Jordan-Wigner transform

We have already seen a few examples in this course of how one type of operator (spin, boson, fermion) can be represented in terms of another. The Holstein-Primakoff representation of spin operators in terms of bosons was one example, and the Schrieffer-Wolff transformation for the low-energy excitations of the Hubbard or single-impurity Anderson model took us from a representation of a fermion $c_{d\sigma}$ to one of a spin.

In one dimension, there is a deep connection between the physics of fermions, bosons, and spins that does not hold in higher dimensions. There are several different hand-waving ways of expressing what is special in one dimension: one way to put it is that particle statistics are defined in terms of exchanges of particles, and in one dimension any exchange requires that the particles pass through each other (collide), which is not true in higher dimensions. One-dimensional systems are realized in carbon nanotubes, some organic compounds, and artificially fabricated "quantum wires." One-dimensional quantum systems are also important because they sometimes describe experimentally important systems in higher dimensions (such as the Kondo effect: the low-energy physics is dominated by the spherically symmetric $s$ channel, so that the problem is effectively radial), and because they provide solvable examples of interacting quantum problems.

The simplest such statistics-changing transformation in 1D is the Jordan-Winger transformation. The $z$ part is local,
$$\begin{equation} S_i^z = {2 c_i^\dagger c_i -1 \over 2}, \end{equation}$$
but we need a nonlocal "string" for raising and lowering operators because spins on different sites commute, while fermions anticommute. The representation we use is
$$\begin{eqnarray} S_i^+ &=& c^\dagger_i \prod_{j < i} (1 - 2 c^\dagger_j c_j), \cr S_i^- &=& \prod_{j < i} (1 - 2 c^\dagger_j c_j) c_i. \end{eqnarray}$$
You can check that these preserve the spin correlation functions: in terms of $\sigma = 2 S$,
$$\begin{eqnarray} [\sigma_i^+,\sigma^-_j] = \delta_{ij} \sigma^Z_i, \quad [\sigma_i^z,\sigma^\pm_j] = \pm 2 \delta_{ij} \sigma^\pm_i \end{eqnarray}$$
and ordinary fermionic commutation relations. The reason the Jordan-Wigner transform works is very simple: the string is cooked up so that it changes sign from +1 to -1 depending on whether the number of fermions to the left of site $i$ is even or odd.

How this connects to numerical methods: one-dimensional systems are also much simpler numerically because of the absence of a "fermion sign problem" in Monte Carlo simulation. Monte Carlo methods essentially calculate an integral for a quantity like the partition function or correlation function by random sampling. The problem with fermions is that, because the wavefunction must change sign under exchange of any two fermions, such a simulation (or a series expansion) must generate terms of both signs. This causes difficulty when the actual quantity to be calculated is much smaller in magnitude than a typical term (as often happens), since the behavior of partial results fluctuates wildly.
If all the resulting values obtained by this sampling process are of the same sign (so that the overall magnitude of the answer is necessarily much larger than that of each individual term) then truncation errors are much less severe.

Other numerical methods like the density-matrix renormalization group, which is essentially an extension of Wilson's iterative numerical RG approach to 1D chains, are also extremely successful in one dimension for low-energy states. This DMRG method has been used to calculate the Haldane gap in the spin-1 chain to many decimal places.

Now we use the above Jordan-Wigner transformation to solve the so-called XX chain, which is like the Heisenberg model but with no $z$ coupling:
$$\begin{equation} H_{XX} = {J \over 4} \sum_i (\sigma_i^+ \sigma_{i+1}^- + \sigma_i^- \sigma_{i+1}^+). \end{equation}$$
Using the Jordan-Wigner transformation gives
$$\begin{eqnarray} H_{XX} &=& {J \over 4} \sum_i \left[\prod_{j < i} (1 - 2 c^\dagger_j c_j)^2 \left(c_i (1-2 c^\dagger_i c_i) c^\dagger_{i+1}+ c^\dagger_i (1-2 c^\dagger_i c_i) c_{i+1} \right)\right]\cr &=& {J \over 4} \sum_i \left(c_i (1-2 c^\dagger_i c_i) c^\dagger_{i+1}+ c^\dagger_i (1-2 c^\dagger_i c_i) c_{i+1}\right). \end{eqnarray}$$
To simplify the terms in parentheses, note that the first $(1-2 c^\dagger_i c_i)$ term gives a $-$ sign. So we are left with
$$\begin{equation} H_{XX} = {J \over 4} \sum_i (c^\dagger_{i+1} c_i +c^\dagger_i c_{i+1} ). \end{equation}$$
This is just a tight-binding model for a single fermion, which we know how to solve: the solution consists of a band of free fermions (in the continuum limit) with energies $ - J / 2 \leq E \leq J / 2 $.

So this very simple mapping tells us that the entire spectrum of the apparently nontrivial XX chain is given just by free fermions! There are two obvious $E=0$ states: the state of all fermion states occupied (minimum $S_z$) and all fermion states empty (maximum $S_z$). The mapping to fermions also shows that not every bond can be optimized for this spin chain in the antiferromagnetic case: if every bond were optimized, the ground state energy would be ${J \over 2}$ per bond, while the spinless fermions have worse energy than this since the average energy of the $N/2$ occupied states ($N$ the number of sites) is greater than $J/2$. The physics of the XX chain is somewhere in between that of the Ising chain and the Heisenberg chain.

Suppose now we consider the $XXZ$ chain, which is obtained by adding a term $\lambda J s^z_i s^z_{i+1}$ to the $XX$ Hamiltonian. Then for $\lambda=0$ we obtain the $XX$ chain and for $\lambda=1$ the ordinary Heisenberg model. The $s^z_i s^z_{i+1}$ interaction becomes a four-fermion interaction in the fermionic representation; these can be treated by a further mapping known as bosonization. It turns out that for all values $0 \leq \lambda \leq 1$, the $XXZ$ chain can be connected to a free bosonic field theory! The main power of bosonization is that, even though it is only valid at low energies, free bosons are such a simple theory that nearly everything can be calculated exactly in the low-energy, long-distance limit. The textbook of Fradkin has a good discussion of this topic.

(For attractive interactions, which are less physically motivated at least for electrons, the system also remains gapless up to the ferromagnetic point at $\lambda = 0$. To understand why this is ferromagnetic, note that we can make a canonical transformation flipping two of the three spin components on one sublattice, which makes this equivalent to the ferromagnet.)

A related fact only realized in the 1980s is that the general state of interacting metallic fermions in 1D is the so-called Tomonaga-Luttinger liquid: interacting fermions become well described by free bosons, and the elementary excitations of the system are bosonic rather than fermionic. Tomonaga and Luttinger independently solved a model with this property much earlier; Haldane showed that this model's behavior was actually quite general, in the sense that many 1D problems were adiabatically continuable to the TL model just as many 2D and 3D problems are continuable to the Fermi gas.