Quantum Mathematics Learning Guide 46: Quantum Spin Chains, Jordan–Wigner Transformation, Free Fermions and Criticality

The Jordan–Wigner transformation is one of the clearest examples of a mathematical change of representation solving a quantum many-body problem. A chain of interacting spins becomes a chain of noninteracting fermionic quasiparticles—provided the nonlocal parity strings and boundary conditions are handled correctly.

Guide 45 introduced many-body Hamiltonians, gaps and criticality. This guide makes those ideas concrete in one dimension. We begin with spin-1/2 Pauli operators, define fermionic creation and annihilation operators through Jordan–Wigner strings, transform the transverse-field Ising chain to a quadratic fermion Hamiltonian, Fourier transform to momentum space, then use a Bogoliubov rotation to expose the excitation spectrum.

The payoff is exact: the thermodynamic excitation gap closes at the critical field, the dynamic exponent is z=1, the correlation-length exponent is ν=1, and the model provides a canonical bridge among lattice spins, free fermions and quantum phase transitions.

Spin chain → nonlocal parity string → fermions → momentum modes → Bogoliubov quasiparticles → exact dispersion → critical scaling.

1. Spin-1/2 chain algebra

At site j, Pauli matrices X_j, Y_j, Z_j satisfy

X_jY_j=iZ_j

and cyclic permutations, while operators on different sites commute.

Define spin raising/lowering operators

σ_j^+=(X_j+iY_j)/2

and

σ_j^-=(X_j-iY_j)/2.

They change the local Z eigenstate. On different sites these spin operators commute, unlike fermionic operators.

2. Fermionic canonical anticommutation relations

Fermionic operators c_j,c_j† must satisfy

• {c_j,c_k}=0;
• {c_j†,c_k†}=0;
• {c_j,c_k†}=δ_jk.

The anticommutator is {A,B}=AB+BA.

The local fermion number is

n_j=c_j†c_j

with eigenvalues 0 and 1 because c_j†c_j†=0 forbids double occupation of one fermionic mode.

3. Jordan–Wigner map: state the convention first

Use the convention

c_j=(∏_{k<j} Z_k)σ_j^+.

Then

c_j†=(∏_{k<j} Z_k)σ_j^-

and

n_j=(1−Z_j)/2, so Z_j=1−2n_j.

Other references swap σ⁺ and σ⁻ or reverse the occupation convention. Those choices change signs in intermediate formulas but not physical spectra when used consistently.

4. Why the parity string is necessary

Without the product ∏_{k<j}Z_k, lowering/raising operators on different spin sites commute. Fermions must anticommute.

The parity string contributes a minus sign whenever one fermionic operator passes another site whose occupation parity matters.

For j<k, the string in c_k contains Z_j, which anticommutes with σ_j⁺. That single anticommutation supplies the required sign:

c_jc_k=−c_kc_j.

The transformation is nonlocal in spins even though it creates local fermionic operators in one dimension.

5. Number operator check

Because the string squares to identity,

n_j=c_j†c_j=σ_j^-σ_j^+=(1−Z_j)/2.

Therefore Z=+1 corresponds to n=0 and Z=−1 corresponds to n=1 under this convention.

This simple identity turns longitudinal Z fields into fermionic chemical-potential terms.

6. Nearest-neighbour strings cancel

The Jordan–Wigner map looks highly nonlocal, but for nearest-neighbour spin interactions most strings cancel.

For adjacent sites j and j+1, the two strings differ only by Z_j. Products such as X_jX_j+1 therefore become quadratic combinations of c_j, c_j†, c_j+1, c_j+1† rather than long nonlocal products.

This cancellation is why one-dimensional nearest-neighbour spin models can become free-fermion problems.

7. Transverse-field Ising chain

Consider the open-chain Hamiltonian

H=−JΣ_{j=1}^{N−1}X_jX_{j+1}−hΣ_{j=1}^{N}Z_j

with J>0.

At h=0, the interaction favours X-aligned ferromagnetic order. At J=0, the field favours the product state with Z=+1 at every site.

The competition between these noncommuting terms produces a quantum critical point at |h/J|=1 in the thermodynamic limit.

