Susceptibilities

A susceptibility states how strongly a chosen detector responds to a chosen weak source. The word does not identify one universal formula. Magnetic, density, compressibility, and pairing susceptibilities differ in their operators, source units, normalizations, symmetry channels, and experimental protocols.

The safest definition is operational. If

$$H_{\mathrm{pert}}(t) = -f(t)B,$$

then the susceptibility of detector $A$ to source $f$ is

$$\chi_{AB} = \left. \frac{ \delta\langle A\rangle_f }{ \delta f } \right|_{f=0}.$$

In real-time linear response, this derivative is represented by a retarded commutator plus any explicit source derivative of the detector. In equilibrium statics, it can instead be a derivative of a thermodynamic potential. Those representations agree only when their ensembles, boundary conditions, and orders of limits describe the same protocol.

Helpful background. The Kubo formula supplies the retarded kernel; fluctuations and susceptibilities explains equilibrium derivatives and imaginary-time covariance.

Defining a susceptibility

Before quoting any susceptibility, specify five ingredients.

1. Source

Write the perturbation explicitly:

$$H_{\mathrm{pert}} = -\sum_b \int d^d r\, f_b(\mathbf r,t) B_b(\mathbf r).$$

The source may be a magnetic field, local chemical-potential shift, force, strain, pairing field, or gauge potential. Its sign and units determine the sign and units of the response.

2. Detector

The measured operator can differ from the source-coupled operator:

$$A_a \ne B_b.$$

The ordered label $\chi_{ab}$ means response of $A_a$ to the field coupled through $B_b$. Interchanging the labels changes a cross-susceptibility unless a reciprocity relation applies.

3. Normalization

State whether the detector is:

• a total extensive quantity;
• a density per physical volume;
• a quantity per lattice site or unit cell;
• a local operator;
• a Fourier mode with a specified transform convention.

A factor of volume can turn a finite bulk susceptibility into an extensive quantity. This is not a cosmetic convention.

4. Protocol

State whether the response is:

• isothermal or adiabatic;
• isolated or bath-equilibrated;
• static or finite frequency;
• uniform or finite wavevector;
• measured before or after a thermodynamic limit;
• defined with internal or externally applied fields.

5. Held-Fixed Variables

An equilibrium derivative is incomplete without its constraints:

$$\left( \frac{\partial n}{\partial\mu} \right)_{T,V} \ne \left( \frac{\partial n}{\partial\mu} \right)_{S,V}.$$

Likewise, fixed density, fixed chemical potential, fixed pressure, fixed strain, and fixed electric boundary conditions can produce different response coefficients.

General Space-Time Response

For several source and detector channels,

$$\delta\langle A_a(\mathbf r,t) \rangle ={} \sum_b \int dt' \int d^d r' \times \chi^{\mathrm R}_{ab} (\mathbf r,t;\mathbf r',t') f_b(\mathbf r',t').$$

When the reference state is stationary and translation invariant,

$$\begin{aligned} \delta\langle A_a(\mathbf q,\omega) \rangle ={}& \sum_b \chi^{\mathrm R}_{ab} (\mathbf q,\omega) f_b(\mathbf q,\omega). \end{aligned}$$

For a detector with no explicit source dependence,

$$\chi^{\mathrm R}_{ab} (\mathbf r,t;\mathbf r',t') ={} \frac{i}{\hbar} \theta(t-t') \times \left\langle \left[ A_a(\mathbf r,t), B_b(\mathbf r',t') \right] \right\rangle_0.$$

The Kubo formula gives this representation. A susceptibility name adds physical meaning only after $A_a$, $B_b$, $f_b$, and all normalizations have been fixed.

Units and Common Normalizations

For a static coefficient or a Fourier-transformed response, the unit rule is

$$[\chi_{AB}] = \frac{[A]}{[f]}.$$

The time-domain kernel carries an additional inverse time; a spatial kernel carries the inverse integration volume as well. These factors disappear only after the corresponding convolution or Fourier transform. Representative static or Fourier-space examples are:

Response	Detector	Source	Typical units
spin response	spin density	spin field in energy units	density per energy
magnetic response	magnetization density	magnetic induction or field	convention dependent
density susceptibility	number density	chemical potential	density per energy
isothermal compressibility	fractional volume or density change	pressure	inverse pressure
pair susceptibility	pair amplitude density	pair field	normalization dependent
cross response	detector $A$	source conjugate to $B$	$[A]/[f_B]$

Electromagnetic unit systems distribute factors such as $\mu_0$, charge, volume, and field conversion differently. The microscopic Hamiltonian coupling is the reliable starting point.

