Date: 2023-11-12 | Article ID: 81040 |

The quantum Ising model is a system of qubits defined on a one-dimensional lattice (a discrete version of one-dimensional space), with an especially simple hamiltonian governing the system's time evolution. It is actually a family of models, parameterized by a real number λ≥ 0. This family is interesting because it has a nontrivial **phase structure**: the model has a symmetry that is respected by the lowest-energy state when λ<1 and that is broken by the lowest-energy state when λ>1, a phenomenon called **spontaneous symmetry breaking (SSB)**. These two phases are strictly distinct from each other only when the lattice is infinite. This article defines the model first on a finite lattice and then explores how the strict distinction between the two phases arises when the lattice becomes infinite. The concept of **superselection sectors**, which is important throughout quantum field theory, is one of the keys to understanding how SSB works. The article also explores how this phase structure manages to coexist with another property of the model called **self-duality**, which is a kind of invariance under λ→ 1/λ.

**2023-11-12**(replaced Hamiltonian with hamiltonian)**2023-04-17**(added the page-range to an item in the bibliography)**2023-02-10**(deleted an extra word in a footnote, added a reference)**2022-10-23**(updated bibliography to include publication data instead of only arxiv link)**2022-06-28**(fixed the proof that local observables can't mix the symmetry-breaking ground states, added a footnote about the cluster property)**2022-06-26**(first version)

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