Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

3.4 exploiting universality 33
In this case, K c,∞ was fixed at the known value of 2/ ln(1 + √ 2). The correction to
scaling exponent tends towards 2 from above, ω = 2 + δ with δ → 0 + as we increase
the lower bound N min used in the fit (δ starts at around 2 · 10 −4 for the full range of
N shown in Fig. 3.9, and it falls below 10 −5 at N min around 400). Since the exponent
of the leading irrelevant operator that breaks rotational invariance is predicted to
be exactly 2 [63], our result is consistent with the conjecture of Ref. [64] that the
only irrelevant operators that appear in the 2d nearest neighbor Ising model are
those due to the lattice breaking of rotational symmetry. This may be tested on
a triangular 2d lattice where the leading irrelevant operator to break rotational
invariance leads to ω = 4 instead, while the leading rotationally invariant operator
would give an isotropic correction to scaling with ω = 2 in either case [64].
On the other hand, our correction exponent for SU(2) is clearly at odds with
ω = 2. In this case, it is possible that there exists an irrelevant operator that is not
present in the 2d Ising model or the corresponding conformal theory. It is more
likely, however, that our exponent is really an effective exponent. When there are
several nearby competing exponents it is extremely difficult to extract the smallest
one from simulations. 6
3.4.4 From coupling to temperature
In 2 + 1 dimensions the coupling g3 2 has the dimension of mass and sets the scale.
The bare lattice coupling is then given by [61],
β = 2N c
ag3B
2 , g3B 2 = g3 2 + c 1 ag3 4 + c 2 a 2 g3 6 + . . . (3.43)
Substituting the expansion into β and using T = 1/(N t a) gives the lattice coupling
in terms of temperature, which at criticality reads,
β c (N t )
2N c
≃ T c
g3
2 1
g3
2 N t − c 1 − c 2 + . . . (3.44)
T c N t
The theory is super-renormalizable with a rapid approach to the continuum limit,
which is reflected here in the leading order linearity of the critical coupling in N t .
Our high precision estimates for β c (N t ) allow us to determine the subleading
corrections to linearity.
It is not clear to the naked eye, but the results in Fig. 3.10 are not well fitted by a
straight line. Indeed, fitting all data points with an acceptable reduced χ 2 requires
three subleading coefficients that are not all well constrained. Restricting to N t ≥ 3
yields a more sensible fit
β c (N t ) = 1.4927(26)N t + 0.868(34) − 1.58(14) 1 N t
+ 1.48(19) 1 N 2 t
+ · · · , (3.45)
6 It was noted in [60] that the observed correction to scaling exponent of 2 +1-dimensional SU(2),
ω = 1.64 in their case or ω = 1.48(4) in ours, agreed well with some predictions for the universality
class of the 2d Ising model that included 1.6 [65]. At the time it was discussed whether such nonintegral
correction exponents could arise in other ferromagnetic models in this class, and whether
the corresponding correction amplitudes happened to vanish identically in the Ising model, see [66].
This appears to be ruled out by a conformal field theory analysis: there is no irrelevant operator with
ω < 2 in any unitary model of the 2d Ising class [67, 64].