O(n) Models: XY and Heisenberg
Overview
The O($n$) model describes a classical spin $\mathbf{S}_i \in \mathbb{R}^n$ with $|\mathbf{S}_i| = 1$ on each lattice site, coupled by the Hamiltonian
\[H = -J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j\]
The global symmetry is the orthogonal group O($n$). Special cases:
| $n$ | Name | Symmetry | Physical realisations |
|---|---|---|---|
| 1 | Ising | $\mathbb{Z}_2$ | Uniaxial magnets, lattice gas |
| 2 | XY | O(2) $\cong$ U(1) | Superfluid $^4$He, thin-film magnets |
| 3 | Heisenberg | O(3) $\cong$ SU(2) | Isotropic magnets (EuO, EuS) |
XY Model ($n = 2$)
$d = 2$ –- Berezinskii-Kosterlitz-Thouless (BKT) Transition
The 2D XY model has no spontaneous symmetry breaking at any $T > 0$ (Mermin-Wagner theorem). Instead, it undergoes a BKT transition at $T_{\mathrm{BKT}}$: below $T_{\mathrm{BKT}}$ correlations decay algebraically (quasi-long-range order), while above they decay exponentially.
The BKT transition is not described by standard power-law critical exponents. Its hallmarks are:
| Property | Value / behaviour | Note |
|---|---|---|
| $\eta(T_{\mathrm{BKT}})$ | $1/4$ | Universal jump at the transition |
| Correlation length above $T_{\mathrm{BKT}}$ | $\xi \sim \exp(b/\sqrt{T - T_{\mathrm{BKT}}})$ | Essential singularity, not power law |
| Superfluid stiffness | Universal jump $\rho_s(T_{\mathrm{BKT}}^-) = 2T_{\mathrm{BKT}}/\pi$ | Nelson-Kosterlitz criterion |
| Vortex-antivortex pairs | Bound for $T < T_{\mathrm{BKT}}$, unbound above | Topological defect mechanism |
The BKT transition has $\nu = \infty$ in the usual sense (essential singularity). Standard exponents $\beta, \gamma, \delta$ are not defined for this transition because there is no true order parameter. QAtlas stores the universal value $\eta = 1/4$ at $T_c$ and flags the BKT nature.
$d = 3$ –- Conformal Bootstrap
In $d = 3$, the XY model has a conventional second-order transition. The most precise exponents come from the O(2) conformal bootstrap.
| Exponent | Value | Uncertainty | Reference |
|---|---|---|---|
| $\alpha$ | $-0.0146(8)$ | — | Chester, Landry, Liu, Poland, Simmons-Duffin, Su, Vichi (2020) |
| $\beta$ | $0.3485(2)$ | — | JHEP 2020, 142 |
| $\gamma$ | $1.3177(5)$ | — | " |
| $\nu$ | $0.6717(1)$ | — | " |
| $\eta$ | $0.0381(2)$ | — | " |
Heisenberg Model ($n = 3$)
$d \leq 2$ –- Mermin-Wagner Theorem
The Mermin-Wagner theorem forbids spontaneous breaking of a continuous symmetry in $d \leq 2$ at $T > 0$ for short-range interactions. Consequently:
- $d = 1$: The 1D Heisenberg chain is disordered at all $T > 0$. At $T = 0$ the quantum spin-1/2 chain is critical with $c = 1$ (Luttinger liquid), described by the SU(2)$_1$ WZW model.
- $d = 2$: The 2D classical Heisenberg model has no finite-$T$ phase transition. (In contrast, the 2D XY model has the BKT transition, which does not break the continuous symmetry.)
$d = 3$ –- Conformal Bootstrap
In $d = 3$, the Heisenberg model has a conventional second-order transition. Best estimates from the O(3) conformal bootstrap:
| Exponent | Value | Uncertainty | Reference |
|---|---|---|---|
| $\alpha$ | $-0.1336(15)$ | — | Chester, Landry, Liu, Poland, Simmons-Duffin, Su, Vichi (2020) |
| $\beta$ | $0.3689(3)$ | — | JHEP 2020, 142 |
| $\gamma$ | $1.3960(9)$ | — | " |
| $\nu$ | $0.7112(5)$ | — | " |
| $\eta$ | $0.0378(3)$ | — | " |
$d \geq 4$ –- Mean-Field
The upper critical dimension for O($n$) models (all $n \geq 1$) is $d_c = 4$. For $d \geq 4$ the exponents take mean-field values: $\beta = 1/2$, $\nu = 1/2$, $\gamma = 1$, $\eta = 0$.
QAtlas API
using QAtlas
# XY d = 3: numerical (Float64 + _err)
e_xy = QAtlas.fetch(Universality(:XY), CriticalExponents(); d=3)
# (β = 0.3485, β_err = 2e-4, ν = 0.6717, ν_err = 1e-4, ...)
# Heisenberg d = 3
e_heis = QAtlas.fetch(Universality(:Heisenberg), CriticalExponents(); d=3)
# (β = 0.3689, β_err = 3e-4, ν = 0.7112, ν_err = 5e-4, ...)
# d ≥ 4: mean-field
e_mf = QAtlas.fetch(Universality(:XY), CriticalExponents(); d=4)
# → same as fetch(MeanField(), CriticalExponents())References
- V. L. Berezinskii, Sov. Phys. JETP 32, 493 (1971) –- BKT transition (part I).
- J. M. Kosterlitz, D. J. Thouless, J. Phys. C 6, 1181 (1973) –- BKT transition.
- N. D. Mermin, H. Wagner, Phys. Rev. Lett. 17, 1133 (1966) –- absence of long-range order in $d \leq 2$.
- S. M. Chester, W. Landry, J. Liu, D. Poland, D. Simmons-Duffin, N. Su, A. Vichi, "Carving out OPE space and precise O(2) model critical exponents", JHEP 2020, 142 –- O(2) and O(3) bootstrap.
- D. R. Nelson, J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977) –- universal superfluid-stiffness jump.
Connections
- Ising: the $n = 1$ case; see Ising.
- Heisenberg chain ($T = 0$): quantum $S = 1/2$ chain with $c = 1$; see Heisenberg model.
- Mean-Field: baseline for $d \geq 4$; see mean-field.