Relativistic Mott Transition

• Relativistic Mott transition is a correlation-driven phenomenon where interactions generate mass gaps in systems with emergent relativistic dispersion, including Dirac semimetals, bosonic lattices, and SOC-coupled oxides.
• Theoretical approaches use Gross–Neveu models, functional renormalization group, and scalar field theories to capture z=1 quantum criticality, mass generation, and fluctuation-induced effects.
• Experimental validations in twisted WSe₂, cold-atom optical lattices, and 4d/5d compounds confirm the emergence of Dirac mass generation, Higgs modes, and altered critical interaction scales.

Searching arXiv for recent and foundational papers on "relativistic Mott transition" across Dirac, bosonic, and spin–orbit-coupled contexts. Relativistic Mott transition denotes a correlation-driven transition whose low-energy description is relativistic in an emergent, effective, or spin–orbit-entangled sense rather than in the literal sense of electrons moving at the speed of light. In current usage, the term spans at least three closely related but nonidentical settings: interaction-driven mass generation in Dirac semimetals, superfluid–Mott criticality at the tips of bosonic Mott lobes with emergent Lorentz invariance, and Mott localization in $4d/5d$ compounds where strong spin–orbit coupling reorganizes the low-energy Hilbert space into $J_{\rm eff}$-type degrees of freedom and qualitatively reshapes the insulating state (Ma et al., 2024, Faccioli et al., 2018, Kim et al., 2016). The common thread is that Mottness is inseparable from a relativistic structure in the effective theory: linear Dirac dispersion and chiral symmetry, $z=1$ quantum criticality, or spin–orbit-entangled local moments and exchange.

1. Scope of the concept

The phrase is not uniform across subfields. In Dirac materials, it usually means a semimetal–insulator transition in which a fermion bilinear condenses and generates a Dirac mass. In cold-atom bosonic systems, it refers to the superfluid–Mott transition at the tip of a Mott lobe, where the effective action becomes Lorentz invariant because the linear time-derivative term vanishes. In spin–orbit-coupled oxides, it denotes a Mott transition or Mott-state evolution in which SOC is of the same order as crystal fields and Coulomb scales, so that the relevant quasispins are spin–orbit entangled rather than ordinary spins (Tolosa-Simeón et al., 6 Mar 2025, Mosca et al., 2023).

Usage	Relativistic structure	Representative setting
Dirac semimetal to insulator	Linear Dirac spectrum, $z=1$, Gross–Neveu–Yukawa criticality	Artificial graphene in twisted WSe$_2$
Bosonic superfluid–Mott transition	Emergent Lorentz invariance at $K_1=0$	Bose–Hubbard model at Mott-lobe tips
Spin–orbit Mott physics	SOC-entangled $J_{\rm eff}$ manifold and anisotropic exchange	Iridates, osmates, honeycomb $t_{2g}^5$ systems

A persistent source of confusion is the adjective “relativistic.” In the bosonic case it refers to the structure of the low-energy Euclidean action, not to microscopic relativistic particles. In the spin–orbit case it refers to relativistic atomic SOC and the resulting $J_{\rm eff}$ basis, not to Dirac cones. In the Dirac-semimetal case, by contrast, the relativistic analogy is direct: the low-energy fermions are massless Dirac quasiparticles, and the insulating phase is obtained by dynamical mass generation.

A second misconception is that every SOC-active insulator is SOC-driven. DFT+DMFT for Ba$_2$NaOsO$J_{\rm eff}$0 shows a counterexample: SOC lowers the critical $J_{\rm eff}$1 and strongly affects multiplet structure, yet the gap remains present for realistic $J_{\rm eff}$2 even without SOC, placing the compound in the Mott-insulator regime rather than in a strictly SOC-driven one (Mosca et al., 2023).

2. Dirac semimetals, dynamical mass generation, and Gross–Neveu criticality

In the Dirac setting, the noninteracting theory is a $J_{\rm eff}$3-dimensional massless Dirac semimetal,

$J_{\rm eff}$4

supplemented by short-range four-fermion interactions or, equivalently, by a Gross–Neveu–Yukawa coupling $J_{\rm eff}$5 to an order-parameter field $J_{\rm eff}$6 (Tolosa-Simeón et al., 6 Mar 2025). For sufficiently strong coupling, $J_{\rm eff}$7, the fermions acquire a mass $J_{\rm eff}$8, and the Dirac semimetal is replaced by a symmetry-broken insulating phase. Depending on the symmetry of $J_{\rm eff}$9, the transition falls into the chiral Ising, chiral XY, or chiral Heisenberg universality class.

