Mean-Field Theory

Mean-field theory replaces the fluctuating environment seen by each degree of freedom with a field determined self-consistently by expectation values of the approximate state. The original interacting problem is thereby reduced to an effective one-body, one-site, or quadratic problem whose coefficients depend on its own solution.

In its simplest form, mean field replaces a product of operators by terms linear in fluctuations around selected averages. If

$$A = \langle A\rangle + \delta A, \qquad B = \langle B\rangle + \delta B,$$

then the exact identity is

$$AB ={} \langle A\rangle B + A\langle B\rangle - \langle A\rangle \langle B\rangle + \delta A\,\delta B.$$

The elementary mean-field decoupling discards the final quadratic fluctuation operator. The retained averages must then equal the averages computed from the resulting effective problem.

This simple rule is powerful but incomplete on its own. A trustworthy mean-field calculation must also identify the trial family or decoupling channel, retain constant subtraction terms, solve every relevant self-consistency equation, compare thermodynamic potentials, test physical stability, and estimate the discarded fluctuations.

Helpful background. The interaction overview explains dimensionless controls, and the variational principle supplies the ground-state energy bound.

Factorization, variation and stationary fields

Here mean field means a static or equilibrium approximation. Dynamical mean-field theory instead retains a frequency-dependent quantum impurity problem. Static mean-field equations can arise in several mathematically related ways. Keeping the viewpoint explicit prevents accidental overclaims.

Fluctuation factorization

Write each selected operator as mean plus fluctuation and neglect products of fluctuations. This gives a transparent algebraic approximation and exposes what was discarded.

Restricted variation

Choose a tractable family of states or density operators, such as product states, Slater determinants, coherent states, or Gaussian states, and make the energy or free-energy functional stationary within that family.

This viewpoint supplies a precise approximation space. At zero temperature, energy minimization over admissible normalized states retains the variational upper-bound property. Other observables do not inherit one-sided bounds automatically.

Saddle-point approximation

Rewrite an interaction using auxiliary fields or collective variables and approximate the functional integral by a stationary configuration. The Hubbard–Stratonovich transformation supplies the exact auxiliary-field identity, with a specified contour and normalization. The stationary auxiliary field is a mean field; fluctuations around it generate Gaussian and higher corrections.

The three routes often produce the same equations when conventions and trial spaces match. They need not do so for arbitrary decouplings. An algebraic closure can fail to come from a thermodynamic functional, and different auxiliary-field channels can produce inequivalent truncated theories.

Generic Interaction Decoupling

Consider a schematic interaction

$$H_{\mathrm{int}} = \frac12 \sum_{a,b} K_{ab}O_aO_b,$$

Here take Hermitian $O_a$ and a real symmetric kernel $K_{ab}=K_{ba}$, with the ordered double sum as written. This also symmetrizes products when the operators do not commute. Define

$$\phi_a = \langle O_a\rangle, \qquad \delta O_a = O_a-\phi_a.$$

Expanding gives

$$H_{\mathrm{int}} = H_{\mathrm{int}}^{\mathrm{MF}}[\phi] + H_{\mathrm{res}}[\phi],$$

with

$$H_{\mathrm{int}}^{\mathrm{MF}}[\phi] = \sum_{a,b} K_{ab}\phi_a O_b - \frac12 \sum_{a,b} K_{ab}\phi_a\phi_b,$$

and

$$H_{\mathrm{res}}[\phi] = \frac12 \sum_{a,b} K_{ab} \delta O_a\delta O_b.$$

Dropping $H_{\mathrm{res}}$ defines this mean-field approximation. The constant term in $H_{\mathrm{int}}^{\mathrm{MF}}$ is essential: it corrects pair double counting and changes energies, free energies, pressures, and phase comparisons even when it does not change the effective eigenvectors.

For fermionic exchange, anomalous pairing, noncommuting local operators, or antisymmetrized matrix elements, the actual decoupling contains channel-specific signs and contractions. The displayed formula is a structural template, not a substitute for those conventions.

