Large-N approaches to quantum magnets – LNAQM

Large-N approaches are powerful methods to study quantum many body problems such as those encountered in solid state physics and correlated electron problems. These methods consist in generalizing the problem to an arbitrary number, N, of particle flavors. When N goes to infinity the different flavors decouple and the interaction terms become solvable (quadratic and mean-field like). The task is then to expend the physical properties in 1/N to approach the initial model of interest. When applied to quantum spin systems in dimension greater than one, this approach has proven to be very fruitful and has provided the basis of our current understanding of the highly quantum phases these systems, and spin liquids in particular. However, one cannot make reliable quantitative predictions about a given microscopic model without including the fluctuation (i.e. finite-N) effects, some of which are known to be related to gauge degrees of freedom.

In this project we will revisit and expand one particular large-N limit of antiferromagnetic Heisenberg model, that based on the Schwinger boson representation [Sp(2N)]. The idea is to start from some systematic numerical exploration of the low-energy landscape defined by the self-consistent mean-field solutions. For N=infinity the system is “locked” into the global energy minimum of this landscape. On the other hand, for finite N, the system can explore the low-energy regions of this landscape, and local minima and saddle-points in particular. Our idea is to gain some insight about the finite-N physics from the detailed knowledge of the mean-field energy landscape. As a first concrete application of this idea, we plan to study some frustrated spin models which have a Z2 spin liquid phase for small enough value of the spin (and large enough N), such as the triangular or kagome Heisenberg models. Such Z2 spin liquids are known to host some particular elementary excitations, dubbed visons, and which correspond to Pi-flux vortices for the (emergent) Z2-gauge field. We will look for these vortex states as saddle-points in the mean-field landscape, study their correlations and energetics (gap). Characterizing the excited mean-field states in magnetically ordered phases is also part of our program. Then, by computing numerically some (imaginary) time-dependent saddle-points of the large-N theory we will have access to the dynamical properties of these visons. Indeed, such instanton calculations should give the tunneling amplitudes for a vison to hop from one lattice plaquette to another, and thus their dispersion relation. This will provide an effective Ising-gauge theory model describing the non-perturbative finite-N fluctuations, and which will allow to address quantitatively some important questions such as the possible critical value of N below which the vison may condense and give rise to quantum phase transition (typically toward a valence-bond crystal).