This framework is the standard field-theoretic realization of a relativistic Mott transition in honeycomb-like Dirac systems. The ordered phase is “Mott-like” because the gap is not a band-structure gap but an interaction-generated one, and the critical point is relativistic because the effective dynamical exponent is $z=1$0. For $z=1$1, functional RG in the LPA$z=1$24 truncation gives $z=1$3, $z=1$4, and $z=1$5 for chiral Ising; $z=1$6, $z=1$7, and $z=1$8 for chiral XY; and $z=1$9, $z=1$0, and $z=1$1 for chiral Heisenberg (Tolosa-Simeón et al., 6 Mar 2025).

The same FRG analysis also clarifies finite-temperature structure. In the symmetric Dirac phase, the thermal correlation length obeys $z=1$2 in the quantum critical fan, with boundaries

$z=1$3

while for small $z=1$4 and low $z=1$5 there is a Dirac-dominated regime with $z=1$6 (Tolosa-Simeón et al., 6 Mar 2025). For continuous $z=1$7 and $z=1$8 symmetries, the same calculation explicitly realizes the Coleman–Hohenberg–Mermin–Wagner theorem: precondensation can occur at intermediate RG scales, but true finite-$z=1$9 long-range order is destroyed in the infrared.

The experimental realization reported in twisted WSe$_2$0 tetralayers gives a concrete condensed-matter implementation of this scenario. There, the first moiré valence band mimics graphene at half filling, with anomalous Landau fans carrying a $_2$1-Berry phase and cyclotron mass $_2$2, and a semimetal–insulator transition appears when the twist angle is reduced below about $_2$3 (Ma et al., 2024). The emergent insulator is compatible with an antiferromagnetic Mott insulator, making the system a candidate realization of the chiral Heisenberg version of relativistic Mott criticality.

3. Spin–orbit-entangled Mott physics

In correlated $_2$4 compounds, “relativistic Mott” refers to a different but equally precise mechanism. Strong atomic SOC, $_2$5, competes with Coulomb repulsion, Hund’s coupling, and crystal-field splitting, and the low-energy states are spin–orbit-entangled multiplets rather than spin-only orbitals. In honeycomb $_2$6 systems, exact diagonalization shows a crossover from a quasimolecular band insulator to a relativistic $_2$7 Mott insulator as either SOC or Coulomb repulsion is increased (Kim et al., 2016). In the QMO limit, the gap is band-like and the hole density in the $_2$8 quasimolecular orbital remains near its ideal value; in the $_2$9 regime, the gap becomes correlation-driven and the characteristic optical scale follows

$K_1=0$0

Optical conductivity evolves from multi-peak inter-QMO structures to a dominant Mott-like peak, while RIXS develops the spin–orbit excitation at $K_1=0$1 and an exciton-like feature tied to $K_1=0$2 doublon–holon physics (Kim et al., 2016).

Ba$K_1=0$3NaOsO$K_1=0$4 illustrates a more restricted sense in which SOC assists rather than drives the transition. In DFT+DMFT with PAW spinor projectors, the paramagnetic cubic phase is insulating both with and without SOC for realistic interactions; the gap is about $K_1=0$5 eV in both cases, while SOC reduces the critical interaction from $K_1=0$6 eV to $K_1=0$7 eV and reorganizes the upper Hubbard band into mixed $K_1=0$8 character (Mosca et al., 2023). The implication is exacting but important: strong SOC does not automatically imply an SOC-driven metal–insulator transition.

