The transverse field Ising model is a quantum version of the classical Ising model. It features a lattice with nearest neighbour interactions determined by the alignment or anti-alignment of spin projections along the z {\displaystyle z} axis, as well as an external magnetic field perpendicular to the z {\displaystyle z} axis (without loss of generality, along the x {\displaystyle x} axis) which creates an energetic bias for one x-axis spin direction over the other. An important feature of this setup is that, in a quantum sense, the spin projection along the x {\displaystyle x} axis and the spin projection along the z {\displaystyle z} axis are not commuting observable quantities. That is, they cannot both be observed simultaneously. This means classical statistical mechanics cannot describe this model, and a quantum treatment is needed. Specifically, the model has the following quantum Hamiltonian:
H = − J ( ∑ ⟨ i , j ⟩ Z i Z j + g ∑ j X j ) {\displaystyle H=-J\left(\sum _{\langle i,j\rangle }Z_{i}Z_{j}+g\sum _{j}X_{j}\right)}
Here, the subscripts refer to lattice sites, and the sum ∑ ⟨ i , j ⟩ {\displaystyle \sum _{\langle i,j\rangle }} is done over pairs of nearest neighbour sites i {\displaystyle i} and j {\displaystyle j} . X j {\displaystyle X_{j}} and Z j {\displaystyle Z_{j}} are representations of elements of the spin algebra (Pauli matrices, in the case of spin 1/2) acting on the spin variables of the corresponding sites. They anti-commute with each other if on the same site and commute with each other if on different sites. J {\displaystyle J} is a prefactor with dimensions of energy, and g {\displaystyle g} is another coupling coefficient that determines the relative strength of the external field compared to the nearest neighbour interaction.
Phases of the 1D transverse field Ising model Below the discussion is restricted to the one dimensional case where each lattice site is a two-dimensional complex Hilbert space (i.e., it represents a spin 1/2 particle). For simplicity here X {\displaystyle X} and Z {\displaystyle Z} are normalised to each have determinant -1. The Hamiltonian possesses a Z 2 {\displaystyle \mathbb {Z} _{2}} symmetry group, as it is invariant under the unitary operation of flipping all of the spins in the z {\displaystyle z} direction. More precisely, the symmetry transformation is given by the unitary ∏ j X j {\displaystyle \prod _{j}X_{j}} . The 1D model admits two phases, depending on whether the ground state (specifically, in the case of degeneracy, a ground state which is not a macroscopically entangled state) breaks or preserves the aforementioned ∏ j X j {\displaystyle \prod _{j}X_{j}} spin-flip symmetry. The sign of J {\displaystyle J} does not impact the dynamics, as the system with positive J {\displaystyle J} can be mapped into the system with negative J {\displaystyle J} by performing a π {\displaystyle \pi } rotation around X j {\displaystyle X_{j}} for every second site j {\displaystyle j} . The model can be exactly solved for all coupling constants. However, in terms of on-site spins the solution is generally very inconvenient to write down explicitly in terms of the spin variables. It is more convenient to write the solution explicitly in terms of fermionic variables defined by Jordan-Wigner transformation, in which case the excited states have a simple quasiparticle or quasihole description.
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