Comparison of algorithms for solving the sign problem in the O(3) model in 1+1 dimensions at finite chemical potential

Abstract

We study three possible ways to circumvent the sign problem in the O(3) nonlinear sigma model in 1+1 dimensions. We compare the results of the worm algorithm to complex Langevin and multiparameter reweighting. Using the worm algorithm, the thermodynamics of the model is investigated, and continuum results are shown for the pressure at different µ/T values in the range 0 − 4. By performing T = 0 simulations using the worm algorithm, the Silver Blaze phenomenon is reproduced. Regarding the complex Langevin, we test various implementations of discretizing the complex Langevin equation. We found that the exponentialized Euler discretization of the Langevin equation gives wrong results for the action and the density at low T /m. By performing a continuum extrapolation, we found that this discrepancy does not disappear and depends slightly on temperature. The discretization with spherical coordinates performs similarly at low µ/T but breaks down also at some higher temperatures at high µ/T . However, a third discretization that uses a constraining force to achieve the φ 2 = 1 condition gives correct results for the action but wrong results for the density at low µ/T .

Finite density 2D O(3) sigma model: Dualization and numerical simulations

“…3. This coincidence is not so manifest at L = 20 because, contrary to the energy density, n has strong size effects [21]. We have verified this assertion by repeating the study for L = 30 and L = 40 at β = 1.2.…”

“…The dual variables appear as flux variables subject to certain constraints, and the dual weight can be proven to be positive for O(N) models [19,20]. The resulting dual theory can be simulated by a worm algorithm, and a number of thermodynamic quantities have been computed along this route [19][20][21].…”

“…Indeed, the results of Ref. [21] show that alternative approaches to the sign problem -the reweighting and the complex Langevin-have certain drawbacks and lead to incorrect results in some regions of the β-µ plane.…”

Spontaneous symmetry breaking in 2D supersphere sigma models and applications to intersecting loop soups

“…(2) q Let us first discuss briefly the Bethe equations of sl(4|2) (2) q . These are, in the second grading [22] with q = e iγ…”

“…with ∂ µ X∂ µ X ≡ −(∂ x X) 2 + (∂ t X) 2 , and κ a normalization factor to make matching with existing literature easier.…”