Total versus bulk response

If

$$A_{\mathrm{tot}} = \int_V d^d r\, a(\mathbf r),$$

then a uniform extensive susceptibility often scales as

$$\chi_{\mathrm{tot}} \sim V\chi_{\mathrm{bulk}}.$$

Near a critical point, this simple extensive scaling can fail because long-range correlations make the bulk susceptibility itself size dependent.

Lattice conventions

For $L$ sites, authors may define

$$A_{\mathbf q} = \sum_j e^{-i\mathbf q\cdot\mathbf r_j} A_j$$

or include $L^{-1/2}$ or $L^{-1}$. The corresponding susceptibility differs by powers of $L$. State the transform before comparing numerical data.

A Taxonomy of Susceptibilities

Several independent adjectives can apply to one response.

Distinction	First object	Second object
spatial	local $\chi_{ij}$	momentum resolved $\chi(\mathbf q)$
wavelength	uniform $\mathbf q=0$	finite $\mathbf q$
time	static	dynamic $\chi(\omega)$
protocol	thermodynamic	isolated retarded
tensor	longitudinal	transverse
channel	diagonal	cross
normalization	total	per volume or site
field	applied	internal or screened

The phrase “the susceptibility” is therefore rarely sufficient in a many-body calculation.

Static Thermodynamic Susceptibility

For sources $f_a$ coupled to observables $B_a$, define

$$\Omega(\mathbf f) = -\frac{1}{\beta} \ln \operatorname{Tr} \exp\left[ -\beta \left( H_0-\sum_a f_aB_a \right) \right].$$

Then

$$\langle B_a\rangle_{\mathbf f} = -\frac{\partial\Omega}{\partial f_a},$$

and the isothermal static susceptibility matrix is

$$\chi^{T}_{ab} = \left. \frac{ \partial\langle B_a\rangle_{\mathbf f} }{ \partial f_b } \right|_{\mathbf f=0} = -\left. \frac{ \partial^2\Omega }{ \partial f_a\partial f_b } \right|_{\mathbf f=0}.$$

For a positive-temperature Gibbs state with a finite, differentiable partition function and Hermitian source channels, this Hessian response is positive semidefinite for every real channel vector $\mathbf v$:

$$\sum_{a,b} v_a \chi^T_{ab} v_b \ge0.$$

If the $B_a$ commute with $H_0$, the matrix reduces to $\beta$ times an ordinary covariance matrix. If they do not commute, imaginary-time Kubo–Mori ordering replaces the ordinary covariance.

Zero-temperature quantum susceptibility

For a nondegenerate ground state and one Hermitian operator $B$, let

$$H(f)=H_0-fB.$$

Second-order perturbation theory gives

$$\chi_{B}(0) = 2 \sum_{m\ne0} \frac{ \lvert\langle m\lvert B\rvert0\rangle\rvert^2 }{ E_m-E_0 }.$$

This expression is nonnegative and shows why low-lying states strongly enhance a static susceptibility. Degeneracy, level crossings, continuous spectra, and spontaneous symmetry breaking require a limiting prescription beyond this elementary formula.

Static Is Not One Limit

For a spatially resolved dynamical susceptibility, at least two limits exist:

$$\chi_{\mathrm{stat}} = \lim_{\mathbf q\to0} \lim_{\omega\to0} \chi^{\mathrm R}(\mathbf q,\omega),$$

and

$$\chi_{\mathrm{unif}} = \lim_{\omega\to0} \lim_{\mathbf q\to0} \chi^{\mathrm R}(\mathbf q,\omega).$$

They need not agree. Taking $\omega\to0$ first allows a finite-wavelength modulation to relax before the wavelength becomes infinite. Taking $\mathbf q\to0$ first can project onto a globally conserved charge whose isolated finite-frequency response vanishes.

Other noncommuting limits include:

$$V\to\infty, \qquad T\to0, \qquad \eta\to0^+, \qquad t_{\mathrm{obs}}\to\infty.$$

Every reported “static susceptibility” should identify the path through these limits.

Response Matrices and Eigenchannels

When several operators share the same symmetry, a source in one channel can induce multiple detectors:

$$\delta\mathbf A = \boldsymbol{\chi}\, \mathbf f.$$

Off-diagonal entries are cross-susceptibilities. Examples include spin–charge response, magnetoelectric response, and coupled density–pair channels in broken-symmetry formalisms.

An eigenvector of a properly normalized static susceptibility matrix identifies a linear combination of operators that responds coherently. A growing largest eigenvalue can reveal the dominant ordering tendency.

Three cautions are essential:

1. Eigenvalues depend on the relative normalization and units of the channels.
2. A large response does not identify the microscopic interaction by itself.
3. For a finite-dimensional system and finite sources, an equilibrium eigenvalue cannot diverge at positive temperature; unbounded Hilbert spaces additionally require a convergent partition function and finite response moments.