The Self-Consistency Condition

Let

$$H_{\mathrm{MF}}[\boldsymbol\phi] = H_0 + H_{\mathrm{int}}^{\mathrm{MF}}[\boldsymbol\phi].$$

At zero temperature, solve for an appropriate ground state

$$H_{\mathrm{MF}}[\boldsymbol\phi] \lvert\Psi_{\mathrm{MF}}\rangle = E_{\mathrm{MF}} \lvert\Psi_{\mathrm{MF}}\rangle.$$

The fields must satisfy

$$\phi_a = \langle\Psi_{\mathrm{MF}}| O_a |\Psi_{\mathrm{MF}}\rangle.$$

At finite temperature, use the mean-field density operator

$$\rho_{\mathrm{MF}} = \frac{ e^{-\beta K_{\mathrm{MF}}[\boldsymbol\phi]} }{ Z_{\mathrm{MF}}[\boldsymbol\phi] },$$

where $K_{\mathrm{MF}}=H_{\mathrm{MF}}-\sum_j\mu_jQ_j$ includes the declared ensemble constraints. Then

$$\phi_a = \operatorname{Tr} \left( \rho_{\mathrm{MF}}O_a \right).$$

Collect these equations into a nonlinear map

$$\boldsymbol\phi = \mathcal F(\boldsymbol\phi).$$

A self-consistent field is a fixed point of this map. It is a candidate approximate equilibrium state, not a certificate of physical validity.

Solving the self-consistency equations

A practical iteration often uses mixing:

$$\boldsymbol\phi^{(n+1)} = (1-\alpha) \boldsymbol\phi^{(n)} + \alpha \mathcal F \left( \boldsymbol\phi^{(n)} \right),$$

with $0<\alpha\le1$ for simple under-relaxation. More advanced solvers use Anderson mixing, quasi-Newton updates, continuation, or direct minimization of a thermodynamic functional.

Convergence should be tested using more than the field increment. Useful residuals include

$$r_a = \mathcal F_a(\boldsymbol\phi) - \phi_a,$$

changes in the thermodynamic potential, symmetry constraints, particle-number constraints, and the stationarity gradient of the underlying functional.

Numerical Convergence Is Not Physical Stability

Linearize the fixed-point map around a solution $\boldsymbol\phi_*$:

$$\delta\boldsymbol\phi^{(n+1)} \simeq J_{\mathcal F} \delta\boldsymbol\phi^{(n)},$$

where

$$(J_{\mathcal F})_{ab} = \left. \frac{\partial\mathcal F_a} {\partial\phi_b} \right|_{\boldsymbol\phi_*}.$$

Plain iteration converges locally when the spectral radius obeys

$$\rho(J_{\mathcal F})<1.$$

Mixing changes the iteration Jacobian to

$$J_{\mathrm{mix}} = (1-\alpha)I + \alpha J_{\mathcal F}.$$

This is a statement about an algorithm. A physically stable equilibrium instead requires the appropriate energy or thermodynamic potential to be locally minimal with respect to admissible variations. A solver can be engineered to converge to a saddle, and a physically stable solution can be difficult for naive iteration.

Always separate:

1. small numerical residual;
2. local thermodynamic stability;
3. global comparison with other self-consistent solutions;
4. validity of the mean-field approximation itself.

Variational Structure at Zero Temperature

Let $\mathcal M$ be a normalized trial manifold. The restricted energy functional is

$$E_{\mathcal M} = \min_{ |\Psi\rangle\in\mathcal M } \langle\Psi|H|\Psi\rangle.$$

For admissible states and a Hamiltonian bounded below,

$$E_{\mathcal M} \ge E_0.$$

A product-state, Slater-determinant, coherent-state, or Gaussian mean field can therefore provide a rigorous upper bound to the ground-state energy when the expectation of the original Hamiltonian is evaluated exactly within the trial state.

The bound does not imply that:

• the state has high fidelity;
• an excitation gap is an upper or lower bound;
• a correlation function has a controlled sign of error;
• a broken-symmetry finite-size state is in the desired exact sector;
• a separately decoupled Hamiltonian retains the same variational guarantee.

Stationarity within a restricted manifold means only that the residual is orthogonal to represented tangent directions. Missing correlation directions can remain large.

