Monopole Operators and Their Symmetries in QED 3 -Gross–Neveu Models
335
treatment. Therefore, the largest scaling dimension (characterizing monopoles with
either S = 0 or S = 1/2) is given, at leading order in 1/N , by the second term
in (12). The gap equation for this expression is solved for M q = 0, which yields
the same scaling dimension as in QED 3 . Monopole operators in QED 3 -cHGN then
have scaling dimensions Δ M q which vary between the lowest monopole scaling
dimension in QED 3 -cHGN and QED 3 ,
Δ
QED 3 -cHGN
q
≤ Δ
QED 3 -cHGN
M q
≤ Δ
QED 3
q
, leading order in 1/N .
(20)
This upper boundary was overestimated in [18] as the possibility of different masses
M q defining each monopole multiplet was not considered.
Conclusion The DSL is a parent state of many quantum phases and exotic nonLandau transitions in frustrated quantum magnets. We characterized monopole
operators in QED 3 -GN models describing various QCPs between this DSL and
other ordered phases. Notably, we obtained the lowest scaling dimension of
monopole operators in the QED 3 -cHGN model. At leading order in 1/N , we find
it is lower than its counterpart in QED 3 . Specifically, for a minimal magnetic
charge, it is given by Δ
QED 3 -cHGN
q=1/2
= 2N × 0.195 + O(1/N 0 ). We discussed how
monopoles are reorganized as irreps of the QED 3 -cHGN QCP reduced symmetry
group SU(2) × SU(N). Monopoles multiplets with the highest SU(2) spin have the
lowest scaling dimension. A more detailed exploration of this monopole hierarchy
at the QCP is reserved for future work.
Acknowledgments We thank Tarun Grover, Joseph Maciejko, David Poland, Zi Yang Meng
and Chong Wang for the interesting discussions and useful observations on our work during the
symposium. We also thank Jaume Gomis and Sergueï Tchoumakov for useful discussions. É.D.
was funded by an Alexander Graham Bell CGS from NSERC. M.B.P. was funded by a Discovery
Grant from NSERC. W.W.-K. was funded by a Discovery Grant from NSERC, a Canada Research
Chair, a grant from the Fondation Courtois, and a “Établissement de nouveaux chercheurs et de
nouvelles chercheuses universitaires” grant from FRQNT.
References
1. A.M. Polyakov, Compact gauge fields and the infrared catastrophe. Phys. Lett. B 59, 82–84
(1975)
2. A.M. Polyakov, Quark confinement and topology of gauge theories. Nuclear Phys. B 120, 429–
458 (1977)
3. V. Borokhov, A. Kapustin, X. Wu, Topological disorder operators in three-dimensional
conformal field theory. J. High Energy Phys. 2002, 049 (2002)
4. M.A. Metlitski, M. Hermele, T. Senthil, M.P.A. Fisher, Monopoles in CP N −1 model via the
state-operator correspondence. Phys. Rev. B 78, 214418 (2008)
5. Y. Ran, M. Hermele, P.A. Lee, X.-G. Wen, Projected-wave-function study of the spin-1/2
heisenberg model on the kagome lattice. Phys. Rev. Lett. 98, 117205 (2007)
6. P. Sindzingre, C. Lhuillier, Low-energy excitations of the kagome antiferromagnet and the
spin-gap issue. Europhys. Lett. 88, 27009 (2009)
335
treatment. Therefore, the largest scaling dimension (characterizing monopoles with
either S = 0 or S = 1/2) is given, at leading order in 1/N , by the second term
in (12). The gap equation for this expression is solved for M q = 0, which yields
the same scaling dimension as in QED 3 . Monopole operators in QED 3 -cHGN then
have scaling dimensions Δ M q which vary between the lowest monopole scaling
dimension in QED 3 -cHGN and QED 3 ,
Δ
QED 3 -cHGN
q
≤ Δ
QED 3 -cHGN
M q
≤ Δ
QED 3
q
, leading order in 1/N .
(20)
This upper boundary was overestimated in [18] as the possibility of different masses
M q defining each monopole multiplet was not considered.
Conclusion The DSL is a parent state of many quantum phases and exotic nonLandau transitions in frustrated quantum magnets. We characterized monopole
operators in QED 3 -GN models describing various QCPs between this DSL and
other ordered phases. Notably, we obtained the lowest scaling dimension of
monopole operators in the QED 3 -cHGN model. At leading order in 1/N , we find
it is lower than its counterpart in QED 3 . Specifically, for a minimal magnetic
charge, it is given by Δ
QED 3 -cHGN
q=1/2
= 2N × 0.195 + O(1/N 0 ). We discussed how
monopoles are reorganized as irreps of the QED 3 -cHGN QCP reduced symmetry
group SU(2) × SU(N). Monopoles multiplets with the highest SU(2) spin have the
lowest scaling dimension. A more detailed exploration of this monopole hierarchy
at the QCP is reserved for future work.
Acknowledgments We thank Tarun Grover, Joseph Maciejko, David Poland, Zi Yang Meng
and Chong Wang for the interesting discussions and useful observations on our work during the
symposium. We also thank Jaume Gomis and Sergueï Tchoumakov for useful discussions. É.D.
was funded by an Alexander Graham Bell CGS from NSERC. M.B.P. was funded by a Discovery
Grant from NSERC. W.W.-K. was funded by a Discovery Grant from NSERC, a Canada Research
Chair, a grant from the Fondation Courtois, and a “Établissement de nouveaux chercheurs et de
nouvelles chercheuses universitaires” grant from FRQNT.
References
1. A.M. Polyakov, Compact gauge fields and the infrared catastrophe. Phys. Lett. B 59, 82–84
(1975)
2. A.M. Polyakov, Quark confinement and topology of gauge theories. Nuclear Phys. B 120, 429–
458 (1977)
3. V. Borokhov, A. Kapustin, X. Wu, Topological disorder operators in three-dimensional
conformal field theory. J. High Energy Phys. 2002, 049 (2002)
4. M.A. Metlitski, M. Hermele, T. Senthil, M.P.A. Fisher, Monopoles in CP N −1 model via the
state-operator correspondence. Phys. Rev. B 78, 214418 (2008)
5. Y. Ran, M. Hermele, P.A. Lee, X.-G. Wen, Projected-wave-function study of the spin-1/2
heisenberg model on the kagome lattice. Phys. Rev. Lett. 98, 117205 (2007)
6. P. Sindzingre, C. Lhuillier, Low-energy excitations of the kagome antiferromagnet and the
spin-gap issue. Europhys. Lett. 88, 27009 (2009)