We study point impurities in non-relativistic quantum field theories, with a focus on scale-invariant fixed points. We establish the framework of conformal defects in Schrödinger field theories and their correspondence to many-body states in a harmonic trap. We discuss a multi-critical quantization of the one-body wave function induced by an impurity, with examples of free and unitary Fermi gases. Our results can potentially be tested in experimental setups.

United States

Avia Raviv-Moshe

Siwei Zhong

Impurities in Schrödinger field theories and s-wave resonance

We study the rich dynamics resulting from introducing static charged particles (Wilson lines) in 2+1 and 3+1 dimensional gauge theories. Depending on the charges of the external particles, there may be multiple defect fixed points with interesting renormalization group flows connecting them, or an exponentially large screening cloud can develop (defining a new emergent length scale), screening the bare charge entirely or partially. We investigate several examples where the dynamics can be solved in various weak coupling or double scaling limits. Sometimes even the elementary Wilson lines, corresponding to the lowest nontrivial charge, are screened. We consider Wilson lines in 3+1 dimensional gauge theories including massless scalar and fermionic QED4, and also in the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 supersymmetric Yang-Mills theory. We also consider Wilson lines in 2+1 dimensional conformal gauge theories such as QED3 with bosons or fermions, Chern-Simons-Matter theories, and the effective theory of graphene. Our results in 2+1 dimensions have potential implications for graphene, second-order superconducting phase transitions, etc. Finally, we comment on magnetic line operators in 3+1 dimensions (’t Hooft lines) and argue that our results for the infrared dynamics of electric and magnetic lines are consistent with non-Abelian electric-magnetic duality.

Israel

Ofer Aharony

Gabriel Cuomo

Zohar Komargodski

United Kingdom

Márk Mezei

Phases of Wilson lines: conformality and screening

We study mass-type surface defects in a free scalar and Wilson-Fisher (WF) O(N) theories. We obtain exact results for the free scalar defect, including its RG flow and defect Weyl anomaly. We classify phases of such defects at the WF fixed point near four dimensions, whose perturbative RG flow is investigated. We propose an IR effective action for the non-perturbative regime and check its self-consistency.

Phases of surface defects in Scalar Field Theories

We study the low-energy limit of Wilson lines (charged impurities) in conformal gauge theories in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> and <math><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions. As a function of the representation of the Wilson line, certain defect operators can become marginal, leading to interesting renormalization group flows and for sufficiently large representations to complete or partial screening by charged fields. This result is universal: in large enough representations, Wilson lines are screened by the charged matter fields. We observe that the onset of the screening instability is associated with fixed-point mergers. We study this phenomenon in a variety of applications. In some cases, the screening of the Wilson lines takes place by dimensional transmutation and the generation of an exponentially large scale. We identify the space of infrared conformal Wilson lines in weakly coupled gauge theories in <math><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions and determine the screening cloud due to bosons or fermions. We also study QED in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions in the large <math><msub><mi>N</mi><mi>f</mi></msub></math> limit and identify the nontrivial conformal Wilson lines. We briefly discuss ‘t Hooft lines in <math><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math>-dimensional gauge theories and find that they are screened in weakly coupled gauge theories with simply connected gauge groups. In non-Abelian gauge theories with <math><mrow><mi>S</mi></mrow></math> duality, this together with our analysis of the Wilson lines gives a compelling picture for the screening of the line operators as a function of the coupling.

Phases of Wilson Lines in Conformal Field Theories

We consider line defects with large quantum numbers in conformal field theories. First, we consider spin impurities, both for a free scalar triplet and in the Wilson-Fisher O(3) model. For the free scalar triplet, we find a rich phase diagram that includes a perturbative fixed point, a new nonperturbative fixed point, and runaway regimes. To obtain these results, we develop a new semiclassical approach. For the Wilson-Fisher model, we propose an alternative description, which becomes weakly coupled in the large spin limit. This allows us to chart the phase diagram and obtain numerous rigorous predictions for large spin impurities in 2 + 1 dimensional magnets. Finally, we also study 1/2-BPS Wilson lines in large representations of the gauge group in rank-1   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal field theories. We contrast the results with the qualitative behavior of large spin impurities in magnets.