Variational Structure at Finite Temperature

At positive temperature, for a normalized density operator $\rho$ on the specified ensemble Hilbert space, define

$$\mathcal F[\rho] = \operatorname{Tr}(\rho H) + k_{\mathrm B}T \operatorname{Tr}(\rho\ln\rho).$$

The exact Gibbs state minimizes this functional, and the minimum is the exact Helmholtz free energy:

$$F = \min_\rho \mathcal F[\rho].$$

Restricting $\rho$ to product or Gaussian density operators gives a variational mean-field free energy. Equivalently, for a solvable trial Hamiltonian $H_0$ with free energy $F_0$,

$$F \le F_0 + \langle H-H_0\rangle_0.$$

Optimizing the right-hand side over parameters in $H_0$ can generate self-consistency equations. This Gibbs–Bogoliubov–Feynman inequality is a free-energy upper bound when both traces exist and all quantities refer to the same Hilbert space, ensemble and positive temperature. A fixed-number trace must stay in its number sector; a full-Fock Gaussian state is a different trial family.

Order Parameters and Mean Fields

A mean field need not be an order parameter.

An order parameter distinguishes phases. The fields used to close a mean-field equation need not do so.

Mean field	Typical operator	Symmetry role
density	$\langle n_i\rangle$	often symmetry preserving
magnetization	$\langle S_i^\alpha\rangle$	may break spin or time-reversal symmetry
condensate amplitude	$\langle b_i\rangle$	phase-referenced $U(1)$ breaking coordinate
pairing amplitude	$\langle c_i c_j\rangle$	anomalous $U(1)$ breaking coordinate
bond field	$\langle c_i^\dagger c_j\rangle$	may preserve or break translation and point-group symmetry
density wave	$\langle n_i\rangle-\bar n$	breaks translation symmetry
sublattice magnetization	$(-1)^i\langle S_i^z\rangle$	detects antiferromagnetic order

A conventional symmetry-breaking order parameter transforms nontrivially under the relevant symmetry and distinguishes its broken branches. A Hartree density can be nonzero in every phase and simply parametrizes an inhomogeneous background.

Channel choice is physical input

The same interaction can be decoupled in density, magnetic, exchange, pairing, or bond channels. A chosen channel restricts which correlations can become self-consistent. If a calculation permits only uniform magnetization, it cannot discover a density wave, spiral, pair-density wave, or larger-unit-cell order.

An unbiased claim therefore requires either:

• a symmetry-complete set of plausible channels;
• a reason other channels are forbidden or parametrically suppressed;
• or an explicit statement that the calculation tests only a selected ansatz.

Algebraic rearrangements of an exact interaction can be equivalent before approximation but inequivalent after different terms are factorized. This is sometimes called a channel or Fierz ambiguity.

Symmetry Breaking Requires a Limit

A mean-field equation can possess nonzero order-parameter solutions even when a finite exact Hamiltonian has a unique symmetric ground state. This is useful, but its meaning must be stated carefully.

For a source $h$ coupled to an order parameter $M$, the many-body broken-symmetry value is characterized by an order such as

$$\lim_{h\to0^+} \lim_{V\to\infty} \langle M\rangle_h.$$

Mean field often works directly with one selected branch and thereby anticipates this limiting state. It does not prove that the finite system literally violates the exact symmetry.

Different mean-field solutions related by an exact symmetry have the same thermodynamic potential when the source vanishes. An optimizer’s preference for one branch can come from initialization, rounding, boundary conditions, or a deliberately applied seed.

For particle-number symmetry, a condensate or BCS state with a definite phase can organize local observables efficiently. The underlying finite-number Hamiltonian may still conserve $N$ exactly. Symmetry projection and number-conserving formulations can recover the exact sector while retaining the same leading bulk physics in an overlap regime.

Thermodynamic potentials and subtraction constants

The one-particle or one-site eigenvalues alone are not generally the total mean-field energy. The subtraction constants correct the double counting of interacting pairs. At a self-consistent state, evaluate the expectation of the original Hamiltonian and its entropy, using the same ensemble for competing solutions.