8. Fermionised Ising Hamiltonian

Under the stated Jordan–Wigner convention, the open-chain Hamiltonian can be written, up to equivalent operator-order signs, as the quadratic form

H=−JΣ_{j=1}^{N−1}(c_j†−c_j)(c_{j+1}†+c_{j+1})−hΣ_j(1−2n_j).

There are hopping terms c†c, pairing terms c†c†+cc and number terms n.

The Hamiltonian is not number conserving because the spin interaction creates and annihilates fermion pairs, but it remains quadratic. Quadratic fermion Hamiltonians are exactly diagonalizable by linear canonical transformations.

9. Periodic boundaries introduce a parity subtlety

If the spin chain is periodic, X_NX₁ crosses the Jordan–Wigner boundary. Its string does not cancel locally.

The boundary term depends on total fermion parity

P_f=(-1)^{Σ_jn_j}=∏_jZ_j.

One must diagonalise even- and odd-parity sectors with corresponding periodic or antiperiodic fermionic boundary conditions.

Dropping this sector dependence can give incorrect finite-size spectra even though the thermodynamic bulk dispersion looks right.

10. Fourier transform

For a translation-invariant bulk system, define momentum fermions

c_j=(1/√N)Σ_k e^{ikj}c_k.

The allowed k values depend on the parity sector and boundary conditions.

Hopping terms become diagonal in k, while pairing couples k and −k. The problem decomposes into independent two-dimensional momentum blocks labelled by {k,−k}.

11. Nambu spinor

Group each momentum pair into a Nambu spinor

Ψ_k=(c_k,c_{−k}†)^T.

Then each {k,−k} block is a 2×2 Bogoliubov–de Gennes matrix. In a common convention with g=h/J, the vector coefficients are proportional to

(sin k, 0, g−cos k).

Diagonalising this 2×2 matrix yields the quasiparticle spectrum.

12. Bogoliubov transformation

Introduce quasiparticles

γ_k=u_k c_k+v_k c_{−k}†

with coefficients chosen so |u_k|²+|v_k|²=1 and the transformed operators satisfy fermionic anticommutation relations.

The Bogoliubov angle θ_k can be defined by

tan(2θ_k)=sin k/(g−cos k)

up to sign and quadrant conventions.

The transformation rotates each momentum pseudospin into its local effective field direction.

13. Exact quasiparticle dispersion

The transverse-field Ising chain has bulk quasiparticle energy

ε(k)=2J√(1+g²−2g cos k), g=h/J.

Equivalently,

ε(k)=2J√[(g−cos k)²+sin²k].

The many-body Hamiltonian becomes

H=E_vac+Σ_k ε(k)γ_k†γ_k

with sector-dependent mode counting and vacuum-energy constants.

14. Gap from the dispersion

For g≥0, the minimum bulk excitation occurs near k=0.

At k=0, cos k=1, so

ε(0)=2J|1−g|.

Thus the thermodynamic quasiparticle gap closes at

g_c=1.

A negative field is related by a simple spin transformation, so |g|=1 marks the corresponding critical scale.

15. Worked gap values

• g=0.5 → Δ=2J(0.5)=J;
• g=0.9 → Δ=0.2J;
• g=1 → Δ=0 in the thermodynamic bulk dispersion;
• g=1.2 → Δ=0.4J.

A finite chain retains a nonzero size-dependent level spacing at the critical coupling, typically scaling as O(1/N) for the lowest conformal modes rather than being exactly zero.

16. Critical dispersion is linear

Set g=1 and expand around small k:

1−cos k≈k²/2

so

ε(k)=2J√(2−2cos k)=4J|sin(k/2)|≈2J|k|.

Energy scales linearly with momentum, which gives dynamical exponent

z=1.

The low-energy theory has an emergent relativistic scaling structure with an effective velocity set by J and lattice spacing.

17. Correlation-length exponent

Near g=1, the gap behaves

Δ∝|g−1|.

Using Δ∼ξ^{-z} and z=1 gives

ξ∼|g−1|^{-1},

so the correlation-length exponent is

ν=1.

This is the universality class of the two-dimensional classical Ising model, reflecting the quantum-to-classical correspondence between one spatial dimension plus imaginary time and two classical dimensions.

18. Ordered and paramagnetic phases

For 0≤g<1, interaction J dominates and the thermodynamic phase has ferromagnetic order in X.