Spin impurities, Wilson lines and semiclassics

We consider line defects in <math><mi>d</mi></math>-dimensional conformal field theories (CFTs). The ambient CFT places nontrivial constraints on renormalization group (RG) flows on such line defects. We show that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the <math><mi>g</mi></math> theorem to line defects in arbitrary dimensions. We demonstrate our results in a flow between Wilson loops in four dimensions.

Renormalization Group Flows on Line Defects

We study operators with large internal charge in boundary conformal field theories (BCFTs) with internal symmetries. Using the state-operator correspondence and the existence of a macroscopic limit, we find a non-trivial relation between the scaling dimension of the lowest dimensional CFT and BCFT charged operators to leading order in the charge. We also construct the superfluid effective field theory for theories with boundaries and use it to systematically calculate the BCFT spectrum in a systematic expansion. We verify explicitly many of the predictions from the EFT analysis in concrete examples including the classical conformal scalar field with a |ϕ|6 interaction in three dimensions and the O(2) Wilson-Fisher model near four dimensions in the presence of boundaries. In the appendices we additionally discuss a systematic background field approach towards Ward identities in general boundary and defect conformal field theories, and clarify its relation with Noether’s theorem in perturbative theories.

Boundary conformal field theory at large charge

Conformal Field Theories (CFTs) have rich dynamics in heavy states. We describe the constraints due to spontaneously broken boost and dilatation symmetries in such states. The spontaneously broken boost symmetries require the existence of new low-lying primaries whose scaling dimension gap, we argue, scales as O(1). We demonstrate these ideas in various states, including fluid, superfluid, mean field theory, and Fermi surface states. We end with some remarks about the large charge limit in 2d and discuss a theory of a single compact boson with an arbitrary conformal anomaly.

Sridip Pal

Spontaneously broken boosts in CFTs

We study the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> model in dimension three (<math><mrow><mn>3</mn><mi>d</mi></mrow></math>) at large and infinite <math><mi>N</mi></math> and show that the line of fixed points found at <math><mi>N</mi><mo>=</mo><mi>∞</mi></math>—the Bardeen-Moshe-Bander (BMB) line—has an intriguing origin at finite <math><mi>N</mi></math>. The large <math><mi>N</mi></math> limit that allows us to find the BMB line must be taken on particular trajectories in the <math><mo>(</mo><mi>d</mi><mo>,</mo><mi>N</mi><mo>)</mo></math> plane: <math><mi>d</mi><mo>=</mo><mn>3</mn><mo>−</mo><mi>α</mi><mo>/</mo><mi>N</mi></math> and not at fixed dimension <math><mi>d</mi><mo>=</mo><mn>3</mn></math>. Our study also reveals that the known BMB line is only half of the true line of fixed points, the second half being made of singular fixed points. The potentials of these singular fixed points show a cusp for a finite value of the field and their finite <math><mi>N</mi></math> counterparts a boundary layer.

France

C. Fleming

B. Delamotte

Japan

S. Yabunaka

Finite <math><mi>N</mi></math> origin of the Bardeen-Moshe-Bander phenomenon and its extension at <math><mi>N</mi><mo>=</mo><mi>∞</mi></math> by singular fixed points

We construct supersymmetric Lifshitz field theories with four real supercharges in a general number of space dimensions. The theories consist of complex bosons and fermions and exhibit a holomorphic structure and non-renormalization properties of the superpotential. We study the theories in a diverse number of space dimensions and for various choices of marginal interactions. We show that there are lines of quantum critical points with an exact Lifshitz scale invariance and a dynamical critical exponent that depends on the coupling constants.