A nonequilibrium extension of the same theme appears in ultrafast studies of relativistic Mott insulators. In a one-band Hubbard model with SOC encoded as spin-dependent hopping, nonequilibrium DMFT shows that photoexcited doublons and holons melt magnetic order through spin–charge coupling, but the outcome depends strongly on lattice geometry and SOC texture (Li et al., 2020). On a square lattice with canted antiferromagnetism, the spin-orbit-induced canting angle remains unchanged after excitation, whereas on a triangular lattice with chiral $K_1=0$9 order, relaxation times are sensitive to SOC. Strictly speaking, this is a photoinduced nonequilibrium analogue of approaching a Mott transition rather than an equilibrium critical point, but it shows that in SOC-entangled systems the route out of the Mott state can itself be “relativistic.”

4. Bosonic relativistic Mott criticality

In the Bose–Hubbard problem, the relativistic Mott transition is the superfluid–Mott transition at integer filling near the tips of the Mott lobes. The coarse-grained Euclidean action is

$J_{\rm eff}$0

and the critical points are the lobe tips where $J_{\rm eff}$1 (Faccioli et al., 2018). At that point the low-energy theory is fully relativistic, with $J_{\rm eff}$2, and the universality class is that of the $J_{\rm eff}$3-dimensional XY model.

The excitation spectrum captures the standard amplitude–phase structure of relativistic symmetry breaking. In the Mott phase there are two gapped particle/hole branches. In the superfluid phase, Gaussian fluctuations produce a gapless Goldstone mode and a gapped Higgs mode. At $J_{\rm eff}$4, the Goldstone branch is linear,

$J_{\rm eff}$5

and the Higgs gap closes as $J_{\rm eff}$6 (Faccioli et al., 2018). Comparison with the quasi-2D optical-lattice experiment of Endres et al. shows good agreement for the excitation gap near the $J_{\rm eff}$7 tip.

The same analysis yields a more subtle result away from the tips. Mean field predicts a second-order quantum phase transition throughout, but Gaussian quantum fluctuations modify the zero-temperature equation of state. At the critical tips, the transition remains continuous in both $J_{\rm eff}$8 and $J_{\rm eff}$9. Away from the tips, where $t_{2g}^5$0, the Gaussian contribution changes the functional dependence of the pressure so that its first derivative becomes discontinuous across the transition, converting the mean-field second-order transition into a fluctuation-induced first-order one (Faccioli et al., 2018). The mechanism is explicitly compared to Coleman–Weinberg-type fluctuation-induced first-order behavior, but here it arises without coupling to gauge fields.

This bosonic usage is often overlooked in discussions dominated by electronic Dirac systems. Yet it provides one of the cleanest realizations of relativistic Mott criticality: a scalar $t_{2g}^5$1 field theory with emergent Lorentz invariance, Higgs and Goldstone modes, and analytically controlled beyond-mean-field corrections.

5. Materials platforms and experimental diagnostics

The most direct experimental realization of a Dirac relativistic Mott transition so far is strongly correlated artificial graphene in twisted WSe$t_{2g}^5$2 tetralayers. Magnetotransport identifies the Dirac semimetal through a Landau fan with the anomalous filling sequence of massless Dirac fermions, a $t_{2g}^5$3-Berry phase, and a cyclotron mass following $t_{2g}^5$4; the interaction strength can be tuned by the twist angle, and the semimetal–insulator transition occurs below about $t_{2g}^5$5, at an estimated critical $t_{2g}^5$6 (Ma et al., 2024). The insulating state is compatible with an antiferromagnetic Mott insulator rather than a single-particle band insulator.

A complementary continuum-plus-Hubbard analysis of twisted double-bilayer WSe$t_{2g}^5$7 sharpens the same picture. The $t_{2g}^5$8-valley moiré valence bands realize Dirac cones at half filling, and the angle-dependent effective Hubbard model on the moiré honeycomb lattice yields a Hartree–Fock critical ratio $t_{2g}^5$9 at the experimentally relevant $J_{\rm eff}$0 (Hawashin et al., 11 Sep 2025). The same work shows that twist-angle tuning can engineer a high-order van Hove singularity in the second-to-topmost moiré band at $J_{\rm eff}$1 in the continuum model and $J_{\rm eff}$2 in the fitted tight-binding description, with

$J_{\rm eff}$3

Near other fillings the mean-field phase diagram contains non-coplanar spin-density waves with nonzero spin chirality and half-metallic uniaxial spin-density waves, indicating that relativistic Mott systems in moiré semiconductors naturally border a broader landscape of VHS-driven magnetism (Hawashin et al., 11 Sep 2025).

