QAtlas (QUAntum Reference Table for Exact Tests) is a Julia package providing a curated dictionary of rigorous results in quantum and statistical physics. Every stored value is traced to a specific publication and cross-validated against independent calculations.
Status and Disclaimer
This package is under active development with AI-assisted code generation. While all stored values cite specific publications and are cross-checked against independent numerical computations (exact diagonalization, brute-force enumeration, automatic differentiation), the maintainer has not personally verified every derivation in equal depth.
Results that the maintainer can independently derive and has verified line-by-line (TFIM exact solution, Onsager critical temperature, Yang magnetization, Heisenberg dimer) carry the highest confidence. Other results (tight-binding Bloch formulas for various lattices, numerical universality exponents transcribed from the literature, E8 mass ratios) are tested against independent computations but the underlying derivations have not been checked by hand.
If you use QAtlas values in a publication, please verify them against the cited original references. If you find a discrepancy, please open an issue.
using QAtlas
# Onsager's critical temperature for the 2D Ising model
Tc = QAtlas.fetch(IsingSquare(), CriticalTemperature()) # 2J / ln(1 + √2)
# Yang's spontaneous magnetization
M = QAtlas.fetch(IsingSquare(), SpontaneousMagnetization(); β=0.5)
# 2D Ising universality class: exact critical exponents
e = QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=2)
# (β = 1//8, ν = 1//1, γ = 7//4, η = 1//4, δ = 15//1, α = 0//1, c = 1//2)
# 3D Ising: numerical estimates with uncertainty
e3 = QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=3)
# (α = 0.11009, α_err = 1.0e-5, β = 0.32642, β_err = 1.0e-5, ...)
# Graphene tight-binding spectrum (honeycomb lattice)
λ = QAtlas.fetch(Graphene(), TightBindingSpectrum(); Lx=3, Ly=3, t=1.0)
# Bethe ansatz: Heisenberg chain ground-state energy density
e0 = QAtlas.fetch(Heisenberg1D(), GroundStateEnergyDensity()) # 1/4 − ln 2
# Classical Ising partition function via transfer matrix
Z = QAtlas.fetch(IsingSquare(), PartitionFunction(); Lx=4, Ly=4, β=0.44)| Model | Quantities | Method |
|---|---|---|
| IsingSquare | Partition function Z(Lx, Ly, β) | Transfer matrix |
| Critical temperature T_c | Onsager (1944) exact | |
| Spontaneous magnetization M(T) | Yang (1952) exact | |
| Graphene (honeycomb TB) | Single-particle spectrum | Bloch Hamiltonian |
| Kagome TB | Spectrum + flat band at +2t | Bloch 3×3 |
| Lieb TB | Spectrum + flat band at E=0 | Bloch 3×3 |
| Triangular TB | Spectrum (frustrated) | Bloch scalar |
| Heisenberg1D | Dimer spectrum {−3J/4, J/4³} | Exact |
| 4-site PBC spectrum | Exact | |
| Ground-state energy density e₀ | Bethe ansatz | |
| TFIM | Energy, free energy, entropy, ... | BdG + quadrature |
Access via Universality{C} with dimension d:
| Class | d | Type | Key exponents |
|---|---|---|---|
| Ising | 2 | Exact (Rational) | β=1/8, ν=1, c=1/2 |
| 3 | Numerical (+err) | Conformal bootstrap | |
| ≥4 | Mean-field | Landau | |
| Percolation | 2 | Exact | β=5/36, ν=4/3 |
| 3-state Potts | 2 | Exact | β=1/9 |
| 4-state Potts | 2 | Exact | β=1/12 |
| XY | 2 | BKT | η=1/4 |
| 3 | Numerical | Bootstrap | |
| Heisenberg | 3 | Numerical | Bootstrap |
| KPZ | 1+1D | Exact | β=1/3, z=3/2 |
| Mean-field | any | Exact | β=1/2, ν=1/2 |
Exact values use Rational{Int} — scaling relations hold with zero floating-point error:
e = QAtlas.fetch(Universality(:Ising), CriticalExponents(); d=2)
e.α + 2*e.β + e.γ == 2 # Rushbrooke: exactly true
e.γ == e.β * (e.δ - 1) # Widom: exactly trueQAtlas is self-verifying: every result in src/ is cross-checked against independent calculations in test/. The test suite (800+ tests) includes:
- Standalone tests: Special values, scaling relations, limiting cases
- Lattice verification: Real-space ED and Bloch diagonalization via Lattice2D.jl on all 8 supported topologies
- AD verification: ForwardDiff-based extraction of thermodynamic quantities from partition functions
- Cross-verification: Universality exponents extracted from model-specific results (e.g., β=1/8 from Yang magnetization, z=1 from TFIM gap scaling)
The partition function and related functions accept ForwardDiff.Dual numbers, enabling automatic differentiation of thermodynamic quantities:
using ForwardDiff
# Internal energy from partition function
E = -ForwardDiff.derivative(β -> log(QAtlas.fetch(
IsingSquare(), PartitionFunction(); Lx=3, Ly=3, β=β)), 0.5)
# Heat capacity via second derivative
C = β² * ForwardDiff.derivative(
β -> ForwardDiff.derivative(β -> log(Z(β)), β), β)using Pkg
Pkg.add("QAtlas")See CONTRIBUTING.md for guidelines on adding new results, writing tests, and the citation standards we follow.
MIT License. See LICENSE for details.