XY Model

One-Sentence Description

The XY model is a spin chain with exchange in the $x$ and $y$ directions, often used as the solvable bridge between Ising-like chains and free fermions.

Physical Setup

Place spin-$1/2$ degrees of freedom on a one-dimensional lattice. The interaction couples only the planar spin components, and a transverse field usually points along $z$.

The model is a standard setting for Jordan–Wigner fermionization, anisotropy, parity sectors, and exactly solvable quantum critical behavior in one dimension.

Hilbert Space

For $N$ spin-$1/2$ sites,

$$\mathcal H = \bigotimes_{j=1}^{N}\mathbb C^2.$$

Open and periodic chains require separate treatment because the Jordan–Wigner transformation handles boundary terms differently.

Hamiltonian

A common anisotropic transverse-field convention is

$$H = -\frac{J}{2} \sum_j \left[ (1+\gamma)\sigma_j^x\sigma_{j+1}^x + (1-\gamma)\sigma_j^y\sigma_{j+1}^y \right] - h\sum_j\sigma_j^z.$$

Here $\gamma$ is the anisotropy. The $\gamma=0$ case is the XX model, while the $\gamma=1$ limit leaves only the $x$ exchange term in this convention and is closely related to the transverse-field Ising chain after relabeling axes.

Parameters

Symbol	Meaning
$J$	planar exchange scale
$\gamma$	anisotropy between $x$ and $y$ exchange
$h$	transverse-field strength
$N$	number of sites

Solvability

The one-dimensional XY chain is exactly solvable by the Jordan–Wigner transformation to fermions. For $\gamma\ne0$, the fermion Hamiltonian includes pairing terms and is diagonalized by a Bogoliubov transformation. For $\gamma=0$, the XX chain maps to hopping fermions with number conservation.

Spectrum and Eigenstates

For a periodic chain, the final quasiparticle spectrum depends on parity and boundary-sector conventions. In the thermodynamic limit, the qualitative single-particle dispersion has the form

$$\varepsilon(k) \sim 2J \sqrt{ \left(\frac{h}{J}-\cos k\right)^2 + \gamma^2\sin^2 k }.$$

The proportionality and allowed $k$ values depend on the Hamiltonian normalization and boundary sector.

Key Observables

• Transverse magnetization.
• Planar spin correlation functions.
• Fermion occupation after Jordan–Wigner mapping.
• Energy gap and critical points.
• Entanglement measures in solvable chains.

What It Teaches

The XY model teaches how spin chains can become quadratic fermion problems, how anisotropy changes critical behavior, and why boundary sectors matter in exactly solvable many-body models.

It is also a bridge from simple Pauli-matrix Hamiltonians to quasiparticles, Bogoliubov transformations, and lattice models used in quantum matter.

Canonical Links

• Transverse-Field Ising Model
• XXZ Spin Chain
• Ising Chain
• Heisenberg Chain
• Spin Operator
• Fermionic Anticommutation Relations
• Tensor Product Ordering

Variants

• XX chain with $\gamma=0$;
• anisotropic XY chain;
• transverse-field XY chain;
• open-chain and periodic-chain versions;
• disordered XY chains;
• long-range XY models.

Common Mistakes

• Forgetting that different books place the field along different axes.
• Treating the periodic-chain Jordan–Wigner boundary term as harmless in finite systems.
• Comparing the anisotropy parameter $\gamma$ without checking the factor of $1/2$ in the Hamiltonian.
• Calling every planar spin model an XY model without specifying whether the spin variables are classical or quantum.

Quick Checks

1. What special simplification occurs at $\gamma=0$?

Solution

The model becomes the XX chain. Under Jordan–Wigner transformation it maps to hopping fermions without pairing terms, so fermion number is conserved.

2. Why are open chains often simpler than periodic chains for a first Jordan–Wigner calculation?

Solution

The Jordan–Wigner string is nonlocal. With periodic boundary conditions it produces a boundary term that depends on fermion parity. Open chains avoid that extra sector bookkeeping.

References

• E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain”, Annals of Physics 16, 407-466, 1961.
• E. Barouch and B. M. McCoy, “Statistical mechanics of the XY model. II. Spin-correlation functions”, Physical Review A 3, 786-804, 1971.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, 2011.