Sometimes, technical points remain unresolved, mainly because they don't impact the main results being written up. Here are a few that still seem worth thinking about-

• Stationary state of discrete Landau-Lifshitz dynamics?

Recently, I collaborated with several others (Das, Dhar, and Kulkarni @ ICTS-TIFR, Huse @ Princeton, Mendl @ Dresden/Munich, & Spohn @ Munich) to test the predictions of nonlinear fluctuating hydrodynamics for the equilibrium spatio-temporal correlations of the classical ferromagnetic XXZ spin chain. I got involved in this off-topic (for me) collaboration in part because I happened to attend Spohn's lectures on fluctuating hydrodynamics at the ICTS program NESP2015. But a significant part of my motivation was also that I had come up with what I thought was a nice algorithm for numerically integrating the equations of motions for precessional dynamics of spins while preserving total angular momentum and total energy exactly (to machine-precision) <<independent>> of integration time-step, in principle allowing efficient access to very-late-time behaviour. To do this, the algorithm (now described in one section of our joint paper) allows for small fluctuations in the length of each spin-vector. These fluctuations go to zero as the integration time step is made smaller and smaller, and, crucially, these fluctuations do not grow in time even when the integration time step is an O(1) number. This algorithm raises the following natural question, which remained unanswered in our joint work, and limited our use of this algorithm to small integration time steps, rendering it no-more-useful than the standard RK4 algorithm for our joint work: If one starts with the classical equilibrium ensemble (for the XXZ chain, or any other short-ranged Hamiltonian of unit-length spins on a bipartite lattice) and evolves each initial condition according to this discrete-time dynamics with O(1) integration time-step, what is the ensemble that describes the system at long times? Can this late-time ensemble be written in terms of a Boltzmann weight corresponding to an energy function with additional <<short-range>> interaction terms?

• Nonlinear sigma-model description of ground state spin texture in quantum antiferromagnets?

In Sambuddha Sanyal's thesis work, we (Sanyal, Banerjee, KD, Sandvik) measured the spatial profile of the net S=1/2 moment that exists in the ground state of the S=1/2 quantum antiferromagnet on a L \times M square lattice with L \times M odd and open boundary conditions using an algorithm which I had developed earlier in collaboration with Argha Banerjee. We repeated this calculation with additional terms in the Hamiltonian that weaken the antiferromagnetic order without destroying it. The central finding was that the staggered Fourier component of this spatial profile, call it nz, was <<universally>> related to the conventional antiferromagnetic order parameter m (defined as the strength of the Bragg peak at (pi,pi) in the infinite volume limit with an even number of spins and periodic boundary conditions). Universal here means that the detailed form of the Hamiltonian does not matter (we tried several different ways of weakening the antiferromagnetism). Given the nature of this observation, it seems very plausible that there ought to be some nonlinear sigma model description not just of the function nz(m), but also of the Fourier modes near (pi, pi) and (0,0) of the full ground state spin texture whose alternating part is nz. How does one formulate this problem in nonlinear sigma model language?

• Direct quantum Monte-Carlo probes of emergent gauge dynamics?

In Argha Banerjee's thesis work, we (Banerjee, Isakov, KD, Kim) provided the first definite Quantum Monte Carlo evidence for a Coulomb phase of bosons, with properties in line with expectations from the original theoretical proposals of Hermele et. al. and Huse et. al. However, in the convention in which the boson number maps to an electric field variable, we could not measure correlators of the magnetic flux. This magnetic flux variable is represented in terms of the ring-exchange operator in the low energy effective Hamiltonian that describes the physics of the microscopic model below the charge gap. However, the Quantum Monte Carlo works with the original bosonic Hamiltonian which does not have any ring-exchange term in the Hamiltonian. Can one identify, in the QMC simulations of the microscopic model, a useful surrogate of the ring-exchange correlation function? The question can be posed more generally: If one is doing QMC on a microscopic model with a big energy gap between low-lying states and everything else, and if the effective Hamiltonian in the low-lying sector has correlated hopping terms of some kind, how does one use the QMC data to obtain correlation functions of the correlated hopping operator (since this is what connects most directly to some effective field theory formulation)?