Spring School on Modern Topics in Condensed Matter

As the first lecture in a mini-course on superconductivity, I will analyse the phenomenon from a thermodynamic perspective. I will discuss the London Equations from a seldom-used thermodynamic viewpoint, and show how much can be deduced abut the necessary ingredients of a microscopic theory (some using 20-20 hindsight but some following the thought processes of Bardeen) from careful analysis of remarkably simple thermodynamic data.

In the first part of this lecture (14.00-14.45), I shall present two key precursors of the BCS theory of superconductivity: the origin of weak attractive forces between electrons (Bardeen and Pines, 1955), and the observation that in the presence of a Fermi surface even weak attraction produces a bound state (Cooper, 1956). In the second part (14.55-15.40), I shall explain how these elements may be combined to write down a candidate many-body ground state for the electronic sector of the superconductor - the so-called BCS wave function (Bardeen, Cooper, and Schrieffer, 1957) - and discuss how it gives rise to the phenomenology of the superconducting state.

Three dimensional magnetic systems promise significant opportunities for both fundamental physics, and technological applications, for example providing higher density devices and new functionalities associated with complex topology and greater degrees of freedom [1,2]. In this lecture, I will describe recent advances in both characterization and nanofabrication techniques that allow us to explore the physics of three dimensional magnetic systems experimentally. In particular, harnessing recent advances in X-ray magnetic tomographic imaging, it is now possible to map both the static configuration [3,4], and dynamical behaviour [5,6], of topological magnetic structures [3-6]. This new insight, combined with recent advances in analytical techniques [7] have led to first experimental observations of 3D magnetic solitons such as nanoscale magnetic vortex rings [7,8]. As well as existing within bulk and thin film systems, 3D spin textures can be introduced via the patterning of complex 3D magnetic nanostructures [9]. Such textures are not limited to the magnetization: recently it has been discovered that highly coupled curvilinear nanosystems can lead to the formation of topological textures within the magnetic stray field [10]. These new experimental capabilities for 3D magnetic systems open the door to complex three-dimensional magnetic structures, and their dynamic behaviour. References [1] Fernández-Pacheco et al., Nature Communications 8, 15756 (2017). [2] C. Donnelly and V. Scagnoli, J. Phys. D: Cond. Matt. 32, 213001 (2020). [3] C. Donnelly et al., Nature 547, 328 (2017). [4] K. Witte, et al., Nano Letters 20, 1305 (2020). [5] C. Donnelly et al., Nature Nanotechnology 15, 356 (2020). [6] S. Finizio et al., Nano Letters (2022) [7] C. Donnelly et al., Nat. Phys. 17, 316 (2020) [8] N. Cooper, PRL. 82, 1554 (1999). [9] L. Skoric et al., Nano Letters 20, 184 (2020). [10] C. Donnelly et al., Nature Nanotechnology 17, 136 (2022)

Quantum phase transitions, i.e., phase transitions taking place at zero temperature, continue to attract the interest of both theorists and experimentalists in condensed matter physics and beyond. The first lecture will describe the phenomenology of quantum phase transitions, introduce general concepts such as the critical scaling and the renormalization group, and discuss simple yet paradigmatic models for quantum phase transitions and their realization in actual materials.

Electronic systems feature several symmetries whose spontaneous breaking can lead to various forms of order, such as magnetism or superconductivity. In this lecture, I will discuss the concept of "intertwined orders" where order parameters are not simple products of individually broken symmetries. This concept encompasses smectic stripe phases and also pair density waves, where antiferromagnetism or superconductivity is intertwined with charge density wave order. I will discuss modern numerical simulations that reveal these orders are ubiquitous in the physics of lightly doped Mott insulators and relate those findings to experiments on the cuprates.

As superconductors maintain coherence over macroscopic length-scales, they form the basis of a growing family of electronic devices which utilize quantum effects. These devices cover a wide range of applications such as the voltage standard in metrology, highly-sensitive magnetometry, and more recently artificial atoms for quantum computing. These devices all make significant use of the Josephson junction, the superconducting analog of the transistor. This lecture will describe the physics behind the Josephson junction, and how it can be used for quantum devices of growing complexity. We will especially focus on superconducting quantum circuits and how superconducting microwave elements can mimic the quantum properties of atomic systems. Towards the end of the lecture, we will discuss the perspectives for novel quantum devices and particularly how unconventional superconductors can be utilized for future quantum devices with unique capabilities.

Quantum phase transitions, i.e., phase transitions taking place at zero temperature, continue to attract the interest of both theorists and experimentalists in condensed matter physics and beyond. The second lecture will discuss selected examples in more detail, in particular the Bose-Einstein condensation of magnons, magnetic quantum phase transitions in metals, Mott-Hubbard transitions, and (time permitting) deconfined quantum criticality.

Assuming, following Polyakov, that critical points are described by conformal field theories (CFTs), we may write the critical exponents in any given universality class in terms of the scaling dimensions of the fields in such a CFT. For CFTs in d=2 spatial dimensions, it is often possible to determine the scaling dimensions analytically (see, for example, Di Francesco, Mathieu, and Senechal, 2012). For the d>2 case, until about 15 years ago, the situation seemed more or less hopeless. In this lecture I will give a brief but self-contained description of an important new approach to determining the scaling dimensions in conformal field theories with d>2: the so-called conformal bootstrap. In the first part (11.00-11.45), I shall introduce the problem and give some important definitions, as well as some key identities that are obeyed by correlation functions in conformal field theories. In the second part (11.55-12.40), I shall explain how these may be exploited to constrain - very efficiently - the possible scaling dimensions of the theory's fields, and thus predict the critical exponents of theories in d>2 with (sometimes, at least) unprecedented precision.

I will give an introduction to recent progress on understanding the physics of strong correlations in the narrow bands of moiré materials, focusing primarily on twisted bilayer graphene. I will primarily focus on analogies to the quantum Hall effect and quantum Hall ferromagnetismbut also touch upon links to the physics of heavy fermions and Hubbard models. The focus will be on developing an understanding of the properties and phenomenology of the parent integer-filling correlated insulators and the ‘normal state’ out of which superconductivity emerges upon doping.

Assuming, following Polyakov, that critical points are described by conformal field theories (CFTs), we may write the critical exponents in any given universality class in terms of the scaling dimensions of the fields in such a CFT. For CFTs in d=2 spatial dimensions, it is often possible to determine the scaling dimensions analytically (see, for example, Di Francesco, Mathieu, and Senechal, 2012). For the d>2 case, until about 15 years ago, the situation seemed more or less hopeless. In this lecture I will give a brief but self-contained description of an important new approach to determining the scaling dimensions in conformal field theories with d>2: the so-called conformal bootstrap. In the first part (11.00-11.45), I shall introduce the problem and give some important definitions, as well as some key identities that are obeyed by correlation functions in conformal field theories. In the second part (11.55-12.40), I shall explain how these may be exploited to constrain - very efficiently - the possible scaling dimensions of the theory's fields, and thus predict the critical exponents of theories in d>2 with (sometimes, at least) unprecedented precision.