Altland Simons Condensed Matter
Raingarda Krzynowek
Scope of Course. This course continues from Many Body 620, and will introduce many body physics needed to understand current research activities in quantum condensed matter, including finite temperature methods, response functions, path integrals, conventional and unconventional superconductivity, strongly correlated electron systems. I will also review essential material and offer additiional tuition to cater to those who were unable to take 620 last semester. Please ask Shirley Hinds for a special permission to register. There will be a lot of discussion and interaction. Please register as soon as possible.
This course will provide a thorough grounding in fundamental aspects of modern theoretical statistical and condensed matter physics. The lectures will cover all necessary formalism, but also emphasize the applicability and usefulness of the methods in the context of contemporary experimental and theoretical research issues.
Contents Starting from basic notions of statistical mechanics and quantum theory, the students will be progressively introduced to the formalisms of operatorial quantization, path integrals and functional field integration. The wide applicability of these methods in condensed matter and statistical physics will be emphasized by addressing topics such as (among others) low-dimensional interacting fermionic and spin systems, spin-charge separation and the Luttinger liquid, the Kondo effect, broken symmetry and Bose-Einstein condensation, superfluidity and superconductivity.
MaterialsClass notes are available here. These are broadly based on the book Condensed Matter Field Theory by B. D. Simons and A. Altland. This book has been selected due to its rather successful attempt to "provide a bridge between formal manipulations and research-oriented thinking", which fits perfectly well in a modern masters-level program aimed at forming a new generation of researchers and thinkers.
Please view this as an indication only: since my coverage of the material will differ somewhat from the book, please check and rely on your class notes to revise the more specific contents we discussed.
Abstract: In this talk, I will review the main progress in multi-band Hubbard model in recent years. Including the orbital degree of freedom in Hubbard model will generate several new features, such as the coexistence of the itinerant and local moment features for the electrons, which are impossible for the single band systems. I will also explain in detail that the interplay between the orbital and spin fluctuation will lead to non-Fermi liquid behavior in these systems in a quite wide temperature range.
Abstract: In this talk, I will show how to discretize the Chern-Simons gauge theory on generic planar lattices/graphs (with or without translational symmetries) embedded in arbitrary 2D closed orientable manifolds. We find that, as long as a one-to-one correspondence between vertices and faces can be defined on the graph such that each face is paired up with a neighboring vertex (and vice versa), a discretized Chern-Simons theory can be constructed consistently. We further verify that all the essential properties of the Chern-Simons gauge theory are preserved in the discretized setup. In addition, we find that the existence of such a one-to-one correspondence is not only a sufficient condition for discretizing a Chern-Simons gauge theory but, for the discretized theory to be nonsingular and to preserve some key properties of the topological field theory, this correspondence is also a necessary one. A specific example will then be provided, in which we discretize the Chern-Simons gauge theory on a tetrahedron.
Abstract: It is known that fractional quantum Hall effect is a star physics platform for the application of Chern-Simons theory that is a typical type of topological quantum field theory (TQFT). In this talk, I would like to explore more TQFT examples from the perspective of generalization of non-local Aharonov-Bohm effect. I will introduce (a) twisted bosonic superconductors; (b) bosonic integer quantum Hall effect; (c) TQFT of bosonic topological insulators where a "cosmological constant term" emerges; (d) towards complete TQFT description of Abelian SPT (symmetry-protected topological phases) in three dimensions. The talk is technique-free and in a plain language.
Abstract: Quantum spin liquids are symmetric ground states of frustrated magnets, featured by fractional excitations called "spinons". Similar to fractional quantum Hall effects where quasiparticles carry fractional charge, spinons and other fractional excitations in a quantum spin liquid can also carry fractional quantum numbers of global and crystal symmetries. This phenomena is coined "symmetry fractionalization".
Motivated by mounting numerical evidence for spin liquid ground states in various two-dimensional frustrated spin models, here we develop systematic methods to measure the global and crystal symmetry quantum numbers of fractional excitations. We show that the symmetry fractionalization patterns in a quantum spin liquid can be measured by a dimensional reduction regime, which relates the two-dimensional symmetric topological orders to one-dimensional symmetry protected topological phases. This general framework is directly applicable to numeric results obtained in 2d DMRG studies, and can be generalized to other gapped topological orders in two dimensions.
