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Details of Grant 

EPSRC Reference: EP/W010283/1
Title: Renormalisation Group Interfaces in Tricritical Ising Model
Principal Investigator: Konechny, Dr A
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: S of Mathematical and Computer Sciences
Organisation: Heriot-Watt University
Scheme: Standard Research - NR1
Starts: 25 September 2022 Ends: 24 September 2023 Value (£): 79,918
EPSRC Research Topic Classifications:
Algebra & Geometry Mathematical Physics
Non-linear Systems Mathematics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
02 Aug 2021 EPSRC Mathematical Sciences Small Grants Panel August 2021 Announced
Summary on Grant Application Form
The renormalisation group (RG) governs the energy scale dependence of a physical system. It is a very important concept and instrument in condensed matter and particle physics. Fixed points of the RG are described by conformal field theories (CFTs). The conformal symmetry group (being a group acting in space-time rather than on internal degrees of freedom) places powerful constraints which can be used to construct and solve such theories. Conformal field theories are generically much simpler than the non-conformal ones and can be used to approximate the non-conformal theories by means of conformal perturbation theory. Conformal symmetry is particularly powerful in two Euclidean dimensions. In this case the state spaces furnish representations of the Virasoro algebra - an infinite-dimensional Lie algebra. This and the associated algebraic structures are behind a lot of the success in constructing and fully solving two-dimensional CFTs.

Near a fixed point the conformal symmetry is broken. The corresponding theories can be described by perturbing the conformal field theory by a linear combination of relevant operators. The corresponding coefficients, i.e. the coupling constants, change with scale which results in a trajectory in theory space also known as an RG flow. The corresponding flow lines end up either at massive theories which have different symmetry properties, e.g. degenerate vacua, or at other CFTs. It is important to have information on whether the flow line passes near (crossover phenomenon) or approaches other fixed points. If that is the case the appropriate conformal perturbation theory can be used.

In condensed matter physics the description of the perturbed theories at large distances amounts to determining a phase diagram. The global structure of such phase diagrams is in general very hard to determine. As a first approximation one usually relies on indications from Landau theory which is a phenomenological approach that can give a rough idea of the phase diagram but is not a reliable method when it comes to precision results. Alternatively one relies on numerics done in the lattice versions of the theory at hand which are quite dependent on computing power available and have other limitations as well. Little has been done to date in terms of determining the global structure of phase diagrams for phases originating from chosen critical points directly in the continuum approach, that is, working with quantum field theories.

In the present project we aim to describe the global structure of the space of all RG flows originating from the CFT describing the two-dimensional tricritical Ising model (TIM). This model has a rich phase diagram exhibiting a line where three phases coexist and three critical lines leading from the original critical point to the critical Ising model. To chart the phase diagram we plan to use a novel approach based on RG interfaces -- physical surfaces separating two scale invariant theories linked by an RG flow.
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Organisation Website: http://www.hw.ac.uk