There is a subtle distinction away from a self-consistent solution. The auxiliary expression

$$\Omega_{\mathrm{dec}}(\phi) =-\beta^{-1}\ln\operatorname{Tr}e^{-\beta K_{\mathrm{MF}}(\phi)}$$

often generates the correct stationarity equations after the subtraction constants are retained. It need not equal the constrained trial-state functional $\operatorname{Tr}(\rho H)-TS[\rho]$ for arbitrary input $\phi$. An auxiliary field may require a saddle rather than a minimum. Consequently its Hessian is not automatically a physical stability test. Test a minimum of the trial-state free energy under admissible variations, or use a fluctuation kernel whose physical meaning has been established. Zero directions can reflect a symmetry or a redundant coordinate.

For example, take two classical antiferromagnetic spins, $H=J s_1s_2$, $J>0$, and restrict a product state to equal means $m$. Put

$$q(m)=\frac{1+m}{2}\ln\frac{1+m}{2} +\frac{1-m}{2}\ln\frac{1-m}{2}.$$

The variational free energy per spin and the decoupled expression are

$$\begin{aligned} f_{\mathrm{var}}(m)&=\frac J2m^2+k_{\mathrm B}Tq(m),\\ f_{\mathrm{dec}}(m)&=-\frac J2m^2 -k_{\mathrm B}T\ln[2\cosh(\beta Jm)]. \end{aligned}$$

Both are stationary at the uniform solution $m=0$, and they agree there. Yet $f_{\mathrm{var}}''(0)=J+k_{\mathrm B}T>0$ while $f_{\mathrm{dec}}''(0)=-J-\beta J^2<0$. The latter maximum does not indicate a uniform physical instability. Allowing independent sublattice means adds another variation; for $k_{\mathrm B}T>J$ the product-state minimum is stable in that direction too.

Two-spin antiferromagnetic example at $k_{\mathrm B}T=2J$, with each curve measured from its value at $m=0$. Their equal stationary value does not make their curvatures interchangeable.

Linear Response and the Instability Criterion

Use a physical source energy $-hM$ and define $m=\langle M\rangle$ per building block. Consider its response to the effective field

$$h_{\mathrm{eff}} = h+J_0m.$$

If the uncoupled building block has response

$$m \simeq \chi_0 h_{\mathrm{eff}},$$

then

$$m = \chi_0(h+J_0m).$$

The mean-field susceptibility is

$$\chi_{\mathrm{MF}} = \frac{\partial m}{\partial h} = \frac{\chi_0} {1-J_0\chi_0}.$$

For positive feedback $J_0>0$, the symmetric branch reaches a linear instability when

$$1-J_0\chi_0 = 0.$$

This structure appears in magnetism, density waves, pairing, and other channels. The sign and matrix ordering depend on conventions, and a divergence of the approximate susceptibility diagnoses instability of the reference state rather than proving the exact phase on its own.

Worked Example: Classical Ising Ferromagnet

Consider spins $s_i=\pm1$ on a regular lattice of coordination number $z$:

$$H = -J \sum_{\langle ij\rangle} s_i s_j - h\sum_i s_i, \qquad J>0.$$

Each bond is counted once. Assume a uniform magnetization

$$m = \langle s_i\rangle.$$

Write

$$s_i s_j ={} m(s_i+s_j) - m^2 + (s_i-m)(s_j-m).$$

Discarding the final fluctuation product and using $Nz/2$ bonds gives

$$H_{\mathrm{MF}}(m) ={} -(zJm+h) \sum_i s_i + \frac{NzJ}{2}m^2.$$

Every spin is now independent in the self-consistent molecular field

$$h_{\mathrm{eff}} = h+zJm.$$

Partition function and free energy

The mean-field partition function is

$$Z_{\mathrm{MF}}(m) ={} e^{-\beta NzJm^2/2} \times \Bigl[ 2\cosh \bigl( \beta(zJm+h) \bigr) \Bigr]^N.$$

The decoupled free energy per site is therefore

$$f_{\mathrm{MF}}(m) ={} \frac{zJ}{2}m^2 - \frac{1}{\beta} \ln\!\Bigl[ 2\cosh \bigl( \beta(zJm+h) \bigr) \Bigr].$$

