Critical statistical mechanics and Conformal Field Theory (CFT) are conjecturally connected since the seminal work of Beliavin et al. (Nucl Phys B 241(2):333–380, 1984). Both exhibit exactly solvable structures in two dimensions. A longstanding question (Itoyama and Thacker in Phys Rev Lett 58:1395–1398, 1987) concerns whether there is a direct link between these structures, that is, whether the Virasoro algebra representations of CFT, the distinctive feature of CFT in two dimensions, can be found within lattice models of statistical mechanics. We give a positive answer to this question for the discrete Gaussian free field and for the Ising model, by connecting the structures of discrete complex analysis in the lattice models with the Virasoro symmetry that is expected to describe their scaling limits. This allows for a tight connection of a number of objects from the lattice model world and the field theory one. In particular, our results link the CFT local fields with lattice local fields introduced in Gheissari et al. (Commun Math Phys 367(3):771–833, 2019) and the probabilistic formulation of the lattice model with the continuum correlation functions. Our construction is a decisive step towards establishing the conjectured correspondence between the correlation functions of the CFT fields and those of the lattice local fields. In particular, together with the upcoming (Chelkak et al. in preparation), our construction will complete the picture initiated in Hongler and Smirnov (Acta Math 211:191–225, 2013), Hongler (Conformal invariance of ising model correlations, 2012) and Chelkak et al. (Annals Math 181(3):1087–1138, 2015), where a number of conjectures relating specific Ising lattice fields and CFT correlations were proven.
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Inspired by the recent work by Delacretaz et. al. [Phys. Rev. Res. 4, 033131 (2022)], we rigorously derive an exact and simple method to bosonize a noninteracting fermionic system with a Fermi surface starting from a microscopic Hamiltonian. In the longwavelength limit, we show that the derived bosonized action is exactly equivalent to the action obtained by Delacretaz et. al. In addition, we propose diagrammatic rules to evaluate correlation functions using our bosonized theory and demonstrate these rules by calculating the three and fourpoint density correlation functions. We also consider a general densitydensity interaction and show that the simplest approximation in our bosonic theory is identical to RPA results.
more » « less Award ID(s):
 2116515
 NSFPAR ID:
 10538921
 Publisher / Repository:
 SciPost
 Date Published:
 Journal Name:
 SciPost Physics
 Volume:
 16
 Issue:
 3
 ISSN:
 25424653
 Page Range / eLocation ID:
 069
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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