Tomáš Brauner (University of Stavanger): "Effective field theory for spontaneously broken symmetry"
This lecture series will give an introduction to the physics of spontaneous symmetry breaking (SSB) with focus on the development and use of effective field theory (EFT). First, I will review the basic features of SSB including the classification of Nambu–Goldstone (NG) bosons, starting from a simple toy model and gradually progressing to a general qualitative discussion of SSB and its consequences in both relativistic and nonrelativistic systems. The next two lectures will focus on the details of the machinery of EFT for NG bosons: the technique of nonlinear realizations of internal symmetry, and its application to the construction of effective Lagrangians for NG bosons. We will define the action of a symmetry group on an arbitrary set of fields and show how it can be brought to a standard form by a suitable choice of field variables. This will equip us with a set of basic building blocks for the construction of EFT for NG bosons. We will see examples of EFTs from both high-energy (particle) physics and low-energy (condensed-matter) physics. The last lecture will be devoted to a specific application: scattering of NG bosons. This will serve to highlight that the constraints imposed by the broken symmetry go well beyond the presence of gapless modes in the spectrum. Time permitting, the lecture will include a brief primer on some of the more advanced features of the modern scattering amplitude program.
Reference T. Brauner,
Effective Field Theory for Spontaneously Broken Symmetry, Lecture Notes in Physics vol. 1023, Springer (2024)
Invited Talks
Tomáš Brauner (University of Stavanger): "Topological dipole symmetries"
Dipole symmetry underpins the restricted mobility of collective degrees of freedom in multiple physical systems including ferromagnets, superfluids, and fractons. In this talk, I will argue that a local dipole conservation law is generally present in bosonic field theory and is intimately connected to conservation of momentum via translation invariance. The integral charge may be nonzero as a consequence of nontrivial topology, either due to the global structure of the classical phase space of the theory, or due to the presence of defects satisfying a nontrivial boundary condition. In both cases, the Lie algebra of spatial translations acquires a central charge. Intriguingly, this may imply the impossibility to construct a local momentum density that would be well-defined on the entire phase space of the system. This issue, also called the “linear momentum problem,” has been known since the 1980s to affect for instance ferromagnets and the superfluid A-phase of helium-3. I will show that its presence leads to a powerful constraint on the low-energy spectrum, predicting the existence of additional light degrees of freedom.
Reference T. Brauner, N. Yamamoto and R. Yokokura,
"Dipole symmetries from the topology of the phase space and the constraints on the low-energy spectrum", SciPost Phys. 16 (2024) 051
Keisuke Fujii (Institute of Science Tokyo): "Induced interaction between impurities from superfluid EFT"
We discuss the long-range behavior of the induced Casimir interaction between two spinless heavy impurities, or polarons, in superfluid cold atomic gases. Using the effective field theory of a Galilean-invariant superfluid, we demonstrate that the induced impurity-impurity potential at long distances universally exhibits a power-law behavior. At zero temperature, the induced potential exhibits an attractive interaction analogous to the relativistic van der Waals force. At finite temperatures, the induced potential becomes complex-valued, with its imaginary part representing decoherence effects, as the superfluid medium acts as a thermal bath of phonons. We show that this imaginary part also exhibits a universal power-law behavior.
Yoshimasa Hidaka (YITP, Kyoto University): "Phase Transition of Vortices in Higgs-Confinement Continuity"
At finite densities of three-flavor QCD, a hadron (confinement) superfluid phase is expected to be realized at low densities, and a color superconducting (Higgs) phase at high densities. It is not well understood whether these two phases are connected with or without a phase transition. In this talk, we consider the Higgs-confinement transition with superfluidity in a U(1)$\times$U(1) lattice model as a simple model. We found that a phase transition occurs on a superfluid vortex, although the bulk system does not exhibit a phase transition. We confirm this phase transition through analytical calculations using weak/strong coupling expansion and Monte Carlo simulations. We also discuss possible scenarios for QCD.
Kentaro Nishimura (Hiroshima University): "Phase diagram of QCD matter with magnetic field: domain-wall Skyrmion chain in chiral soliton lattice"
QCD matter in a strong magnetic field exhibits a rich phase structure. In the presence of an external magnetic field, the chiral Lagrangian for two flavors is accompanied by the Wess-Zumino-Witten (WZW) term containing an anomalous coupling of the neutral pion to the magnetic field via the chiral anomaly. Due to this term, the ground state is inhomogeneous in the form of either chiral soliton lattice (CSL), an array of solitons in the direction of the magnetic field, or domain-wall Skyrmion (DWSk) phase in which Skyrmions supported by $\pi_3(SU(2))\sim \mathbb{Z}_2$ appear inside the solitons as topological lumps supported by $\pi_2(S^2)\sim \mathbb{Z}_2$ in the effective worldvolume theory of the soliton. In this paper, we determine the phase boundary between the CSL and DWSk phases beyond the single-soliton approximation, within the leading order of chiral perturbation theory.
Naoki Yamamoto (Keio University): "Quantum Hall liquids in high-density QCD"
In this talk, we discuss quantum Hall liquids for two-flavor color superconductivity (2SC) in QCD at large baryon density. Due to the coupling of the flavor-singlet meson (eta) to confined SU(2) gluons in the 2SC phase, the effective theory on the eta domain wall is described by the SU(2)_{−1} Chern-Simons theory, which is dual to the U(1)_2 Chern-Simons theory. This theory has a spin-1 droplet excitation that does not carry baryon number, which is identified as a vector meson. We also discuss the effective theories and baryonic droplet excitations on the eta domain walls in the superfluid phases of QCD at large isospin density and two-color QCD at large baryon density.
Ryo Yokokura (Keio University): "Selection rules of topological solitons from non-invertible symmetries in axion electrodynamics"
We investigate a relation between non-invertible symmetries and selection rules of topological solitons such as axionic domain walls and magnetic strings in the $(3+1)$-dimensional axion electrodynamics with a massive axion or a massive photon. In the low-energy limit of the phases where either the axion or the photon is massive, we identify non-invertible 0- or 1-form symmetry generators as axionic domain walls or magnetic strings, respectively. By non-invertible transformations on magnetic monopoles or axionic strings, we give constraints on possible configurations of topological solitons in the presence of the monopoles or axionic strings. Our results are consistent with a solution to the axionic domain wall problem by the magnetic monopole. Further, we give a new constraint on a linked configuration of the magnetic and axionic strings.