Stationarity gives

$$\begin{aligned} 0 &= \frac{1}{zJ} \frac{\partial f_{\mathrm{MF}}}{\partial m} \\ &= m - \tanh \bigl( \beta(zJm+h) \bigr). \end{aligned}$$

Thus the self-consistency equation is

$$m = \tanh \bigl( \beta(zJm+h) \bigr).$$

For comparison, the constrained product-state functional is

$$f_{\mathrm{var}}(m)=-\frac{zJ}{2}m^2-hm+k_{\mathrm B}Tq(m).$$

Its stationary equation is the same, and its value agrees with $f_{\mathrm{MF}}$ at a self-consistent solution. Away from a solution the two functions differ. In this uniform ferromagnetic channel their local stability signs agree at stationary points.

Critical temperature

At $h=0$ and small $m$,

$$\tanh(\beta zJm) ={} \beta zJm - \frac{(\beta zJm)^3}{3} + O(m^5).$$

The symmetric solution loses stability when

$$\beta_c zJ = 1,$$

so

$$k_{\mathrm B}T_c^{\mathrm{MF}} = zJ.$$

For $T<T_c^{\mathrm{MF}}$, two symmetry-related nonzero solutions appear at zero field.

Landau expansion and critical exponent

Expanding the free energy at $h=0$ gives

$$f_{\mathrm{MF}}(m) ={} f_0 + \frac{zJ}{2} (1-\beta zJ)m^2 + \frac{\beta^3(zJ)^4}{12} m^4 + O(m^6).$$

Close below $T_c^{\mathrm{MF}}$,

$$m^2 \simeq 3 \frac{T_c^{\mathrm{MF}}-T} {T_c^{\mathrm{MF}}}.$$

Hence

$$m \propto (T_c^{\mathrm{MF}}-T)^{1/2}.$$

The mean-field order-parameter critical exponent is $1/2$. It is not the exact exponent for short-range Ising systems below the upper critical dimension.

Susceptibility above the transition

Differentiate the self-consistency equation with respect to $h$. In the symmetric phase,

$$\chi = \frac{\partial m}{\partial h} = \beta(zJ\chi+1),$$

and therefore

$$\chi_{\mathrm{MF}} = \frac{\beta} {1-\beta zJ} = \frac{1} {k_{\mathrm B}(T-T_c^{\mathrm{MF}})}.$$

This is the Curie–Weiss divergence in the normalization used here.

What the Ising Example Gets Right and Wrong

The calculation correctly illustrates:

• a self-consistent molecular field;
• entropy competing with interaction energy;
• symmetric and symmetry-broken stationary points;
• susceptibility as a stability diagnostic;
• a continuous bifurcation in a scalar order parameter;
• an exactly solvable effective one-site problem.

It is quantitatively or qualitatively wrong when neglected spatial fluctuations dominate. The nearest-neighbor one-dimensional Ising model has no nonzero-temperature phase transition, while the mean-field equation predicts $k_{\mathrm B}T_c=2J$ because $z=2$. In two and three dimensions, mean field predicts the wrong short-range critical exponents and shifted transition temperatures. Critical Exponents and Scaling derives the mean-field exponent benchmark and explains its upper-critical-dimension and hyperscaling limits.

For an infinite-range Curie–Weiss model with properly scaled coupling, the magnetization concentrates on the minimizing branches in the thermodynamic limit (on both symmetry-related branches at zero source below the transition), and the same saddle-point equation becomes asymptotically exact for the bulk free energy.

