# Jordan-Wigner transformation

1. Jul 23, 2008

### yukawa

How to go from this Hamiltonian :

$$H= \sum^{N}_{j=1}[\sigma^{+}_{j}\sigma^{-}_{j+1}+\sigma^{-}_{j}\sigma^{+}_{j+1}]$$

to the following Hamiltonain:

$$H = \sum^{N-1}_{j=1}[c^{*}_{j}c_{j+1}+c^{*}_{j+1}c_{j}]-(c^{*}_{1}c_{N}+c^{*}_{N}c_{1})exp[i\pi\sum^{N}_{j=1}c^{*}_{j}c_{j}]$$

(where c* is the Hermitian conjugate of c)

using the Jordan-Wigner transformation:
$$\sigma^{+}_{j} = exp[i\pi\sum^{j-1}_{n=1}c^{*}_{n}c_{n}] c_{j}$$

$$\sigma^{-}_{j} = exp[-i\pi\sum^{j-1}_{n=1}c^{*}_{n}c_{n}] c^{*}_{j}$$

where are fermionic operators.

The following is what I have calculated, please correct my mistakes if there are.

$$\sigma^{+}_{j}\sigma^{-}_{j+1}=c_{j}exp[-i\pi c^{*}_{j}c_{j}]c^{*}_{j+1}=c_{j}(1-2c^{*}_{j}c_{j})c^{*}_{j+1}$$

$$\sigma^{-}_{j}\sigma^{+}_{j+1}=c^{*}_{j}exp[i\pi c^{*}_{j}c_{j}]c_{j+1}=c^{*}_{j}(1-2c^{*}_{j}c_{j})c_{j+1}$$

putting them into the first Hamiltonian yeilds:

$$\sum^{N}_{j=1}c_{j}(1-2c^{*}_{j}c_{j})c^{*}_{j+1}+c^{*}_{j}(1-2c^{*}_{j}c_{j})c_{j+1}$$

how to proceed further in order to arrive at the second Hamiltonian?