For g>1, the transverse field dominates and the ground state is continuously connected to the Z-polarised product state.

The transition at g=1 separates two gapped phases whose symmetry realisation differs.

19. Exact spontaneous magnetisation

In the thermodynamic ordered phase, the spontaneous X magnetisation is

m_x=(1−g²)^{1/8}, 0≤g<1

and m_x=0 for g≥1.

As g→1⁻,

m_x∼(1−g)^{1/8}

up to a constant factor, so the order-parameter critical exponent is β=1/8.

Finite symmetry-preserving chains can still have ⟨X⟩=0; spontaneous magnetisation refers to the thermodynamic symmetry-broken limit.

20. Jordan–Wigner reveals domain walls as fermions

In the ordered phase, low-energy excitations can be viewed as domain walls between regions of opposite X magnetisation.

The fermionic quasiparticles created by γ_k† encode these collective excitations rather than literal electrons.

“Fermionisation” is therefore a change of mathematical variables: spin excitations obey an emergent fermionic algebra even when the underlying hardware is a chain of qubits or magnetic moments.

21. XY chain generalisation

The anisotropic XY chain

H=−Σ_j[(J_x X_jX_{j+1}+J_yY_jY_{j+1})/2+hZ_j]

also maps to a quadratic fermion Hamiltonian.

The combination J_x+J_y produces hopping while J_x−J_y produces pairing. The isotropic XX limit has no pairing and conserves fermion number.

This connects spin chains to one-dimensional superconducting/Kitaev-chain mathematics.

22. Majorana operators

Each complex fermion can be split into two Hermitian Majorana operators:

γ_{2j−1}=c_j+c_j†

and

γ_{2j}=i(c_j†−c_j).

They satisfy {γ_a,γ_b}=2δ_ab.

Quadratic fermion Hamiltonians become bilinear forms (i/4)ΣA_abγ_aγ_b with real antisymmetric A. Diagonalisation reduces to canonical blocks of this antisymmetric matrix.

23. Edge Majoranas and open chains

In the ordered/topological regime of the equivalent Kitaev-chain description, an open chain supports Majorana modes localised near its ends in the thermodynamic limit.

Their overlap falls exponentially with chain length, producing an exponentially small splitting between parity sectors.

This gives another explanation for the near-degenerate ferromagnetic ground-state pair of a long open Ising chain.

24. Wick’s theorem

Ground states of quadratic fermion Hamiltonians are Gaussian fermionic states.

Wick’s theorem expresses higher-order fermion correlators as sums of products of two-point correlators.

Thus a large class of observables can be computed from the covariance matrix

Γ_ab=(i/2)⟨[γ_a,γ_b]⟩.

This finite matrix replaces explicit storage of 2^N wavefunction amplitudes for free-fermion states.

25. Integrability

Because the transformed Hamiltonian decomposes into independent quasiparticle occupations, every occupation number γ_k†γ_k is conserved.

This extensive set of conserved quantities makes the model integrable.

Guide 47 contrasts this with quantum-chaotic models, where generic symmetry-resolved level statistics resemble random-matrix ensembles and local observables often satisfy eigenstate thermalisation.

26. Quench dynamics

Suppose the chain begins in the ground state of g_i and the field is suddenly changed to g_f.

The initial state is not generally the new quasiparticle vacuum. It contains pairs of ±k excitations relative to the final Hamiltonian.

Because each k pair evolves independently, time-dependent correlators can be computed exactly from mode phases e^{-iε_k t}.

Dephasing among many frequencies produces apparent relaxation of local observables even though the global state evolves unitarily.

27. Kibble–Zurek scaling

If g is ramped slowly through the critical point, the closing gap prevents perfectly adiabatic evolution for any finite ramp rate in the thermodynamic limit.

Kibble–Zurek reasoning predicts a freeze-out length scale

ξ_hat∼τ_Q^{ν/(1+zν)}

for a linear ramp timescale τ_Q.

With Ising exponents z=ν=1,

ξ_hat∼τ_Q^{1/2}

and defect density scales roughly as τ_Q^−1/2 in one dimension under the ideal scaling assumptions.

28. Worked critical-mode estimate

At g=1, take a small momentum k=0.05 and J=1.