Quantum Spin Mean Field

For the transverse-field Ising Hamiltonian

$$H = -J \sum_{\langle ij\rangle} \sigma_i^z\sigma_j^z - \Gamma \sum_i\sigma_i^x,$$

take

$$m = \langle\sigma_i^z\rangle.$$

The one-site mean-field Hamiltonian is

$$h_{\mathrm{MF}} = -zJm\,\sigma^z - \Gamma\sigma^x + \frac{zJ}{2}m^2.$$

Define

$$E(m) = \sqrt{ (zJm)^2 + \Gamma^2 }.$$

The finite-temperature self-consistency equation is

$$m = \frac{zJm}{E(m)} \tanh\bigl(\beta E(m)\bigr).$$

At zero temperature, a nonzero solution exists in this approximation when

$$\Gamma<zJ.$$

Thus

$$\Gamma_c^{\mathrm{MF}} = zJ.$$

The transverse-field Ising model gives the exact one-dimensional solution in the same Pauli-spin normalization. The comparison makes the mean-field limitation concrete: solving the local quantum problem exactly does not restore the spatial entanglement discarded by site factorization.

Mean Field and Gaussian Fluctuations

Mean field is often the first term in a hierarchy rather than the final theory. After finding a stationary configuration, restore small fluctuations:

$$O_a = \phi_a + \delta O_a.$$

Keeping terms quadratic in $\delta O_a$ produces a Gaussian theory. Depending on the system, this leads to:

• spin-wave theory around magnetic order;
• Bogoliubov modes around a condensate;
• random-phase response around a density or magnetic mean field;
• amplitude and phase modes around a pairing saddle;
• one-loop corrections to a thermodynamic potential.

The fluctuation spectrum also tests stability. A negative quadratic mode signals that the candidate saddle is unstable in a direction not stabilized by the current solution. A zero mode can be required by a continuous broken symmetry, but only if the approximation respects the associated conservation identities.

When Mean Field Can Be Controlled

Mean field is not controlled by “many particles” alone. Useful control can arise from specific structures.

Infinite-range or collective coupling

With interactions scaled so the energy is extensive, collective fields can obey a law of large numbers and their relative fluctuations can vanish as the system grows.

Large coordination

If each site couples weakly to many neighbors, individual neighbor fluctuations can average out. A standard large-$z$ limit scales a bond coupling as $J/z$ so that the total local field remains finite.

Large component number

In large-N or large-flavor theories, the effective action can scale as $N$, making saddle-point fluctuations parametrically small. The physical finite-$N$ correction must still be checked.

Dilute weakly interacting condensates

For a three-dimensional Bose gas, the gas parameter $na_s^3\ll1$ controls depletion and beyond-mean-field corrections. Macroscopic occupation helps identify the field, while diluteness supplies the actual small parameter.

High dimension or distance from criticality

Short-range fluctuations become less important in sufficiently high dimension. Even where mean field is qualitatively useful, it generally fails inside a fluctuation-dominated critical region whose size is estimated in Landau–Ginzburg Theory Preview by the Ginzburg criterion.

Restricted observables

Energy density or a coarse order parameter may be accurate while entanglement, spectral linewidths, or short-distance correlations are not. Control must be stated observable by observable.

Why Critical Regions Are Dangerous

Near a continuous transition, the correlation length grows and fluctuations become coherent over large regions. The assumption that each degree of freedom sees a narrowly distributed average environment then becomes least reliable.

Landau theory develops the uniform analytic potential and its critical exponents. Below the upper critical dimension, long-wavelength fluctuations generally renormalize those exponents. At the upper critical dimension, logarithmic corrections can survive. Universality explains how dimension and order-parameter structure determine when a short-range critical point shares the mean-field fingerprint.

For short-range systems with a continuous symmetry, infrared fluctuations can forbid finite-temperature long-range order in low dimension under the hypotheses of the Hohenberg–Mermin–Wagner results. A static mean field can nevertheless produce a nonzero order parameter because it suppresses precisely those long-wavelength fluctuations.

This is not a small numerical error. It is a qualitative failure of the approximation’s fluctuation content.

Failure Modes Beyond Criticality

Mean field can also fail far from a conventional critical point.

Strong local correlations

A product or determinant state can misrepresent double occupancy, local moments, Mott localization, and multiplet structure even when no symmetry changes.

Low-dimensional entanglement

One-dimensional ground states can possess strong quantum correlations that no site-product state captures. A correct local expectation does not certify the correlation length or excitation spectrum.

Frustration and competing orders

Many nearly degenerate patterns can create a rugged free-energy landscape. A small unit cell or single-channel ansatz can select an artificial order.

