
Transverse-field Ising model

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 formula_1 axis, as well as an external magnetic field perpendicular to the formula_1 axis (without loss of generality, along the formula_3 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 formula_3 axis and the spin projection along the formula_1 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:
Here, the subscripts refer to lattice sites, and the sum formula_7 is done over pairs of nearest neighbour sites formula_8 and formula_9. formula_10 and formula_11 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. formula_12 is a prefactor with dimensions of energy, and formula_13 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 formula_14 and formula_15 are normalised to each have determinant -1. The Hamiltonian possesses a formula_16 symmetry group, as it is invariant under the unitary operation of flipping all of the spins in the formula_1 direction. More precisely, the symmetry transformation is given by the unitary formula_18.
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 formula_18 spin-flip symmetry. The sign of formula_12 does not impact the dynamics, as the system with positive formula_12 can be mapped into the system with negative formula_12 by performing a formula_23 rotation around formula_10 for every second site formula_9.
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.
Ordered phase.
When formula_26, the system is said to be in the ordered phase. In this phase the ground state breaks the spin-flip symmetry. Thus, the ground state is in fact two-fold degenerate. For formula_27 this phase exhibits ferromagnetic ordering, while for formula_28 antiferromagnetic ordering exists.
Precisely, if formula_29 is a ground state of the Hamiltonian, then formula_30 is also a ground state, and together formula_31 and formula_32 span the degenerate ground state space. As a simple example, when formula_33 and formula_34, the ground states are formula_35 and formula_36, that is, with all the spins aligned along the formula_1 axis.
This is a gapped phase, meaning that the lowest energy excited state(s) have an energy higher than the ground state energy by a nonzero amount (nonvanishing in the thermodynamic limit). In particular, this energy gap is formula_38.
Disordered phase.
In contrast, when formula_39, the system is said to be in the disordered phase. The ground state preserves the spin-flip symmetry, and is nondegenerate. As a simple example, when formula_13 is infinity, the ground state is formula_41, that is with the spin in the formula_42 direction on each site.
This is also a gapped phase. The energy gap is formula_43
Gapless phase.
When formula_44, the system undergoes a quantum phase transition. At this value of formula_45, the system has gapless excitations and its low-energy behaviour is described by the two-dimensional Ising conformal field theory. This conformal theory has central charge formula_46, and is the simplest of the unitary minimal models with central charge less than 1. Besides the identity operator, the theory has two primary fields, one with scaling dimensions formula_47 and another one with scaling dimensions formula_48.
Jordan-Wigner transformation.
It is possible to rewrite the spin variables as fermionic variables, using a highly nonlocal transformation known as the Jordan-Wigner Transformation.
A fermion creation operator on site formula_49 can be defined as formula_50. 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. formula_51This Hamiltonian fails to conserve total fermion number and does not have the associated formula_52 global continuous symmetry, due to the presence of the formula_53 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 formula_54 and formula_55, the Hamiltonian takes on an even simpler form, formula_56
Kramers-Wannier duality.
A nonlocal mapping of Pauli matrices known as the Kramers–Wannier duality transformation can be done as follows:
formula_57
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 formula_58. This indicates that the model with coupling parameter formula_13 is dual to the model with coupling parameter formula_60, 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 formula_61.
Note that there are some subtle considerations at the boundaries of the Ising chain; as a result of these, the degeneracy and formula_62 symmetry properties of the ordered and disordered phases are changed under the Kramers-Wannier duality.
Generalisations.
The q-state quantum Potts model and the formula_63 quantum clock model are generalisations of the transverse field Ising model to lattice systems with formula_64 states per site. The transverse field Ising model represents the case where formula_65 .
Classical Ising Model.
The quantum transverse field Ising model in formula_66 dimensions is dual to an anisotropic classical Ising model in formula_67 dimensions.