Dynamics of strongly correlated quantum matter via the advanced recursion method

Dynamics of strongly correlated quantum matter have long been considered to be quantitatively intractable beyond phenomenological or perturbative approaches. However, currently the field is being transformed by new ideas revolving around the phenomenon of operator growth. I will report an advanced implementation of the recursion method – a technique to solve coupled Heisenberg equations in the tridiagonal operator basis – and its application to near- and far-from-equilibrium dynamics of strongly correlated spin-1/2 systems on one-, two- and three-dimensional lattices. From the computational point of view, the core routine of the method is the symbolic computation of nested commutators between the system Hamiltonian and the observable of interest. The result of this computation is a colossal computer formula containing Pauli matrices and parameters of the Hamiltonian. After this formula is obtained and stored in computer memory, various post-processing techniques are applied to convert it to correlation functions, transport coefficients or time dependence of the observable after a quantum quench. Thanks to the symbolic nature of computation, a single run of the core routine covers the whole parameter space of the microscopic model, as well as the whole space of initial states for the quench problem, resulting in a drastic gain in the efficiency of the method.