Topological and fractionalized phases

Conventional local order parameters can miss long-range entanglement, gauge constraints, and fractional excitations. Parton mean fields may still be useful, but gauge fluctuations and projection are then central rather than optional details.

First-order transitions and metastability

Multiple local minima produce hysteresis and initialization dependence. A spinodal where one minimum disappears is not the same as the coexistence point where free energies cross.

Conservation-law violations

An inconsistent truncation can violate particle number, gauge covariance, Ward identities, or sum rules. A self-consistent approximation is not automatically conserving.

Dynamics

Static self-consistency does not supply collision integrals, damping, thermalization, or memory. Time-dependent mean-field equations can capture coherent collective motion but often miss scattering-induced relaxation.

Exercises

Curie–Weiss susceptibility

Differentiate

$$m = \tanh \bigl[ \beta(zJm+h) \bigr]$$

and derive the susceptibility for arbitrary self-consistent $m$.

Solution

Let $\chi=\partial m/\partial h$. Differentiation gives

$$\chi = \beta \operatorname{sech}^2 \bigl[ \beta(zJm+h) \bigr] (zJ\chi+1).$$

At a self-consistent solution,

$$\operatorname{sech}^2 \bigl[ \beta(zJm+h) \bigr] = 1-m^2.$$

Therefore

$$\chi = \frac{ \beta(1-m^2) }{ 1-\beta zJ(1-m^2) }.$$

For $m=0$, this reduces to $\beta/(1-\beta zJ)$.

Why the subtraction constant matters

Suppose one incorrectly drops $zJm^2/2$ from the Ising free energy. Show that the correct self-consistency equation is no longer its stationarity condition.

Solution

The incorrect per-site expression would be

$$\begin{aligned} f_{\mathrm{bad}}(m) &= -\frac{1}{\beta} \ln \mathcal C(m), \\ \mathcal C(m) &:= 2\cosh \bigl( \beta(zJm+h) \bigr). \end{aligned}$$

Its derivative is

$$\frac{df_{\mathrm{bad}}}{dm} = -zJ \tanh \bigl( \beta(zJm+h) \bigr).$$

At a self-consistent point this equals $-zJm$, which vanishes only for $m=0$. The nonzero self-consistent states are therefore not stationary points of the incorrectly truncated free energy. Restoring $zJm^2/2$ adds $zJm$ to the derivative and repairs the variational structure.

Bipartite antiferromagnetic mean field

For

$$H = J \sum_{\langle ij\rangle} s_i s_j, \qquad J>0,$$

on a bipartite lattice, take $m_A=-m_B=m$ at zero field. Derive the self-consistency equation and transition temperature.

Solution

An $A$-sublattice spin sees $z$ neighbors with mean $m_B=-m$. Its effective energy is

$$Jz s_A m_B = -zJm s_A.$$

Thus

$$m_A = \tanh(\beta zJm).$$

The $B$ equation gives the opposite sign consistently. Therefore

$$m = \tanh(\beta zJm),$$

and the linear instability occurs at

$$k_{\mathrm B}T_N^{\mathrm{MF}} = zJ.$$

A uniform one-field ansatz would miss this ordered pattern, illustrating why the allowed unit cell is part of the approximation.

Mixing and local convergence

For a scalar fixed-point equation $\phi=\mathcal F(\phi)$, show how linear mixing changes the local convergence factor near $\phi_*$.

Solution

Let $\delta_n=\phi_n-\phi_*$. Linearizing gives

$$\mathcal F(\phi_n) - \phi_* \simeq \mathcal F'(\phi_*)\delta_n.$$

The mixed update produces

$$\delta_{n+1} \simeq \left[ 1-\alpha + \alpha\mathcal F'(\phi_*) \right] \delta_n.$$

The local numerical convergence condition is therefore

$$\left| 1-\alpha + \alpha\mathcal F'(\phi_*) \right| <1.$$

Changing $\alpha$ can stabilize the iteration without changing the underlying fixed point or its thermodynamic Hessian. Algorithmic and physical stability remain separate questions.

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