In one dimension, the Ising model consists simply of a line of spins with nearest-neighbor interactions that tend to align the spins spontaneously in a ferromagnetic order. The one-dimensional model readily yields an exact solution, and the system has no phase transition at finite temperatures. The Ising model for a 2D array of spins is far more challenging, and its theoretical description, which includes a second-order phase transition, leads to far richer physics. The 1944 work that Lars Onsager did on the 2D Ising model paved the way for the subsequent study of critical phenomena that occur at phase transitions.

Many decades later, the Ising model retains its interest for studies of critical phenomena, but such studies are now largely focused on quantum critical points (QCPs), where phase transitions occur at a temperature of absolute zero. For example, as shown in figure 1(a), the 1D Ising chain, when...

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