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Non-Perturbative Quantization of Topological Excitations in Quantum Field Theory and Quantum Spin Systems

English title Non-Perturbative Quantization of Topological Excitations in Quantum Field Theory and Quantum Spin Systems
Applicant Wiese Uwe-Jens
Number 200424
Funding scheme Project funding
Research institution Institut für Theoretische Physik Universität Bern
Institution of higher education University of Berne - BE
Main discipline Theoretical Physics
Start/End 01.04.2021 - 31.01.2024
Approved amount 528'475.00
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Keywords (3)

quantum spin systems; vortices; infraparticles

Lay Summary (German)

Lead
Im Rahmen des Projekts "Nonperturbative Quantization of Topological Excitations in Quantum Field Theory and Quantum Spin Systems" werden physikalische Systeme mit vielen stark gekoppelten Freiheitsgraden untersucht. Solche Systeme spielen sowohl in der Teilchenphysik als auch in der Physik der kondensierten Materie eine zentrale Rolle. Insbesondere sollen kollektive topologische Anregungen untersucht werden, die ein komplexes Zusammenspiel der stark gekoppelten Freiheitsgrade widerspiegeln.
Lay summary

Das Projekt gliedert sich in drei Teilprojekte, die im Folgenden kurz beschrieben werden.

Vortizes treten zum Beispiel als topologische Anregungen in Supraflüssigkeiten und in ultrakalten Quantengasen auf. Sie manifestieren sich dort als Wirbel im Flüssigkeitsstrom. Ein solcher Wirbel kann nicht durch eine kontinuierliche Deformation zum Verschwinden gebracht werden, da er (ähnlich wie ein Knoten) in sich verwunden ist. Topologische Anregungen werden meist im Rahmen der klassischen Physik diskutiert und dann nur störungstheoretisch approximativ quantisiert. In diesem Projekt sollen Vortizes jenseits der Störungstheorie quantisiert werden. Zu diesem Zweck wird die zugrunde liegende Quantenfeldtheorie auf einem raum-zeitlichen Gitter formuliert und dann mit numerischen Monte Carlo Simulationsmethoden sehr genau studiert.

Quantenspinsysteme sind einerseits stark korrelierte Systeme der Physik der kondensierten Materie. Andererseits führt ihr Verhalten bei niedrigen Energien auf effektive Quantenfeldtheorien, die auch in der Teilchenphysik eine zentrale Rolle spielen. Solche Systeme haben den grossen Vorteil, dass sie auch in sogenannten Quantensimulatoren mit ultrakalten Atomen realisiert werden können. Diese sind zwar auf spezielle Anwendungen beschränkt, bilden aber eine interessante Vorstufe zum bisher noch hypothetischen universellen Quantencomputer.

Es ist bisher noch nicht gelungen, Vortizes in Quantenspinsystemen mathematisch zufriedenstellend zu konstruieren. Mit Hilfe einer sogenannten Dualisierung soll dies in diesem Projekt ermöglicht werden. Sobald eine zufriedenstellende Konstruktion gelungen ist, soll diese wiederum mit Monte Carlo Methoden im Detail studiert werden. Die Ergebnisse können dann bei der korrekten Interpretation von Quantensimulator Experimenten sehr hilfreich sein.

Sogenannte Eichfeldtheorien spielen eine zentrale Rolle im Standardmodell der Teilchenphysik. In Eichtheorien sind geladene Teilchen von einer Wolke von Eichteilchen umgeben. So sind geladene Elektronen von einer Wolke masseloser Photonen, den Eichteilchen des Elektromagnetismus, umgeben. Elektrisch geladene Teilchen werden auch als Abelsche Infrateilchen bezeichnet. In der Teilchenphysik stellt das Verständnis der starken Wechselwirkung zwischen Quarks und Gluonen eine grosse Herausforderung dar. Dort sind die Gluonen sogenannte nicht-Abelsche Eichteilchen. Einzelne Quarks kommen in der Natur nicht vor, da sie als nicht-Abelsche Infrateilchen unendlich viel Energie erfordern würden. Hier soll untersucht werden, ob sich solche Teilchen mit endlichem Energieaufwand in einem endlichen Volumen aufhalten können. Diese Untersuchung soll unser Verständnis der starken Wechselwirkung weiter vertiefen.

Direct link to Lay Summary Last update: 31.03.2021

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Associated projects

Number Title Start Funding scheme
172616 Nonperturbative Problems in Particle, Condensed Matter, and Quantum Information Physics 01.04.2017 Project funding