The exact dispersion is

ε=2√(2−2cos0.05).

Since 2−2cos0.05≈0.00249948,

ε≈2(0.0499948)=0.09999.

The linear approximation 2|k|=0.10 is already accurate to roughly 10⁻⁴ relative scale here.

29. Common misconception: Jordan–Wigner makes every spin model free

The map always converts spins to fermions in one dimension, but generic spin interactions can become quartic or higher-order fermion interactions. Exact free-fermion solvability is special to Hamiltonians whose mapped form remains quadratic.

30. Common misconception: the parity string is an optional sign convention

The string is essential for fermionic anticommutation. Different conventions can move signs around, but some nonlocal parity bookkeeping must remain.

31. Common misconception: periodic spins mean periodic fermions

Jordan–Wigner boundary conditions depend on total fermion parity. Periodic spin chains split into parity sectors with different fermionic momentum quantisation.

32. Worked synthesis problem

For the thermodynamic transverse-field Ising model with J=1 and g=0.96:

Step 1: Gap. Δ=2|1−0.96|=0.08.

Step 2: Characteristic time. In ℏ=1 units, 1/Δ=12.5 is the natural inverse-gap scale.

Step 3: Correlation length. Critical scaling with ν=1 suggests ξ∝|1−g|^{-1}≈25 up to a nonuniversal prefactor.

Step 4: Order parameter. m_x=(1−0.96²)^{1/8}=(0.0784)^{1/8}≈0.727 in the thermodynamic symmetry-broken convention.

Step 5: Finite chain. If N is comparable with or smaller than ξ, finite-size effects are strong and thermodynamic formulas should not be interpreted as exact finite-N observables.

33. Practice set

1. State the fermionic canonical anticommutation relations.
2. Write the Jordan–Wigner map used in this guide.
3. What is n_j in terms of Z_j?
4. Why is the Z-string needed?
5. Why do nearest-neighbour strings mostly cancel?
6. Write the transverse-field Ising Hamiltonian.
7. What subtlety appears for periodic boundaries?
8. What does the Fourier transform accomplish?
9. Why is a Bogoliubov transform needed?
10. State the Ising quasiparticle dispersion.
11. Where does the thermodynamic gap close?
12. What are z and ν for the Ising critical point?

Answers

1. {c_j,c_k}=0, {c_j†,c_k†}=0, {c_j,c_k†}=δ_jk.
2. c_j=(∏_{k<j}Z_k)σ_j^+.
3. (1−Z_j)/2.
4. It generates the minus signs required when fermionic operators on different sites are exchanged.
5. The two adjacent strings differ by only one local parity operator, leaving a quadratic local fermion expression.
6. −JΣX_jX_{j+1}−hΣZ_j.
7. The boundary term depends on total fermion parity, producing different fermionic boundary conditions in different sectors.
8. It diagonalises translation-invariant hopping and reduces the problem to momentum pairs k and −k.
9. The fermion Hamiltonian contains pairing terms mixing c_k with c_{−k}†.
10. 2J√(1+g²−2gcosk).
11. At |g|=1.
12. z=1 and ν=1.

Sources and further study

[1] E. Lieb, T. Schultz and D. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics 16, 407–466 (1961). The classic exact spin-chain fermionisation treatment.

[2] P. Jordan and E. Wigner, Über das Paulische Äquivalenzverbot, Zeitschrift für Physik 47, 631–651 (1928). The original Jordan–Wigner transformation.

[3] Pierre Pfeuty, The one-dimensional Ising model with a transverse field, Annals of Physics 57, 79–90 (1970). Exact ground-state and correlation analysis of the transverse-field Ising chain.

[4] Subir Sachdev, Quantum Phase Transitions. The Ising chain as a central example of quantum criticality and universality.

Continue through Quantum Mathematics

Guide 45: Quantum Many-Body Hamiltonians, Ground States, Correlation Functions and Phase Transitions supplies the general phase language. Guide 47: Quantum Chaos, Random Matrices, Level Statistics and Scrambling contrasts integrable free-fermion spectra with chaotic many-body spectra. Guide 48: Quantum Thermodynamics, Gibbs States, Free Energy, Work and Resource Theories asks how energy, entropy and thermalisation constrain usable work.

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