. For Nf = 4 four-component Dirac fermions, relevant to moir & eacute; bilayer graphene, we estimate 1/v = 0.916(5), q phi = 0.926(13), and q psi = 0.0404(13). For Nf = 2, potentially relevant to recent tetralayer WSe2 experiments, the Gross-Neveu-XY fixed point may be unstable owing to a fixed-point collision at Nf,c, with Nf,c = 1 + 2 + O(E) in the expansion around the lower critical dimension.

The discovery of a charge density wave in FeGe extends the discussion of the nature of charge order in kagome metals to a magnetic compound. Motivated by this observation, we combine density functional theory (DFT) and functional-renormalization-group calculations to study interaction-induced Fermi-surface instabilities of the magnetic state of FeGe. We argue that the leading intra-band contribution to electronic correlations are approximately two-dimensional (2D) and come from Van Hove points at the projected M points. By varying parameters around DFT values, we determine a phase diagram for the quasi-2D scenario as function of on-site and nearest-neighbor interactions. We discuss universal aspects in the electronic mechanisms for the resulting phases, as well as the role of SU(2) symmetry breaking. We find FeGe to be in a regime of strong competition between p-wave charge density wave, f-wave pairing, and d-wave spin Pomeranchuk instabilities. This interplay can be influenced in favor of superconducting pairing for slightly increased nearest-neighbor interaction, suggesting a potential to induce superconductivity in FeGe.

Spontaneous symmetry breaking can persist at all temperatures in certain biconical O(N)×Z2 vector models when the underlying field theories are ultraviolet complete. So far, the existence of such theories has been established in fractional dimensions for local but nonunitary models or in 2+1 dimensions but for nonlocal models. Here, we study local models at zero and finite temperature directly in 2+1 dimensions employing functional methods. At zero temperature, we establish that our approach describes the quantum critical behavior with good accuracy for all N≥2. We then exhibit the mechanism of discrete symmetry breaking from O(N)×Z2→O(N) for increasing temperature near the biconical critical point when N is finite but large. We calculate the corresponding finite-temperature phase diagram and further show that the Hohenberg-Mermin-Wagner theorem is fully respected within this approach, i.e., symmetry breaking only occurs in the Z2 sector. Finally, we determine the critical N above which this phenomenon can be observed to be Nc≈15. Published by the American Physical Society 2025

Moir\'e 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 moir\'e 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 moir\'e 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 moir\'e 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.

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 Néel 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≥ 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.

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 moiré 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.

The discovery of correlated insulating states in moire 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 moire heterostructures also exhibit interactions that break the SU(4) symmetry down to SU(2)spin & OTIMES; U(1)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 moire unit cell. We explore a broad regime of couplings-including XXZ anisotropies, Dzyaloshinskii-Moriya exchange, and on-site Hund's couplings-using semiclassical 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 pseudofermion 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.

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.

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.