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arXiv:2511.16496

$c=-2$ conformal field theory in quadratic band touching

Rintaro Masaoka

Published Nov 20, 2025• 0 citations
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Abstract

Quadratic band touching in fermionic systems defines a universality class distinct from that of linear Dirac points, yet its characterization as a quantum critical point remains incomplete. In this work, I show that a $(d+1)$-dimensional free-fermion model with quadratic band touching exhibits spatial conformal invariance, and that its equal-time ground-state correlation functions are exactly captured by the $d$-dimensional symplectic fermion theory. I establish this correspondence by constructing explicit mappings between physical fermionic operators and the fields of the symplectic fermion theory. I further explore the implications of this correspondence in two spatial dimensions, where the symplectic fermion theory is a logarithmic conformal field theory with central charge $c=-2$. In the corresponding $(2+1)$-dimensional systems, I identify anyonic excitations originating from the underlying symplectic fermion theory, even though the Hamiltonian is gapless. Transporting these excitations along non-contractible loops generates transitions among topologically degenerate ground states, in close analogy with those in topologically ordered phases. Moreover, the action of a $2π$ rotation on these excitations is represented by a Jordan block, reflecting the logarithmic character of the associated conformal field theory.

DOI
10.48550/arxiv.2511.16496

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c=−2c=-2c=−2 conformal field theory in quadratic band touching
Quadratic band touching in fermionic systems defines a universality class distinct from that of linear Dirac points, yet its characterization as a quantum critical point remains incomplete. In this work, I show that a (d+1)(d+1)(d+1)-dimensional free-fermion model with quadratic band touching exhibits spatial conformal invariance, and that its equal-time ground-state correlation functions are exactly captured by the ddd-dimensional symplectic fermion theory. I establish this correspondence by constructing explicit mappings between physical fermionic operators and the fields of the symplectic fermion theory. I further explore the implications of this correspondence in two spatial dimensions, where the symplectic fermion theory is a logarithmic conformal field theory with central charge c=−2c=-2c=−2. In the corresponding (2+1)(2+1)(2+1)-dimensional systems, I identify anyonic excitations originating from the underlying symplectic fermion theory, even though the Hamiltonian is gapless. Transporting these excitations along non-contractible loops generates transitions among topologically degenerate ground states, in close analogy with those in topologically ordered phases. Moreover, the action of a 2π2π2π rotation on these excitations is represented by a Jordan block, reflecting the logarithmic character of the associated conformal field theory.
The internal low-energy symmetry of the massless Lorentz-invariant Dirac Hamiltonian in 2+12+12+1 dimensions is known to be O(2N)O(2N)O(2N), where NNN is the number of two-component Dirac fermions. Here we point out that there exists an analogous internal symmetry of the single-particle quadratic-band-touching Hamiltonian in two spatial dimensions, and it is the unitary symplectic group, USp(2N)USp(2N)USp(2N). All fermionic bilinears belong to one of the three small irreducible representations of this group. The interacting theory that respects the USp(2N)USp(2N)USp(2N) symmetry and the spatial rotations is constructed and found to allow two independent interaction terms. When these interactions are infrared-relevant the symplectic symmetry either remains preserved or becomes spontaneously broken to USp(N)×USp(N)USp(N) \times USp(N)USp(N)×USp(N). The symmetry in the lattices such as honeycomb to infinite order in the dispersion's expansion in powers of local momentum is given by the overlap of the symplectic and the orthogonal groups. We show that this overlap is O(2N)⋂USp(2N)=U(N)O(2N) \bigcap USp(2N) = U(N)O(2N)⋂USp(2N)=U(N).
Conformal invariance often accompanies criticality in Hermitian systems. However, its fate in non-Hermitian settings is less clear, especially near exceptional points where the Hamiltonian becomes non-diagonalizable. Here we investigate whether a 1+1-dimensional gapless non-Hermitian system can admit a conformal description, focusing on a PT-symmetric free-fermion field theory. Working in the biorthogonal formalism, we identify the conformal structure of this theory by constructing a traceless energy-momentum tensor whose Fourier modes generate a Virasoro algebra with central charge c=−2c=-2c=−2. This yields a non-Hermitian, biorthogonal realization of a logarithmic conformal field theory, in which correlation functions exhibit logarithmic scaling and the spectrum forms Virasoro staggered modules that are characterized by universal indecomposability parameters. We further present a microscopic construction and show how the same conformal data (with finite-size corrections) can be extracted from the lattice model at exceptional-point criticality, thereby supporting the field-theory prediction.
