Dynamical Correlation Function

In this demo, we will look at the quantity $$G(x,t) := \langle S^z_x(t) S^z_0(0) \rangle$$ for the $$S=1/2$$ XXZ chain with Hamiltonian $$H = \sum_i S^z_i S^z_{i+1} + \Delta\left[S^x_i S^x_{i+1} + S^y_i S^y_{i+1}\right].$$

In the XXZ model, the total magnetization $$M := \sum_x S^z_x$$ is a conserved quantity, and $$G(x,t)$$ is the simplest non-trivial object that probes the flow of the spin density corresponding to M. There are three regimes in the XXZ model for the dynamical properties at high temperatures: $$\Delta < 1$$ with ballistic transport, $$\Delta > 1$$ with spin diffusion, and $$\Delta = 1$$ where KPZ hydrodynamics [1] is present [2]. In the ground state, this model with $$\Delta = 1$$ is described within the Tomonaga-Luttinger liquid framework, and the dynamics is governed by ballistic transport and not by KPZ hydrodynamics.

This demo displays $$G(x,t)$$ as a color plot, as well as the auto-correlation given by $$G(x=0,t)$$. These different dynamical regimes can be seen in this demo by looking at the color plot generated by the simulation. In the case of ballistic transport, the color plot will show a sharp light-cone like structure, indicating that the excitations move freely. In the case of KPZ and diffusion, there will not be high resolution maximum intensity lines. Distinguishing KPZ and diffusion requires more computational resources than are present in this demo, and I point the interested reader to explore the following references [2, 3, 4].

For the simulation, you can adjust the system size $$L$$, the anisotropy parameter in the Hamiltonian $$\Delta$$, the maximum time $$t$$, the bond-dimension $$\chi$$, and whether to look at the ground state (T=0) or a thermal state (T=$$\infty$$). To have sufficient resolution to see the different structures in the dynamical correlation function, it is recommended to use $$L = 64$$, $$t=10$$, and $$\chi=8$$ in the ground state, or $$L = 32$$, $$t=10$$, and $$\chi=32$$ for a thermal state. The time needed for the code to run scales as $$tL\chi^3$$, so it may take up to a few mintues to run a task that has not been calculated previously.