Nonperturbative physics in sigma models
Final Report Abstract
The studies we performed within this grant can only be small steps towards a more complete knowledge of strongly coupled systems, on the one hand QCD and on the other hand asymptotically free sigma models. For the former, magnetic fields have served as benchmarks of our understanding of the finite temperature transition. Over the last decade, this topic has witnessed lots of activities, however, with falling tendency lately. We clarified the role of Landau levels in QCD, which are surprisingly effective even at realistic coupling and in four dimensions. Low-energy models and effective theories can be checked, adjusted or improved with our findings. In nature, however, there might not be experimental setups for this theoretical exercise: although magnetic fields of the strength of ΛQCD might (have) exist(ed) in the early universe, strongly magnetised neutron stars and in present heavy-ion collisions, it is not sure at all whether they are constant over the necessary QCD scales in space and time. Moreover, such fields can be measured only very indirectly and in heavyion collisions the assumption of quark matter to be in thermal equilibrium is questionable. For quark matter at nonzero density, our study of scalar QCD dualised at strong coupling has revealed some interesting technical results, but is as far away as any other current approach from giving realistic answers. Two-dimensional sigma models are astonishingly vital to date, mainly as testing grounds for theoretical developments and the corresponding numerical techniques/simulations. The renormalon picture, that we confirmed from first principles in P C(N ) models — for the first time beyond gauge theories — has retrieved attentions recently in the context of resurgence/trans-series. This in turn is related to the thimble method, motivated by which we have written down the first complexified classical solutions for sigma models at nonzero µ. Whether the resurgence framework is more than an academic observation is yet to be proven. One physical goal would be to see how the dynamical mass gap at µ = m (at low temperatures) shows up starting from those complex solutions. The MPS approach is very attractive for this region of the phase space, as it addresses the ground state (i.e., zero temperature) without a sign problem. We have shown that truncations in the angular momentum of the O(3) model rotors are feasible for a numerical approach that captures asymptotic freedom, the dynamical mass generation and the central charge from the entanglement entropy. I am convinced that more can be learned about the nature of the ground state on both sides of the quantum phase transition. Another nice application of the MPS approach could be to solve the long-standing sign problem induced by a theta angle, and especially to varify Haldane’s conjecture at θ = π. While such tensor networks work well for low-dimensions, it is by no means clear whether they can successfully be applied to higher dimensional field theories.
Publications
- “Two-dimensional O(3) model at nonzero density: From dual lattice simulations to repulsive bosons,” Phys. Rev. D94 (2016) no.11, 114503
F. Bruckmann, C. Gattringer, T. Kloiber and T. Sulejmanpasic
(See online at https://doi.org/10.1103/PhysRevD.94.114503) - “Landau levels in QCD,” Phys. Rev. D96 (2017) no.7, 074506
F. Bruckmann, G. Endrödi, M. Giordano, S. D. Katz, T. G. Kovacs, F. Pittler and J. Wellnhofer
(See online at https://doi.org/10.1103/PhysRevD.96.074506) - “A Lattice Study of Renormalons in Asymptotically Free Sigma Models,” PoS LATTICE2018 (2018) 237, 1811
F. Bruckmann and M. Puhr
(See online at https://doi.org/10.48550/arXiv.1811.02836) - “Complex instantons in sigma models with chemical potential,” Phys. Rev. D98 (2018) no.6, 065005
F. Bruckmann and S. Lochner
(See online at https://doi.org/10.1103/PhysRevD.98.065005) - “Diagrammatic representation of scalar QCD and sign problem at nonzero chemical potential,” Phys. Rev. D97 (2018) no.1, 014501
F. Bruckmann and J. Wellnhofer
(See online at https://doi.org/10.1103/PhysRevD.97.014501) - “Landau levels in QCD in an external magnetic field,” EPJ Web Conf. 175 (2018) 07014
F. Bruckmann, G. Endrödi, M. Giordano, S. D. Katz, T. G. Kovacs, F. Pittler and J. Wellnhofer
(See online at https://doi.org/10.1051/epjconf/201817507014) - “Rigorous construction of coherent state path integrals through dualization”
F. Bruckmann and J. D. Urbina
(See online at https://doi.org/10.48550/arXiv.1807.10462) - “O(3) nonlinear sigma model in 1+1 dimensions with matrix product states,” Phys. Rev. D99 (2019) no.7, 074501
F. Bruckmann, K. Jansen and S. Kühn
(See online at https://doi.org/10.1103/PhysRevD.99.074501) - “Universal Renormalons in Principal Chiral Models,” Phys. Rev. D101 (2020) no.3, 034513
F. Bruckmann and M. Puhr
(See online at https://doi.org/10.1103/PhysRevD.101.034513)
