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): "Generalized chiral instabilities, linking numbers, and non-invertible symmetries"
We demonstrate a universal mechanism of a class of instabilities in infrared regions for massless Abelian p-form gauge theories with topological interactions, which we call generalized chiral instabilities. Such instabilities occur in the presence of initial electric fields for the p-form gauge fields. We show that the dynamically generated magnetic fields tend to decrease the initial electric fields and result in configurations with linking numbers, which can be characterized by non-invertible global symmetries. The so-called chiral plasma instability and instabilities of the axion electrodynamics and (4+1)-dimensional Maxwell-Chern-Simons theory in electric fields can be described by the generalized chiral instabilities in a unified manner. We also illustrate this mechanism in the (2+1)-dimensional Goldstone-Maxwell model in electric field.
Student Talks
Towa Takahashi (Niigata University): "Flavino dark matter in the A4 flavor model"
We talk a relic density of the “flavino” dark matter in an A4 model which is respecting the SU (2)L × A4 × Z3 × U (1)R symmetry. In estimating the relic density, we consider the relevant interactions from the Lagrangian that realize the vacuum expectation value alignments and charged lepton masses where we assume that the supersymmetry breaking effects are small for “flavon” sector. As a result, we find the degenerate masses between the lightest “flavon” and “flavino”, and only two parameters in the potential determines the relic density. Then the allowed parameter space of these parameters are estimated from the relic density calculation. This presentation is based on JHEP 09 (2024), 036.
Shota Kaneko (Niigata University): "Complex-valued potential between heavy quarkonia in a thermal pion gas"
In my talk, I will introduce the imaginary part of the potential between heavy quarkonia due to interactions with thermal pions. In particular, I will explain that the imaginary part of the potential shows the $r^{-2}$ behavior at long distances.
Daiki Miura (Niigata University): "One-pion exchange potential in a strong magnetic field"
The properties of QCD matter in a strong magnetic field have attracted much attention because of their relevance to the physics of relativistic heavy-ion collisions and magnetars. For example, the effects of magnetic fields are intensely studied in both single-body and many-body problems, such as modifications of the hadron mass spectrum and the QCD phase diagram. On the other hand, recent progress in lattice QCD and femtoscopy has enabled more direct investigations of hadron-hadron interactions. Given these developments, it is now timely to investigate the hadron-hadron interaction in the presence of strong magnetic fields. In this study, we analyze how a strong magnetic field affects the long-range behavior of the nuclear force, specifically the one-pion exchange potential(OPEP). Based on chiral perturbation theory in magnetic fields, we derive the OPEP and demonstrate how the potential between the proton and neutron behaves.
