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arXiv:2210.12370 · Physical review. D/Physical review. D.

Evidence for web of dualities from monopole operators

Shai M. Chester, Éric Dupuis, William Witczak‐Krempa

Published Jul 5, 2023• 11 citations
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Abstract

We give evidence for the web of 3d bosonization dualities in conformal field theories (CFTs) by computing monopole operator scaling dimensions in ($2+1$)-dimensional quantum electrodynamics (QED3) with Chern-Simons level $k$ and $N$ complex bosons in a large-$N$, $k$ expansion. We first consider the $k=0$ case, where we show that scaling dimensions previously computed to subleading order in $1/N$ can be extrapolated to $N=1$ and matched to $O(2)$ Wilson-Fisher CFT scaling dimensions with around 5% error, which is evidence for particle-vortex duality. We then generalize the subleading calculation to large $N$, $k$ and fixed $k/N$, extrapolate to $N=k=1$, and consider monopole operators that are conjectured to be dual to nondegenerate scalar operators in a theory of a single Dirac fermion. We find matches typically with 1% error or less, which is strong evidence of this so-called ``seed'' duality that implies a web of 3d bosonization dualities among CFTs.

DOI
10.1103/physrevd.108.l021701

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We give evidence for the web of 3d bosonization dualities in conformal field theories (CFTs) by computing monopole operator scaling dimensions in (2+12+12+1)-dimensional quantum electrodynamics (QED3) with Chern-Simons level kkk and NNN complex bosons in a large-NNN, kkk expansion. We first consider the k=0k=0k=0 case, where we show that scaling dimensions previously computed to subleading order in 1/N1/N1/N can be extrapolated to N=1N=1N=1 and matched to O(2)O(2)O(2) Wilson-Fisher CFT scaling dimensions with around 5% error, which is evidence for particle-vortex duality. We then generalize the subleading calculation to large NNN, kkk and fixed k/Nk/Nk/N, extrapolate to N=k=1N=k=1N=k=1, and consider monopole operators that are conjectured to be dual to nondegenerate scalar operators in a theory of a single Dirac fermion. We find matches typically with 1% error or less, which is strong evidence of this so-called ``seed'' duality that implies a web of 3d bosonization dualities among CFTs.
We grow the web of dualities for conformal field theories (CFTs) in 2+1 spacetime dimensions to include quantum critical transitions of Dirac fermions. Our construction uses the seed duality, equating a free Dirac fermion with a complex boson coupled to a Chern-Simons gauge field, to express various Gross-Neveu-Yukawa (GNY) critical points of N fermions in terms of the same number of complex bosons and partner gauge fields. In some instances, monopole operators appear explicitly in the dual action, such as for the superconducting transition of a single Dirac fermion, which is a CFT with emergent supersymmetry. We match phase diagrams, symmetries and their anomalies. By further gauging certain symmetries, we derive offspring dualities. For example, starting with the duality for the regular GNY transition of 2 Dirac fermions, we arrive at the duality between a Neel-to-VBS deconfined quantum critical point and a gauged-Gross-Neveu transition. Finally, we use a large-N expansion for monopole operators, and compare with known scaling dimensions in the simplest single-flavour GNY transition.
Monopole operators are studied in a large family of quantum critical points between Dirac spin liquids and topological quantum spin liquids (QSLs): chiral and Z 2 QSLs.These quantum phase transitions are described by conformal field theories (CFTs): quantum electrodynamics in 2 þ 1 dimensions with 2N flavors of two-component massless Dirac fermions and a four-fermion interaction term.For the transition to a chiral spin liquid, it is the Gross-Neveu interaction (QED 3 -GN), while for the transitions to Z 2 QSLs, it is a superconducting pairing term with general spin and valley structure (generalized QED 3 -Z 2 GN).Using the state-operator correspondence, we obtain monopole scaling dimensions to subleading order in 1=N.For monopoles with a minimal topological charge q ¼ 1=2, the scaling dimension is 2N × 0.26510 at leading order, with the quantum correction being 0.118911(7) for the chiral spin liquid, and 0.102846(9) for the simplest Z 2 case (the expression is also given for a general pairing term).Although these two anomalous dimensions are nearly equal, the underlying quantum fluctuations possess distinct origins.The analogous result in QED 3 is also obtained, and we find a subleading contribution of -0.038138ð5Þ, which is slightly different from the value -0.0383 first