**Siddhartha Lal Reserach Group : Frustration shapes multi-channel Kondo physics: a star graph perspective**

Frustration. We think we understand it, and how to deal with it. But what does frustration refer to in a quantum system? Consider classical spins (i.e., spins that have only two configurations, say, pointing up and pointing down) placed on an equilateral triangle and interacting with one another through a nearest neighbour antiferromagnetic Ising exchange interaction. It is easy to see that Neel (anti-parallel) ordering is no longer possible: for any two spins that are anti-aligned with one another, the third is left confused on which direction to choose (see Fig.1). This confusion is labelled as the frustration of the classical Neel order. Frustration can also been seen in quantum mechanical systems: while two quantum spin 1/2s with an antiferromagnetic Heisenberg interaction will form a maximally-entangled singlet state, introducing a third spin-1/2 creates a dilemma (see Fig.2) - the two-spin singlet cannot accommodate another spin (often curiously referred to as entanglement monogamy). Since the spin-flip quantum fluctuations of the system will want to lower the energy of the system by entangling all three spins, no spin can be left free and the two-spin singlet cannot be the true ground-state.

The multichannel Kondo problem involves a local antiferromagnetic Heisenberg interaction between a single spin-1/2 impurity and the electrons of several conduction bath channels (see Fig.3). Had there been only one conduction bath, the impurity moment would form a singlet together with a “cloud” of electrons from the bath. We refer to this as the screening of the impurity moment (as the singlet has no magnetisation). However, in the multichannel Kondo problem described above, the formation of a singlet between the impurity and electron from one of the conduction channels is frustrated. As a result, the so-called Kondo screening of the impurity spin’s magnetic moment is hampered. Indeed, if the total spin of the conduction bath is greater than the spin of the impurity, the multichannel Kondo problem is said to be over-screened. The screening process, as well as its breakdown, are truly many-body in nature: a macroscopic number of conduction electrons interact with a single quantum impurity, and are therefore “aware” of one another. A proper description of the physics thus requires a field-theoretic treatment of the impurity-bath interactions, and the problem has been studied using a wide variety of powerful analytic and numerical methods.

Our contribution in this work was to show that the fascinating properties of the N-channel Kondo problem could be linked to those of the associated skeletal problem: a central quantum spin-1/2 coupled to N quantum spin-1/2s (corresponding to the N conduction channels) through identical antiferromagnetic Heisenberg exchange couplings. Such a model is often referred to as a star graph, and it can be identified as a limit of the multichannel problem in which the kinetic energy of the itinerant electrons has been switched off. We show in our work that certain properties of the star graph, such as the ground-state degeneracy and the magnetisation, are linked to bulk thermodynamic properties. The star graph also sets the scattering phase shift of the conduction electrons, and the scattering phase shift then dictates how the quantum fluctuations resolve themselves in order to lead to novel features. In this way, the quantum frustration inherent in the underlying simple quantum mechanical problem is seen to offer great insights into a many-body problem which looks quite daunting otherwise. Please read our work to find out more.

Authors: Siddhartha Patra, Abhirup Mukherjee, Anirban Mukherjee, N. S. Vidhyadhiraja, A. Taraphder, and Siddhartha Lal Journal Reference: Journal of Physics: Condensed Matter 35, 315601 (2023) Link: https://iopscience.iop.org/article/10.1088/1361-648X/acd09c

__#Research Highlight__

*Posted on: May 23rd, 2023*