Critical Exponents for the
Valence-Bond-Solid Transition in Lattice
Quantum Electrodynamics
Rufus Boyack and Joseph Maciejko
Abstract Recent sign-problem-free quantum Monte Carlo simulations of (2+1)dimensional lattice quantum electrodynamics (QED 3 ) with N f flavors of fermions
on the square lattice have found evidence of continuous quantum phase transitions
between a critical phase and a gapped valence-bond-solid (VBS) phase for flavor
numbers N f = 4, 6, and 8. We derive the critical theory for these transitions, the
chiral O(2) QED 3 -Gross–Neveu model, and show that the latter is equivalent to the
gauged Nambu–Jona-Lasinio model. Using known large-N f results for the latter,
we estimate the order parameter anomalous dimension and the correlation length
exponent for the transitions mentioned above. We obtain large-N f results for the
dimensions of fermion bilinear operators, in both the gauged and ungauged chiral
O(2) Gross–Neveu models, which, respectively, describe the long-distance powerlaw decay of two-particle correlation functions at the VBS transition in lattice QED 3
and the Kekulé-VBS transition for correlated fermions on the honeycomb lattice.
Keywords Lattice gauge theory · Valence-bond solid · Quantum phase
transition · Quantum electrodynamics · Gross–Neveu model ·
Nambu–Jona-Lasinio model · Conformal field theory
Quantum phase transitions that involve fractionalized degrees of freedom fall
outside the traditional Landau paradigm and have been the focus of much interest
in recent years. The classic example is deconfined quantum critical points between
conventional phases of quantum antiferromagnets [1, 2], where emergent fractionalized matter fields and gauge fields appear at the critical point but are confined in the
phases themselves. A class of transitions comparatively less studied, but also beyond
the Landau paradigm, are transitions between phases supporting fractionalized
excitations, such as different types of spin liquids, or between a fractionalized
R. Boyack · J. Maciejko ()
Department of Physics and Theoretical Physics Institute, University of Alberta, Edmonton,
AB, Canada
e-mail: boyack@ualberta.ca; maciejko@ualberta.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_32
337
Valence-Bond-Solid Transition in Lattice
Quantum Electrodynamics
Rufus Boyack and Joseph Maciejko
Abstract Recent sign-problem-free quantum Monte Carlo simulations of (2+1)dimensional lattice quantum electrodynamics (QED 3 ) with N f flavors of fermions
on the square lattice have found evidence of continuous quantum phase transitions
between a critical phase and a gapped valence-bond-solid (VBS) phase for flavor
numbers N f = 4, 6, and 8. We derive the critical theory for these transitions, the
chiral O(2) QED 3 -Gross–Neveu model, and show that the latter is equivalent to the
gauged Nambu–Jona-Lasinio model. Using known large-N f results for the latter,
we estimate the order parameter anomalous dimension and the correlation length
exponent for the transitions mentioned above. We obtain large-N f results for the
dimensions of fermion bilinear operators, in both the gauged and ungauged chiral
O(2) Gross–Neveu models, which, respectively, describe the long-distance powerlaw decay of two-particle correlation functions at the VBS transition in lattice QED 3
and the Kekulé-VBS transition for correlated fermions on the honeycomb lattice.
Keywords Lattice gauge theory · Valence-bond solid · Quantum phase
transition · Quantum electrodynamics · Gross–Neveu model ·
Nambu–Jona-Lasinio model · Conformal field theory
Quantum phase transitions that involve fractionalized degrees of freedom fall
outside the traditional Landau paradigm and have been the focus of much interest
in recent years. The classic example is deconfined quantum critical points between
conventional phases of quantum antiferromagnets [1, 2], where emergent fractionalized matter fields and gauge fields appear at the critical point but are confined in the
phases themselves. A class of transitions comparatively less studied, but also beyond
the Landau paradigm, are transitions between phases supporting fractionalized
excitations, such as different types of spin liquids, or between a fractionalized
R. Boyack · J. Maciejko ()
Department of Physics and Theoretical Physics Institute, University of Alberta, Edmonton,
AB, Canada
e-mail: boyack@ualberta.ca; maciejko@ualberta.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_32
337