We construct and discuss the field theory for tensorial nematic order parameter coupled to gapless four-component fermions at the quadratic band touching point in three (spatial) dimensions. Within a properly formulated epsilon-expansion this theory is found to have a quantum critical point, which describes the (presumably continuous) transition from the semimetal into a (nematic) Mott insulator. The latter phase breaks the rotational, but not the time-reversal, symmetry and may be relevant to materials such as gray tin or mercury telluride at low temperatures. The critical point represents a simple quantum analog of the familiar classical isotropic-to-nematic transition in liquid crystals. The properties and the consequences of this quantum critical point are discussed. Its existence supports the scenario of the ``fixed-point collision,'' according to which three-dimensional Fermi systems with quadratic band touching and long-range Coulomb interactions are unstable towards the gapped nematic ground state at low temperatures.
We develop a general theory of frustration-free free-fermion systems and derive the necessary and sufficient conditions for such Hamiltonians. Assuming locality and translation invariance, we find that any band touching between the valence and the conduction bands is always quadratic or softer, which rules out the possibility of describing Dirac and Weyl semimetals using frustration-free local Hamiltonians. We further construct a frustration-free free-fermion model on the honeycomb lattice and show that its density fluctuations acquire an anomalous gap originating from the diverging quantum metric associated with the quadratic band-touching points. Nevertheless, an O ( 1 / L 2 ) finite-size scaling of the charge-neutral excitation gap can be verified even in the presence of interactions, consistent with the more general results we derive in an accompanying work [Masaoka , ].
Abstract Interaction-driven topological phases significantly enrich the class of topological materials and thus are of great importance. Here, we study the phase diagram of interacting spinless fermions filling the two-dimensional checkerboard lattice with a quadratic band touching (QBT) point. By developing new diagnosis based on the state-of-the-art density-matrix renormalization group and exact diagonalization, we determine accurate quantum phase diagram for such a system at half-filling with three distinct phases. For weak nearest-neighboring interactions, we demonstrate the instability of the QBT towards an interaction-driven spontaneous quantum anomalous Hall (QAH) effect. For strong interactions, the system breaks the rotational symmetry realizing a nematic charge-density-wave (CDW) phase. Interestingly, for intermediate interactions we discover a symmetry-broken bond-ordered critical phase sandwiched in between the QAH and CDW phases, which splits the QBT into two Dirac points driven by interaction. Instead of the direct transition between QAH and CDW phases, our identification of an intermediate phase sheds new light on the theoretical understanding of the interaction-driven phases in QBT systems.
Despite great successes in the study of gapped phases, a comprehensive understanding of the gapless phases and their transitions is still under development.In this paper, we study a general phenomenon in the space of (1 + 1)-dimensional critical phases with fermionic degrees of freedom described by a continuous family of conformal field theories (CFTs), also known as the conformal manifold.Along a one-dimensional locus on the conformal manifold, there can be a transition point, across which the fermionic CFTs on the two sides differ by stacking an invertible fermionic topological order (IFTO), point by point along the locus.At every point on the conformal manifold, the order and disorder operators have power-law two-point functions, but their critical exponents cross over with each other at the transition point, where stacking the IFTO leaves the fermionic CFT unchanged.We call this continuous transition on the fermionic conformal manifold a topological transition.By gauging the fermion parity, the IFTO stacking becomes a Kramers-Wannier duality between the corresponding bosonic CFTs.Both the IFTO stacking and the Kramers-Wannier duality are induced by the electromagnetic duality of the (2 + 1)-dimensional Z 2 topological order.We provide several examples of topological transitions, including the familiar Luttinger model of spinless fermions (i.e., the c = 1 massless Dirac fermion with the Thirring interaction) and a class of c = 2 examples describing U(1) × SU(2)-protected gapless phases.