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arXiv:2510.13964 · Journal of High Energy Physics

Subdimensional disorder and logarithmic defect

Soichiro Shimamori, Yifan Wang

Published Jul 15, 2026• 1 citation
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Abstract

A bstract We study quenched disorder localized on a p -dimensional subspacetime in a d -dimensional conformal field theory. Motivated by the logarithmic behavior often associated with disorder, we introduce a defect setup in which bulk local operators transform in ordinary conformal representations, while defect local operators assemble into logarithmic multiplets. We refer to such objects as logarithmic defects and investigate their model-independent properties dictated solely by conformal symmetry and its representation theory, including correlation functions, logarithmic defect operator expansions, and conformal blocks. As a concrete example, we analyze the free scalar theory with a generalized pinning defect subject to random coupling fluctuations, and we identify a half-line of fixed points describing the corresponding logarithmic conformal defects. Along the way, we propose a candidate monotone governing defect renormalization group flows induced by subdimensional disorder, and demonstrate its consistency with such flows in the free theory across various dimensions. We further comment on generalizations and outline a broader program for bootstrapping logarithmic defects.

DOI
10.1007/jhep07(2026)129

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A bstract We study quenched disorder localized on a p -dimensional subspacetime in a d -dimensional conformal field theory. Motivated by the logarithmic behavior often associated with disorder, we introduce a defect setup in which bulk local operators transform in ordinary conformal representations, while defect local operators assemble into logarithmic multiplets. We refer to such objects as logarithmic defects and investigate their model-independent properties dictated solely by conformal symmetry and its representation theory, including correlation functions, logarithmic defect operator expansions, and conformal blocks. As a concrete example, we analyze the free scalar theory with a generalized pinning defect subject to random coupling fluctuations, and we identify a half-line of fixed points describing the corresponding logarithmic conformal defects. Along the way, we propose a candidate monotone governing defect renormalization group flows induced by subdimensional disorder, and demonstrate its consistency with such flows in the free theory across various dimensions. We further comment on generalizations and outline a broader program for bootstrapping logarithmic defects.
A bstract Defects in conformal field theories are interesting objects to study from both formal and applied points of view. In this paper, we construct conformal defects in free scalar field CFTs in diverse dimensions. After discussing the possible quadratic defects, we explore interacting setups. These are realized by coupling the bulk free scalar to lower-dimensional theories, including the unitary family of minimal models M(m,m+1) \mathcal{M}\left(m,m+1\right) M(m,m+1) M m m + 1 . Another example involves coupling to a two-dimensional free scalar field, from which we construct surface defects for the bulk dimensions three and five. Additionally, we consider monodromy defects associated with a global U(1) flavour symmetry. In these theories, we study both self-defect interactions and couplings to Minimal Models, finding new IR defect fixed points. For all our examples, we provide results for correlation functions, such as those involving the bulk stress tensor and the displacement operator, and for the defect central charges.
We consider defect conformal field theories (DCFTs) at finite temperature, where a ppp-dimensional conformal defect wraps the thermal circle. Extending the previous study of line defects to defects of general dimensions, we determine the general forms of the thermal one-point functions of bulk scalars, conserved currents and the stress tensor by imposing the residual symmetry and the conservation laws. The low temperature expansion of the thermal one-point functions is shown to be reproduced by the bulk-defect operator expansion, allowing us to read off the thermal one-point coefficients of defect operators. We examine the general results in four classes of examples with free bulk theories: the trivial defect, free scalar and fermion theories with a boundary, the free scalar theory with a localized ϕϕϕ deformation in d=2 p+2d=2\,p+2d=2p+2 dimensions and the free O(N)\mathrm{O}(N)O(N) model with a localized ϕ2ϕ^2ϕ2 deformation in d=p+2−εd=p+2-εd=p+2−ε dimensions. In the last example with ε=1ε= 1ε=1, where the defect becomes an interface, we find that the thermal one-point functions coincide with those of the free scalar theory with the Dirichlet boundary condition, in accordance with the conjectured factorization of an interface CFT into two decoupled boundary CFTs.
In this paper, we analyze the renormalization group (RG) flow of field theories with quenched disorder, in which the couplings vary randomly in space.We analyze both classical (Euclidean) disorder and quantum disorder, emphasizing general properties rather than specific cases.The RG flow of the disorderaveraged theories takes place in the space of their coupling constants and also in the space of distributions for the disordered couplings, and the two mix together.We write down a generalization of the Callan-Symanzik equation for the flow of disorder-averaged correlation functions.We find that local operators can mix with the response of the theory to local changes in the disorder distribution and that the generalized Callan-Symanzik equation mixes the disorder averages of several different correlation functions.For classical disorder, we show that this can lead to new types of anomalous dimensions and to logarithmic behavior at fixed points.For quantum disorder, we find that the RG flow always generates a rescaling of time relative to space, which at a fixed point generically leads to Lifshitz scaling.The dynamical scaling exponent z behaves as an anomalous dimension (as in other nonrelativistic RG flows), and we compute it at leading order in perturbation theory in the disorder for a general theory.Our results agree with a previous perturbative computation by Boyanovsky and Cardy, and with a holographic disorder computation of Hartnoll and Santos.We also find in quantum disorder that local operators mix with nonlocal (in time) operators under the RG, and that there are critical exponents associated with the disorder distribution that have not previously been discussed.In large-N theories, the disorder averages may be computed exactly, and we verify that they are consistent with the generalized Callan-Symanzik equations.
Abstract In this paper we study the constraints imposed by conformal invariance on extended objects a.k.a. defects in a conformal field theory. We identify a particularly nice class of defects that is closed under conformal transformations. Correlation function of the defect with a bulk local operator is fixed by conformal invariance up to an overall constant. This gives rise to the notion of defect expansion, where the defect itself is expanded in terms of local operators. This expansion generalizes the idea of the boundary state. We will show how one can fix the correlation function of two defects from the knowledge of the defect expansion. The defect correlator admits a number of conformal cross-ratios depending on their dimensionality. We find the differential equation obeyed by the conformal block and solve them in certain special cases.
We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane.If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs.For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect.We compare the value of the boundary entropy that our formula gives for the flows between nearby Virasoro minimal models  m with the boundary entropy of the defect constructed by Gaiotto (2012 J. High Energy Phys. JHEP12(2012) 103) and find a match at the first two orders in the m 1 expansion.
We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane. If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs. For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect. We compare the value of the boundary entropy that our formula gives for the flows between nearby Virasoro minimal models Mm with the boundary entropy of the defect constructed by Gaiotto in [1] and find a match at the first two orders in the 1/m expansion.