In SOC-entangled oxides, the dominant diagnostics are different. Honeycomb $J_{\rm eff}$4 materials such as Na$J_{\rm eff}$5IrO$J_{\rm eff}$6 and $J_{\rm eff}$7-RuCl$J_{\rm eff}$8 are placed far from the quasimolecular band-insulating limit and on the relativistic Mott side of the QMO–$J_{\rm eff}$9 crossover by optical conductivity and RIXS, especially through the emergence of the $_2$0 local $_2$1–$_2$2 excitation and the $_2$3 Mott-scale optical feature (Kim et al., 2016). In Ba$_2$4NaOsO$_2$5, by contrast, spectroscopy and DMFT distinguish a Mott insulator with strong SOC from a truly SOC-driven insulator (Mosca et al., 2023).

Cold-atom optical lattices provide the most controlled bosonic platform. Near the tip of the $_2$6 Mott lobe, lattice modulation and related probes detect the Higgs amplitude mode and confirm the relativistic GL description of the superfluid–Mott transition (Faccioli et al., 2018). Across platforms, the diagnostics differ, but the logic is common: identify the relativistic low-energy sector, locate the interaction scale at which a gap opens, and determine whether the gap reflects symmetry-breaking mass generation, correlation-induced Hubbard splitting, or both.

6. Finite-temperature structure, nearby instabilities, and open problems

Finite temperature does not merely broaden relativistic Mott transitions; it reorganizes them. In the GNY description, the quantum critical fan occupies a finite region above the zero-temperature QCP, with $_2$7, while continuous-symmetry cases exhibit precondensation: order develops at intermediate RG scales and is later destroyed by Goldstone fluctuations in the infrared (Tolosa-Simeón et al., 6 Mar 2025). In the chiral XY case, FRG also reveals BKT-like signatures even in the presence of strong Dirac-fermion fluctuations, including rapid growth of the correlation length and a quasi-constant phase stiffness over wide RG intervals (Tolosa-Simeón et al., 6 Mar 2025).

Another nearby phenomenon is superconductivity out of an incoherent Dirac metal. A SYK-like large-$_2$8 treatment of Gross–Neveu criticality shows that critical bosons can mediate pairing only once the fermions become sufficiently incoherent, with a threshold anomalous dimension $_2$9 (Stangier et al., 7 Oct 2025). In twisted double-bilayer WSe$J_{\rm eff}$00, all time-reversal-even gap-opening collective modes promote pairing whereas time-reversal-odd ones do not; in a Dirac model for twisted bilayer graphene, an inter-valley-coherent relativistic Mott transition yields multiple degenerate or nearly degenerate superconducting channels (Stangier et al., 7 Oct 2025). The same algebraic structure that controls boson-mediated pairing also controls the existence of Wess–Zumino–Witten terms and charge-carrying skyrmions in the proximate insulating state, tying superconductivity to topological defects rather than treating it as an unrelated instability.

An additional open direction concerns microscopic reformulations of Mott criticality itself. A slave-fermion treatment of the nonrelativistic Hubbard model proposes that the Mott transition can be viewed as a dynamic charge Kondo effect of local doublon and holon states and argues that the formalism can be ported to relativistic dispersions by replacing the bare auxiliary-fermion band structure with a Dirac or Weyl one (Long et al., 2022). This suggests a possible alternative language for relativistic Mott transitions, but such an extension remains inferential rather than canonical.

Two controversies therefore remain central. First, the label “relativistic Mott transition” is not universal: it can mean Gross–Neveu mass generation, emergent Lorentz-invariant bosonic criticality, or SOC-entangled Mottness. Second, even within one usage, the role of the additional “relativistic” ingredient varies. In some iridates and moiré Dirac systems it is indispensable to the very existence of the transition; in others, such as Ba$J_{\rm eff}$01NaOsO$J_{\rm eff}$02, it primarily lowers $J_{\rm eff}$03, restructures excitations, and controls magnetism without being the primary origin of the gap (Mosca et al., 2023). The phrase is thus best treated not as a single universal mechanism but as a family of Mott phenomena in which relativistic structure is part of the low-energy definition of the problem.