Dr Guo Chuan Thiang
|Position||ARC DECRA Fellow|
|Org Unit||Mathematical Sciences|
|Telephone||+61 8 8313 4762|
I am an ARC DECRA Fellow (2017-2020) specialising in mathematical physics.
I was awarded a University of Adelaide Research Fellowship for 2018.
In 2015-2016, I was an ARC Postdoctoral Research Associate at the University of Adelaide.
I completed a DPhil in mathematics at the University of Oxford in December 2014 (conferred March 2016). Prior to this, I studied physics and mathematics at the National University of Singapore and the University of Cambridge.
In 2010, I was a research assistant at the Centre for Quantum Technologies, Singapore.
I work on applications of K-theory, operator algebras, index theory, algebraic and differential topology, and noncommutative geometry to the phenomena of topological phases of matter in condensed matter physics and dualities in string theory.
Current interests: discovering new mathematical dualities from physical phenomena, and in reverse, understanding certain physical systems through dualities.
One type of duality is the bulk-boundary correspondence. I recently proved rigorously that the Chern class invariant for topological insulators leads dually to quantised unidirectional metallic boundary currents which can follow the edge of a material around arbitrary corners and imperfections, as theoretically suggested and experimentally verified by physicists. This duality is a physical manifestation of the index theory of certain semigroup C*-algebras.
K. Gomi and I recently discovered the notion of crystallographic T-duality. This is closely related to a super-version of the Baum-Connes conjecture, and implements a duality of twisted equivariant K-theories with interesting computational consequences when complemented with "traditional" spectral sequence methods. The intuition for this duality came from studying position and momentum space versions of topological invariants in solid state physics, and the general concept of T-dualities as topological Fourier transforms.
In return, I was able to show with M. Ludewig that the K-theory of a group C*-algebra has the interpretation of obstructions to existence of "good atomic" models in solid state physics. This topology-localisation duality is a basic tenet of "topological quantum chemistry". In this work, and in an ongoing collaboration with K. Gomi and Y. Kubota, I gave the first direct physical interpretation of the Baum-Connes assembly map which computes the above K-theory obstruction from physically determined K-homology data.
An exciting new result, published in Physical Review Letters (Japanese press release), which I obtained in collaboration with theoretical and experimental physicists in spintronics, proves that the classical phenomenon of magnetostatic spin waves (MSSWs), used in the Nobel-winning discovery of Giant Magnetoresistence, has a differential-topological origin.
Earlier work on mathematics of topological insulators, semimetals, and T-dualities: My earlier contributions include a rigorous mathematical analysis of the general classification problem for topological insulating phases, the classification of topological semimetal phases, and the formulation of bulk-boundary correspondences. Roughly speaking, topological phases are equivalence classes of physical systems subject to certain spectral and symmetry conditions, labelled by intereting topological invariants. These invariants are typically "invisible" until boundary conditions are introduced, whence the they manifest as analytic boundary-localised zero modes via some index theorem. Some notes for a lecture series given in Feb-Mar '17 in Leiden are available here, and notes for a lecture series given in Feb-March in Seoul and Taiwan are available here.
As general tools, I also study the mathematics of T-duality, which has historically found deep applications in the analysis of D-branes in string theory. In particular, I seek to apply concepts from string dualities to the condensed matter setting. For instance, I introduced the notion of T-duality of topological insulators in a paper with V. Mathai. Together with K. Hannabuss, we demonstrated the conceptual and computational utility of T-duality in simplifying and providing geometric intuition for bulk-boundary correspondence for topological insulators.
Euclidean symmetry of non-relativistic dynamics (of an electron) can be broken into crystallographic group symmetry, with far-reaching consequences that have recently gained attention in the form of topological crystalline phases. Utilising a concrete physical model and applying the heuristics of a "crystallographic bulk-edge correspondence", I discovered, with K. Gomi, a new mod 2 "super-index theorem".