Abstract: The discovery of quantum materials with non-trivial topological electronic structures has recently ignited intensive interest in physics and materials science. Topological Weyl semimetals (TWSs) represent a new state that that not only possesses Weyl fermions in the bulk and unique Fermi-arcs generated by topological surface states, but also exhibits appealing physical properties such as extremely large magnetoresistance and ultra-high carrier mobility.
In this talk, I will show our angle-resolved photoemission spectroscopy (ARPES) measurement on a recently proposed materials (represented by TaAs), we directly observed the band structures with characteristic Fermi-arcs. In addition, by systematically investigating different compounds from the same material family, we observed the Fermiology evolution with spin-orbit coupling (SOC) strength. The discovery of this family of TWSs provides a rich material base for exploring many exotic physical phenomena and novel future applications.
Abstract: This talk discusses the concepts of the local Hamiltonian problem, the quantum marginal problem, and the relationship to the geometry of the reduced density matrices. We then discuss how this geometry is related to quantum phase and phase transitions.
Abstract: Liquid 3He becomes superfluid at low temperatures, and has two different phases, A phase and B phase. Since 3He atoms are neutral, there is no Meissner effect, but atoms form pairing like the Copper pairs of electrons in superconductors. Atoms also avoid the singlet pairing, as in metals, and tend to pair in the form of spin triplet. The 3He A phase is an equal spin pairing state with gapless excitations. Weyl semimetals are three-dimensional topological states of matter, in a sense that they host paired monopole and anti-monopole of Berry curvature in momentum space, leading to the chiral anomaly. The chiral anomaly has long been believed to give a positive magnetoconductivity in strong and parallel fields.
In this talk, I start with the 3He A phase, and show the connection between the phase and Weyl semimetals. Weyl semimetals and Weyl superconductors/superfluids can be described by the same mathematical model as the 3He A phase.
Abstract: We study the spontaneous spin textures induced by magnetic dipole-dipole interaction in ferromagnetic spinor condensates under various trap geometries. At the mean-field level, we show the system is dual to a Dirac string gas with a negative string tension in which the ground state spin texture can be easily determined. We find that three-dimensional condensates prefer a meron-like vortex texture, quasi-one-dimensional condensates prefer the axially polarized flare texture, while condensates in quasi-two-dimensions exhibit either a meron texture or an in-plane polarized texture.
Abstract: In the Landau paradigm, classical orders are characterized by local order parameters. It is one of the most prevailing framework for understanding classical orders in different phases of matter. Yet the use local order parameters leads to a certain limitation of this paradigm. In the detection of the orders in a given phase, some knowledge of the specific form of the order parameters, which could sometime be hard to obtain, is often require a priori. Also, this paradigm fails to capture the topological order which are not described local order parameters. In this talk, we propose to use the mutual information, as an alternative to local order parameters, as a generalized probe that describes both classical and topological orders in the same language. We will illustrate the behavior of the mutual information in both classically and topologically ordered systems in various of spacetime dimensions. We will propose the topological uncertainty principle of the mutual information which serves as the unique features of topological orders in general dimensions.
Abstract: The eigenstate thermalization hypothesis (ETH) attempts to bridge the gap between quantum mechanical and statistical mechanical descriptions of isolated quantum systems. Here, we define unbiased measures for how well the ETH works in various regimes, by mapping general interacting quantum systems on regular lattices onto a single particle living on a high-dimensional graph. By numerically analyzing deviations from ETH behavior in the non-integrable Ising model, we propose a quantity that we call the weight to democratically characterize the average deviations for all operators residing on a given number of sites, irrespective of their spatial structure. It appears to have a simple scaling form, that we conjecture to hold true for all non-integrable systems. A closely related quantity, that we term the distinguishability, tells us how well two states can be distinguished if only site operators are measured. Along the way, we discover that complicated operators on average are worse than simple ones at distinguishing between neighboring eigenstates, contrary to the naive intuition created by the usual statements of the ETH that few-body (many-body) operators acquire the same (different) expectation values in nearby eigenstates at finite energy density. Finally, we sketch heuristic arguments that the ETH originates from the limited ability of simple operators to distinguish between quantum states of a system, especially when the states are subject to constraints such as roughly fixed energy with respect to a local Hamiltonian.
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