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arXiv:2602.12341

Lifshitz critical points meet Zamolodchikov perturbation theory

António Antunes

Published Feb 12, 2026• 0 citations
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Abstract

Critical points of classical and quantum lattice models are often described by scale-invariant Lifshitz theories which are anisotropic in the continuum limit, as characterized by a dynamical critical exponent $z\neq1$. This type of critical behavior can in principle be studied by deforming ordinary $z=1$ conformal field theories (CFTs) by relevant vector operators breaking the rotational/Lorentz symmetry. In this short note, we consider a two-dimensional system of coupled minimal model CFTs $\mathcal{M}_{m,m+1}$ which realizes this perspective in a controlled fashion via Zamolodchikov's large $m$ expansion. The model turns out to exhibit interesting properties, including a manifold of interacting Lifshitz fixed points and emergent rotational symmetry in the infrared.

DOI
10.48550/arxiv.2602.12341

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Critical points of classical and quantum lattice models are often described by scale-invariant Lifshitz theories which are anisotropic in the continuum limit, as characterized by a dynamical critical exponent z≠1z\neq1z=1. This type of critical behavior can in principle be studied by deforming ordinary z=1z=1z=1 conformal field theories (CFTs) by relevant vector operators breaking the rotational/Lorentz symmetry. In this short note, we consider a two-dimensional system of coupled minimal model CFTs Mm,m+1\mathcal{M}_{m,m+1}Mm,m+1​ which realizes this perspective in a controlled fashion via Zamolodchikov's large mmm expansion. The model turns out to exhibit interesting properties, including a manifold of interacting Lifshitz fixed points and emergent rotational symmetry in the infrared.
We explore the dynamical behavior at and near a special class of two-dimensional quantum critical points. Each is a conformal quantum critical point (CQCP), where in the scaling limit the equal-time correlators are those of a two-dimensional conformal field theory. The critical theories include the square-lattice quantum dimer model, the quantum Lifshitz theory, and a deformed toric code model. We show that under generic perturbation the latter flows toward the ordinary Lorentz-invariant (2+12+12+1)-dimensional Ising critical point, illustrating that CQCPs are generically unstable. We exploit a correspondence between the classical and quantum-dynamical behavior in such systems to perform an extensive numerical study of two lines of CQCPs in a quantum eight-vertex model or, equivalently, two coupled deformed toric codes. We find that the dynamical critical exponent zzz remains 2 along the U(1)-symmetric quantum Lifshitz line, while it continuously varies along the line with only Z2{\mathbb{Z}}_{2}Z2​ symmetry. This illustrates how two CQCPs can have very different dynamical properties, despite identical equal-time ground-state correlators. Our results equally apply to the dynamics of the corresponding purely classical models.
We consider Lifshitz criticality (LC) with the dynamical critical exponent z=2z=2z=2 in one-dimensional interacting fermion systems with a filled Dirac sea. We report that interactions have crucial effects on Lifshitz criticality. Single-particle excitations are destabilized by an interaction and decay into the particle-hole continuum, which is reflected in the logarithmic divergence in the imaginary part of one-loop self-energy. We show that the system is sensitive to the sign of the interaction. The random-phase approximation shows that the collective particle-hole excitations emerge only when the interaction is repulsive. The dispersion of collective modes is gapless and linear. If the interaction is attractive, the one-loop renormalization group (RG) shows that there may exist a stable RG fixed point described by two coupling constants. We also show that the on-site interaction (without any other perturbations at the UV scale) would always turn on the relevant velocity perturbation to the quadratic Lagrangian in the RG flow, driving the system flow to the conformal-invariant criticality. In the numerical simulations of the lattice model at half filling, we find that, for either on-site positive or negative interactions, the dynamical critical exponent becomes z=1z=1z=1 in the infrared limit and the entanglement entropy is a logarithmic function of the system size LLL. The work paves the way to study one-dimensional interacting LCs.
An introduction to the theory of critical behavior at Lifshitz points is given, and the recent progress made in applying the field-theoretic renormalization group (RG) approach to ϕ4ϕ^4ϕ4 nnn-vector models representing universality classes of mmm-axial Lifshitz points is surveyed. The origins of the difficulties that had hindered a full two-loop RG analysis near the upper critical dimension for more than 20 years and produced long-standing contradictory εεε-expansion results are discussed. It is outlined how to cope with them. The pivotal role the considered class of continuum models might play in a systematic investigation of anisotropic scale invariance within the context of thermal equilibrium systems is emphasized. This could shed light on the question of whether anisotropic scale invariance implies an even larger invariance, as recently claimed in the literature.
The critical behaviour of semi-infinite d -dimensional systems with short-range interactions and an O( n ) invariant Hamiltonian is investigated at an m -axial Lifshitz point with an isotropic wave-vector instability in an m -dimensional subspace of d parallel to the surface. Continuum |ϕ| 4 models representing the associated universality classes of surface critical behaviour are constructed. In the boundary parts of their Hamiltonians quadratic derivative terms (involving a dimensionless coupling constant λ) must be included in addition to the familiar ones ∝ϕ 2 . Beyond one-loop order the infrared-stable fixed points describing the ordinary, special and extraordinary transitions in d = 4 + m /2 − dimensions (with > 0) are located at λ = λ* = O( ). At second order in , the surface critical exponents of both the ordinary and the special transitions start to deviate from their m = 0 analogues. Results to order 2 are presented for the surface critical exponent β ord 1 of the ordinary transition. The scaling dimension of the surface energy density is shown to be given exactly by d + m (θ − 1), where θ = ν l 4 /ν l 2 is the bulk anisotropy exponent.
Field-theoretic renormalization-group methods are developed to describe in a unified fashion the critical exponents of an m-fold Lifshitz point at the two-loop order in the anisotropic $(m\ensuremath{\ne}d)$ and isotropic (m=d(m=d(m=d close to 8) situations. The general theory is illustrated for the N-vector ${\ensuremath{\varphi}}^{4}$ model describing a d-dimensional system. A new regularization and renormalization procedure is presented for both types of Lifshitz behavior. The anisotropic cases are formulated with two independent renormalization-group transformations. The description of the isotropic behavior requires only one type of renormalization-group transformation. We point out the conceptual advantages implicit in this picture and show how this framework is related to other previous renormalization-group treatments for the Lifshitz problem. The Feynman diagrams of arbitrary loop order can be performed analytically provided these integrals are considered to be homogeneous functions of the external momenta scales. The anisotropic universality class (N,d,m)(N,d,m)(N,d,m) reduces easily to the Ising-like (N,d)(N,d)(N,d) when m=0.m=0.m=0. We show that the isotropic universality class (N,m)(N,m)(N,m) when m is close to 8 cannot be obtained from the anisotropic one in the limit $\stackrel{\ensuremath{\rightarrow}}{d}m$ near 8. The exponents for the uniaxial case d=3,N=m=1d=3,N=m=1d=3,N=m=1 are in good agreement with recent Monte Carlo simulations for the axial next-nearest-neighbor Ising model.