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 ${\displaystyle X}$ and ${\displaystyle Z}$ are normalised to each have determinant -1. The Hamiltonian possesses a ${\displaystyle \mathbb {Z} _{2}}$ symmetry group, as it is invariant under the unitary operation of flipping all of the spins in the ${\displaystyle z}$ direction. More precisely, the symmetry transformation is given by the unitary ${\displaystyle \prod _{j}X_{j}}$.

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 ${\displaystyle \prod _{j}X_{j}}$ spin-flip symmetry. The sign of ${\displaystyle J}$ does not impact the dynamics, as the system with positive ${\displaystyle J}$ can be mapped into the system with negative ${\displaystyle J}$ by performing a ${\displaystyle \pi }$ rotation around ${\displaystyle X_{j}}$ for every second site ${\displaystyle j}$.

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.

It is possible to rewrite the spin variables as fermionic variables, using a highly nonlocal transformation known as the Jordan-Wigner Transformation.[3]

A fermion creation operator on site ${\displaystyle j}$ can be defined as ${\displaystyle c_{j}^{\dagger }={\frac {1}{2}}(Z_{j}+iY_{j})\prod _{k<j}X_{k}}$. Then the transverse field Ising Hamiltonian (assuming an infinite chain and ignoring boundary effects) can be expressed entirely as a sum of local quadratic terms containing creation and annihilation operators.

${\displaystyle H=-J\sum _{j}(c_{j}^{\dagger }c_{j+1}+c_{j+1}^{\dagger }c_{j}+c_{j}^{\dagger }c_{j+1}^{\dagger }+c_{j+1}c_{j}-2g(c_{j}^{\dagger }c_{j}-1/2))}$

This Hamiltonian fails to conserve total fermion number and does not have the associated ${\displaystyle U(1)}$ global continuous symmetry, due to the presence of the ${\displaystyle c_{j}^{\dagger }c_{j+1}^{\dagger }+c_{j+1}c_{j}}$ term. However, it does conserve fermion parity. That is, the Hamiltonian commutes with the quantum operator that indicates whether the total number of fermions is even or odd, and this parity does not change under time evolution of the system. The Hamiltonian is mathematically identical to that of a superconductor in the mean field Bogoliubov deGennes formalism and can be completely understood in the same standard way. The exact excitation spectrum and eigenvalues can be determined by Fourier transforming into momentum space and diagonalising the Hamiltonian. In terms of Majorana fermions ${\displaystyle a_{j}=c_{j}^{\dagger }+c_{j}}$ and ${\displaystyle b_{j}=-i(c_{j}^{\dagger }-c_{j})}$, the Hamiltonian takes on an even simpler form,

${\displaystyle H=i\sum _{j}J(a_{j+1}b_{j}+gb_{j}a_{j})}$

A nonlocal mapping of Pauli matrices known as the Kramers–Wannier duality transformation can be done as follows:[4]

$${\displaystyle {\begin{aligned}{\tilde {X_{j}}}&=Z_{j}Z_{j+1}\\{\tilde {Z}}_{j}{\tilde {Z}}_{j+1}&=X_{j+1}\end{aligned}}}$$

Then, in terms of the newly-defined Pauli matrices with tildes, which obey the same algebraic relations as the original Pauli matrices, the Hamiltonian is simply ${\displaystyle H=-Jg\sum _{j}({\tilde {Z}}_{j}{\tilde {Z}}_{j+1}+g^{-1}{\tilde {X}}_{j})}$. This indicates that the model with coupling parameter ${\displaystyle g}$ is dual to the model with coupling parameter ${\displaystyle g^{-1}}$, and establishes a duality between the ordered phase and the disordered phase. In terms of the Majorana fermions mentioned above, this duality is more obviously manifested in the trivial relabeling ${\displaystyle a_{j}\to b_{j},b_{j}\to a_{j+1}}$.

Note that there are some subtle considerations at the boundaries of the Ising chain; as a result of these, the degeneracy and ${\displaystyle \mathbb {Z} _{2}}$ symmetry properties of the ordered and disordered phases are changed under the Kramers-Wannier duality.

The q-state quantum Potts model and the ${\displaystyle Z_{q}}$ quantum clock model are generalisations of the transverse field Ising model to lattice systems with ${\displaystyle q}$ states per site. The transverse field Ising model represents the case where ${\displaystyle q=2}$ .

The quantum transverse field Ising model in ${\displaystyle d}$ dimensions is dual to an anisotropic classical Ising model in ${\displaystyle d+1}$ dimensions.[5]