Abstract

Topological excitations play an important role in several field theories, with applications in both particle and condensed matter physics. This includes magnetic monopoles, cosmic strings, or Skyrmions in particle physics, as well as vortices and baby-Skyrmions in condensed matter physics. A priori, topological excitations are objects of classical field theory in which the fields are continuous and may give rise to the concept of topological winding numbers. In the context of quantum field theory, in which fields are typically neither differentiable nor continuous, it is a priori not clear whether topological excitations persist in the presence of violent quantum fluctuations. In the past, topological excitations in quantum field theories have been investigated in great detail using semi-classical quantization methods. Then a topologically non-trivial classical solution is perturbed by small quantum fluctuations, which have calculable effects on its physical properties. However, this approach is limited to perturbation theory and may thus not yield reliable results in strongly coupled quantum field theories.Around 1985, Froehlich and Marchetti have achieved a breakthrough in the rigorous non-perturbative construction of topological sectors in strongly coupled quantum field theories. This includes magnetic monopoles in compact U(1) lattice gauge theory as well as vortices in the Abelian Higgs model, anyons in Chern-Simons gauge theories, and solitons in other quantum field theories. Based on these insights, a long time ago I have applied these constructions to numerical simulations of magnetic monopoles and have extended them to cosmic strings.Magnetic monopoles in the Coulomb phase of compact U(1) lattice gauge theory are related to the charged particles in scalar quantum electrodynamics by an exact duality transformation. Charged particles are surrounded by a soft cloud of massless photons. This implies that they are so-called infraparticles, which do not simultaneously have a definite charge and definite mass. These results, which are based on rigorous considerations in the framework of algebraic quantum field theory, have important implications for lattice calculations of charged particles. In particular, the commonly used periodic boundary conditions are inappropriate in this context, because the Gauss law implies that a torus is always neutral. In lattice studies of magnetic monopoles I have introduced so-called C-periodic boundary conditions, which endow the torus with a charge conjugation twist, and thus make charged particles accessible to numerical lattice calculations in a finite volume. With periodic boundary conditions, on the other hand, due to Gauss' law a single charged particle is absent from the spectrum of the theory in a finite volume.The vortices in a Bose-Einstein condensate or in a superfluid are condensed matter analogs of the monopoles in particle physics. In particular, the (2+1)-dimensional O(2)-symmetric quantum field theory is again dual to scalar QED. The superfluid Goldstone boson then manifests itself as a dual massless photon, and the vortices are dual to charged scalar infraparticles. In this case, the soft cloud surrounding the vortex infraparticle consists of massless Goldstone bosons. Since in (2+1) dimensions the Coulomb potential is logarithmically divergent, the mass of the vortex is infrared divergent. In the context of condensed matter physics, this has given rise to various attempts to develop other concepts of a vortex mass. However, endowed with C-periodic boundary conditions the vortex mass can be defined rigorously. Here it is proposed to calculate it precisely in a finite volume. This provides a unique opportunity to derive new universal quantities at the O(2) Wilson-Fisher fixed point.Defining the vortex mass is a subtle issue. Since it diverges logarithmically in the infinite volume limit, in condensed matter physics the concepts of Kopnin mass and Baym-Chandler mass have been introduced, which are both associated with the vortex core. However, these do not properly measure the inertia of a moving vortex. C-periodic boundary conditions provide an infrared regularization of the vortex mass and allow us to rigorously define the inertia of the vortex. Since it is an infraparticle, it spontaneously breaks Lorentz invariance. Consequently, the rest mass and the inertial mass of the vortex do not necessarily coincide. Here it is proposed to determine these two masses first by semi-classical quantization and then fully non-perturbatively.In the massive phase of the (2+1)-d O(2) model, which is dual to the Higgs phase of scalar QED, the vortices condense, which endows the photon with a mass. In 1991, my collaborators and I have determined the condensate of monopoles in 4-d compact U(1) lattice gauge theory, which has a first order phase transition that is not associated with universal behavior. Here it is proposed to determine the vortex condensate as yet another universal quantity when one approaches the O(2) Wilson-Fisher fixed point from the symmetric phase. Finally, the vortex mass and the vortex condensate shall also be investigated in the conformal field theory directly at the fixed point.At low temperatures, the (2+1)-d quantum XY model has a spontaneously broken O(2) symmetry with massless Goldstone bosons. The corresponding low-energy effective theory is the (2+1)-d O(2)-symmetric non-linear sigma-model. This naturally raises the subtle question whether the (2+1)-d quantum XY model also supports vortex excitations. Due to the strong quantum fluctuations of spins 1/2, there is no obvious notion of continuity and hence of topology in the corresponding field theory. Work in the spirit of Froehlich and Marchetti is required in order to decide whether vortex superselection sectors can be constructed rigorously in the quantum XY model. For this purpose, it is proposed to perform an exact dualization of the quantum XY model in a C-periodic volume for arbitrary spin. At least in the limit of large spin, one may expect that vortex superselection sectors exist.The issue of non-Abelian infraparticles is an unresolved problem in theoretical physics. Quarks are confined and thus isolated quark states, which would qualify as non-Abelian infraparticles, are removed to infinite energy. Although the same is true for the vortex in the (2+1)-d O(2)-symmetric quantum field theory, it can very well be investigated in detail in a finite C-periodic volume. Here it is proposed to take a similar approach to non-Abelian charged states. While they again cannot exist on a torus, a priori there is no reason why they should not fit into a C-periodic box. In this context it is vital to construct a state with finite volume quantum numbers that can not be realized with ordinary neutral states in a non-Abelian gauge theory. This is most promising for SU(N) gauge theories with even N, including the SU(2) gauge-Higgs model or SU(2) QCD. In the latter, all confined states (including the baryons) are bosons. Just as the mass of the vortex in the (2+1)-d O(2)-symmetric quantum field theory, due to confinement the mass of a non-Abelian infraparticle diverges in the infinite volume limit.My research is characterized by an interdisciplinary approach that aims at connecting apparently disconnected areas of theoretical physics. In this way, we have gained new insights that would otherwise seem difficult to obtain. Again, this is the key to the research proposed here. In particular, insights into the subtle dynamics of infraparticles obtained in algebraic quantum field theory shall be used to shed light on vortices in condensed matter systems. C-periodic boundary conditions, which were originally developed for the study of magnetic monopoles, provide an infrared regularization that is very useful in the investigations of other charged excitations as well, both in particle and in condensed matter physics.
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