I have also studied the global topology of semimetals through Poincare duality, or "Dirac stringy" methods. Topological semimetals have the potential to realise exotic topologically stable fermions which are characterised by subtle topological invariants. In particular, there are intriguing links between semimetal topology, and Seiberg-Witten invariants, Kervaire semicharacteristics, and torsions of manifolds. In the presence of time-reversal symmetry, semimetals realise a new exotic type of monopole which acts as a charge for the famous mod 2 invariant of Kane-Mele.
Another general direction is the study of topological phases in different geometries. The idea that many-body effects can change the effective geometry "felt" by a single electron had previously been used to model the fractional quantum Hall effect. Utilising a variant of T-duality for Riemann surfaces, I formulated a bulk-boundary correspondence for fractional indices for the first time.
Some years ago, I dabbled in algebraic quantum field theory, and was a researcher in quantum information theory at the Centre for Quantum Technologies, National University of Singapore.
In September 2016, I organised a conference on mathematical topics at the interface of string theory, condensed matter physics, K-theory, operator algebras, and geometry. [Website]
In 2017, I co-organised workshops on string geometries and dualities [Website], gauge theory and higher geometry [Website], and the Australia-China conference in noncommutative geometry and related areas. [Website]
In 2018 and 2019, I am the convenor of the Differential Geometry Seminar in the University of Adelaide [Website]
- Edge-following topological states. [1908.09559]
- Good Wannier functions in Hilbert modules associated to topological insulators (with M. Ludewig). [1904.13051]
- Topological phases on the hyperbolic plane: fractional bulk-boundary correpondence (with V. Mathai). Advances in Theoretical and Mathematical Physics, 23(3), to appear (2019) [1712.02952]
- Topological characterization of classical waves: the topological origin of magnetostatic surface spin waves (with K. Yamamoto, P. Pirro, K.-W. Kim, K. Everschor-Sitte, E. Saitoh). Physical Review Letters, 122 217201 (2019) [1905.07909]
- Crystallographic T-duality (with K. Gomi). Journal of Geometry and Physics, 139 50-77 (2019) [1806.11385]
- Crystallographic bulk-edge correspondence: glide reflections and twisted mod 2 indices (with K. Gomi). Letters in Mathematical Physics, 109(4) (857-904) (2019) [1804.03945]
- T-duality simplifies bulk-boundary correspondence: the noncommutative case (with K. Hannabuss and V. Mathai). Letters in Mathematical Physics 108(5) 1163-1201 (2018) [1603.00116]
- Fu-Kane-Mele monopoles in semimetals (with K. Sato and K. Gomi). Nuclear Physics B 923 107-125 (2017) [1705.06657]
- Differential topology of semimetals (with V. Mathai). Communications in Mathematical Physics 355(2) 561-602 (2017) [1611.08961]
Degree of Separability of Bipartite Quantum States. Physical Review A 82(1) 012332 (2010)
Optimal Lewenstein-Sanpera Decomposition for two-qubit states using Semidefinite Programming (with B.-G. Englert and P. Raynal). Physical Review A 80(5) 052313 (2009)
Refereed Conference Proceedings
- T-duality and K-theory: a view from condensed matter physics. In: Noncommutative Geometry and Physics IV, proceedings for TFC thematic year 2015 on "Fundamental Problems in Quantum Physics: Strings, Black Holes and Quantum Information", pp. 279-314 (2017)
- On the K-theoretic classification of topological phases of matter (conspectus). In: Proceedings of Frontiers of Fundamental Physics 14 (2014)
- String geometries, dualities and topological matter (with V. Mathai, P. Hekmati, H. Bursztyn, P. Bouwknegt, D. Baraglia). Journal of Geometry and Physics 138 (2019)
- Lecture notes on topological phases and K-theory (updated 2 May 17) - Leiden_Lectures_2_May.pdf [492.2K] (application/pdf)
- CV - August 2019 - CV-Aug_19.pdf [102.1K] (application/pdf)
- Seoul lectures on K-theory and T-duality of topological phases - Seoul_lectures.pdf [2.6MB] (application/pdf)
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Entry last updated: Wednesday, 4 Sep 2019
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