obtained in the literature.The scaling dimension of a QED 3 -GN monopole with minimal charge is very close to the scaling dimensions of other operators predicted to be equal by a conjectured duality between QED 3 -GN with 2N ¼ 2 flavors and the CP 1 model.Additionally, nonminimally charged monopoles with equal charges on both sides of the duality have similar scaling dimensions.By studying the large-q asymptotics of the scaling dimensions in QED 3 , QED 3 -GN, and QED 3 -Z 2 GN, we verify that the constant Oðq 0 Þ coefficient precisely matches the universal nonperturbative prediction for CFTs with a global U(1) symmetry.Finally, we identify numerous open questions regarding the fate of monopoles and their hierarchies at transitions to spin liquids and ordered phases.
We study the properties of 2+1d conformal field theories (CFTs) in a background magnetic field. Using generalized particle-vortex duality, we argue that in many cases of interest the theory becomes gapped, which allows us to make a number of predictions for the magnetic response, background monopole operators, and more. Explicit calculations at large N for Wilson-Fisher and Gross-Neveu CFTs support our claim, and yield the spectrum of background (defect) monopole operators. Finally, we point out that other possibilities exist: certain CFTs can become metallic in a magnetic field. Such a scenario occurs, for example, with a Dirac fermion coupled to a Chern-Simons gauge field, where a non-Fermi liquid is argued to emerge. Published by the American Physical Society 2024
We construct a dual-bosonized description of a massless Majorana fermion in (2+1)ddd. In contrast to Dirac fermions, for which a bosonized description can be constructed using a flux attachment procedure, neutral Majorana fermions call for a different approach.We argue that the dual theory is an SO(N)1\mathrm{SO}{(N)}_{1}SO(N)1​ Chern-Simons gauge theory with a critical SO(N)\mathrm{SO}(N)SO(N) vector bosonic matter field ($N\ensuremath{\ge}3$). The monopole of the SO(N)\mathrm{SO}(N)SO(N) gauge field is identified with the Majorana fermion. We provide evidence for the duality by establishing the correspondence of adjacent gapped phases and by a parton construction. We also propose a generalization of the duality to Nf{N}_{f}Nf​ flavors of Majorana fermions, and discuss possible resolutions of a caveat associated with an emergent global Z2{Z}_{2}Z2​ symmetry. Finally, we conjecture a dual description of an N=1\mathcal{N}=1N=1 supersymmetric fixed point in (2+1)ddd, which is realized by tuning a single flavor of Majorana fermions to an interacting (Gross-Neveu) critical point.
We present a practical Monte Carlo determination of the scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{Q}$ of flux QQQ Abelian monopoles through finite-size scaling analysis of the free energy to introduce the background field of classical Dirac monopole-antimonopole pair at critical points of three-dimensional lattice theories. We validate the method in free fermion theory, and by verifying the particle-vortex duality between the monopole scaling dimension at the inverse-XY fixed point and the charge scaling dimension at the XY fixed point. At the O(2)O(2)O(2) Wilson-Fisher fixed point of the XY model, we determine the critical exponents ${\mathrm{\ensuremath{\Delta}}}_{1}=0.13(2)$, ${\mathrm{\ensuremath{\Delta}}}_{2}=0.29(1)$ and ${\mathrm{\ensuremath{\Delta}}}_{3}=0.47(2)$, which we find to be proportional to the finite-size critical spectrum of monopoles on square torus.
We study the quantum phase transition from a Dirac spin liquid to an antiferromagnet driven by condensing monopoles with spin quantum numbers. We describe the transition in field theory by tuning a fermion interaction to condense a spin-Hall mass, which in turn allows the appropriate monopole operators to proliferate and confine the fermions. We compute various critical exponents at the quantum critical point (QCP), including the scaling dimensions of monopole operators by using the state-operator correspondence of conformal field theory. We find that the degeneracy of monopoles in QED3{\mathrm{QED}}_{3}QED3​ is lifted and yields a nontrivial monopole hierarchy at the QCP. In particular, the lowest monopole dimension is found to be smaller than that of QED3{\mathrm{QED}}_{3}QED3​ using a large-Nf{N}_{f}Nf​ expansion, where 2Nf2{N}_{f}2Nf​ is the number of fermion flavors. For the minimal magnetic charge, this dimension is 0.39Nf0.39{N}_{f}0.39Nf​ at leading order. We also study the QCP between Dirac and chiral spin liquids, which allows us to test a conjectured duality to a bosonic CP1{\mathrm{CP}}^{1}CP1 theory. Finally, we discuss the implications of our results for quantum magnets on the kagome lattice.