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arXiv:1808.08970 · Physical review. D/Physical review. D.

Monopole scaling dimension using Monte-Carlo simulation

Nikhil Karthik

Published Oct 31, 2018• 10 citations
arXivHTMLPDF

Abstract

We present a practical Monte Carlo determination of the scaling dimensions ${\mathrm{\ensuremath{\Delta}}}_{Q}$ of flux $Q$ 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 $O(2)$ 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.

DOI
10.1103/physrevd.98.074513

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We present a practical Monte Carlo determination of the scaling dimensions \ensuremathΔQ{\mathrm{\ensuremath{\Delta}}}_{Q}\ensuremathΔQ​ 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 \ensuremathΔ1=0.13(2){\mathrm{\ensuremath{\Delta}}}_{1}=0.13(2)\ensuremathΔ1​=0.13(2), \ensuremathΔ2=0.29(1){\mathrm{\ensuremath{\Delta}}}_{2}=0.29(1)\ensuremathΔ2​=0.29(1) and \ensuremathΔ3=0.47(2){\mathrm{\ensuremath{\Delta}}}_{3}=0.47(2)\ensuremathΔ3​=0.47(2), which we find to be proportional to the finite-size critical spectrum of monopoles on square torus.
We present a finite size scaling technique to calculate the central charge and some critical exponents of two-dimensional critical models with a Monte Carlo simulation. We use systems with dimensions $L\ifmmode\times\else\texttimes\fi{}M,$ and focus on the scaling behavior in M/L.M/L.M/L. The finite size scaling relation that we use is the universal expression for the stress tensor on the torus. The stress tensor is the operator that governs the anisotropy of the system, and stems for the theory of conformal invariance. We show that a lattice representation of the stress tensor can easily be constructed, such that its expectation value on the torus can be calculated in a Monte Carlo simulation. In doing so, we observe that the stress tensor turns out to be remarkably insensitive to critical slowing down. We show that the typical simulation time scales with the linear system dimension LLL roughly as L4,{L}^{4},L4, and that this scaling holds for the ordinary Metropolis algorithm as well as for more sophisticated cluster algorithms, such that it is fruitless to invoke the latter. We test the method for the Ising model (with central charge c=c=c=12\frac{1}{2}21​), the Ashkin-Teller model (c=1),(c=1),(c=1), and the F model (also c=1c=1c=1).
We numerically investigate the critical behavior of the Hubbard model on the honeycomb and the $\ensuremath{\pi}$-flux lattice, which exhibits a direct transition from a Dirac semimetal to an antiferromagnetically ordered Mott insulator. We use projective auxiliary-field quantum Monte Carlo simulations and a careful finite-size scaling analysis that exploits approximately improved renormalization-group-invariant observables. This approach, which is successfully verified for the three-dimensional XY transition of the Kane-Mele-Hubbard model, allows us to extract estimates for the critical couplings and the critical exponents. The results confirm that the critical behavior for the semimetal to Mott insulator transition in the Hubbard model belongs to the Gross-Neveu-Heisenberg universality class on both lattices.
The classical cubic dimer model has a columnar ordering transition that is continuous and described by a critical Anderson-Higgs theory containing an SU(2)\mathrm{SU}(2)SU(2)-symmetric complex field minimally coupled to a noncompact U(1)\mathrm{U}(1)U(1) gauge theory. Defects in the dimer constraints correspond to monopoles of the gauge theory, with charge determined by the deviation from unity of the dimer occupancy. By introducing such defects into Monte Carlo simulations of the dimer model at its critical point, we determine the scaling dimensions ${y}_{2}=1.48\ifmmode\pm\else\textpm\fi{}0.07$ and ${y}_{3}=0.20\ifmmode\pm\else\textpm\fi{}0.03$ for the operators corresponding to defects of charge q=2q=2q=2 and 3, respectively. These results, which constitute the first direct determination of the scaling dimensions, shed light on the deconfined critical point of spin-12\frac{1}{2}21​ quantum antiferromagnets, thought to belong to the same universality class. In particular, the positive value of y3{y}_{3}y3​ implies that the transition in the JQJQJQ model on the honeycomb lattice is of first order.
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 study the cubic fixed point for N=3N=3N=3 and 4 by using finite-size scaling applied to data obtained from Monte Carlo simulations of the NNN-component ${\ensuremath{\phi}}^{4}$ model on the simple cubic lattice. We generalize the idea of improved models to a two-parameter family of models. The two-parameter space is scanned for the point, where the amplitudes of the two leading corrections to scaling vanish. To this end, a dimensionless quantity is introduced that monitors the breaking of the O(N)O(N)O(N) invariance. For N=4N=4N=4, we determine the correction exponents ${\ensuremath{\omega}}_{1}=0.763(24)$ and ${\ensuremath{\omega}}_{2}=0.082(5)$. In the case of N=3N=3N=3, we obtain Y4=0.0142(6){Y}_{4}=0.0142(6)Y4​=0.0142(6) for the renormalization group exponent of the cubic perturbation at the O(3)O(3)O(3)-invariant fixed point, while the correction exponent ${\ensuremath{\omega}}_{2}=0.0133(8)$ at the cubic fixed point. Simulations close to the improved point result in the estimates $\ensuremath{\nu}=0.7202(7)$ and $\ensuremath{\eta}=0.0371(2)$ of the critical exponents of the cubic fixed point for N=4N=4N=4. For N=3N=3N=3, at the cubic fixed point, the O(3)O(3)O(3) symmetry is only mildly broken and the critical exponents differ only by little from those of the O(3)O(3)O(3)-invariant fixed point. We find $\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}01⪅{\ensuremath{\eta}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\eta}}_{O(3)}⪅0.000\phantom{\rule{0.16em}{0ex}}07$ and ${\ensuremath{\nu}}_{\text{cubic}}\ensuremath{-}{\ensuremath{\nu}}_{O(3)}=\ensuremath{-}0.000\phantom{\rule{0.16em}{0ex}}61(10)$.