Symmetry can force cross entries to vanish. Reciprocity can relate $\chi_{ab}$ and $\chi_{ba}$ only after time-reversal parities and time-reversal-odd control fields have been transformed correctly.

Magnetic Susceptibility

Let the microscopic magnetic coupling be

$$H_Z(t) = -\int d^d r\, \mathbf B(\mathbf r,t) \cdot \mathbf M(\mathbf r),$$

where $\mathbf M$ is the magnetic-moment density in the chosen unit convention. Then

$$\delta\langle M_i(\mathbf q,\omega) \rangle = \sum_j \chi^M_{ij} (\mathbf q,\omega) B_j(\mathbf q,\omega).$$

The susceptibility is generally a tensor. Crystal symmetry, spin–orbit coupling, external fields, and magnetic order determine which components are independent.

Microscopic field versus reported bulk field

In SI macroscopic notation, experiments often report

$$\mathbf M = \boldsymbol{\chi}_H \mathbf H_{\mathrm{int}},$$

while a microscopic Zeeman Hamiltonian is naturally written using magnetic induction $\mathbf B$. Since

$$\mathbf B = \mu_0 \left( \mathbf H+\mathbf M \right),$$

field conversion and demagnetization geometry matter. A susceptibility inferred from the applied field is not automatically the intrinsic response to the internal field. Magnetic Anisotropy defines the SI demagnetizing tensor, self-energy, and ellipsoidal limits.

Longitudinal and transverse spin response

Relative to a unit vector $\hat{\mathbf m}$ set by the field or ordered moment,

$$\chi_L = \hat m_i \chi_{ij} \hat m_j,$$

whereas the transverse tensor is projected with

$$P^T_{ij} = \delta_{ij} - \hat m_i\hat m_j.$$

Longitudinal response changes the moment magnitude. Transverse response rotates it and can contain precession or Goldstone poles. In an isotropic unpolarized state, the tensor reduces to a scalar multiple of $\delta_{ij}$.

Independent spin-1/2 benchmark

For one spin-$1/2$ magnetic moment,

$$H_Z = -g\mu_{\mathrm B}B \frac{\sigma_z}{2}.$$

The moment is

$$\langle\mu_z\rangle = \frac{g\mu_{\mathrm B}}{2} \tanh\left( \frac{\beta g\mu_{\mathrm B}B}{2} \right).$$

Therefore

$$\chi_B = \frac{ \partial\langle\mu_z\rangle }{ \partial B } = \frac{ \beta(g\mu_{\mathrm B})^2 }{4} \operatorname{sech}^2 \left( \frac{\beta g\mu_{\mathrm B}B}{2} \right).$$

At weak field,

$$\chi_B(0) = \frac{ (g\mu_{\mathrm B})^2 }{ 4k_{\mathrm B}T }.$$

Multiplying by the spin density gives the bulk Curie law for independent moments. Interactions, crystal-field multiplets, Kondo screening, and collective order can change this behavior qualitatively.

Pauli benchmark

For a spin-degenerate noninteracting Fermi gas at zero temperature, with a finite smooth density of states at the Fermi energy and no orbital magnetic coupling, let

$$\mathcal D(E_{\mathrm F})$$

be the total density of one-particle states per volume and per energy, including both spin species at zero field. The response to magnetic induction is

$$\chi_{\mathrm P,B} = \frac{(g\mu_{\mathrm B})^2}{4} \mathcal D(E_{\mathrm F}).$$

If the density of states is quoted per spin instead, the prefactor changes accordingly. In SI and weak magnetization, the dimensionless susceptibility to $\mathbf H$ introduces an additional factor of $\mu_0$.

The Curie and Pauli laws describe different physics: localized thermal moments versus redistribution near a Fermi surface.

Density Susceptibility

Use a local chemical-potential source:

$$H_{\mathrm{pert}}(t) = -\int d^d r\, \delta\mu(\mathbf r,t) n(\mathbf r).$$

The density susceptibility is defined by

$$\delta n(\mathbf q,\omega) = \chi^R_{nn}(\mathbf q,\omega) \delta\mu(\mathbf q,\omega).$$

For a stable compressible phase, the static long-wavelength chemical-potential response is nonnegative under this convention.

Particle–Hole Excitations shows how a one-body density vertex creates occupied-to-empty fermion promotions and how their phase space sets the absorptive support of the independent susceptibility.

Potential-energy convention

Electronic-response literature often introduces a potential energy $U$ through

$$H_{\mathrm{pert}} = +\int d^d r\, U(\mathbf r,t)n(\mathbf r).$$

Define

$$\delta n = \Pi^R\delta U.$$

Since

$$U=-\delta\mu,$$

the two response functions satisfy

$$\Pi^R = -\chi^R_{nn}.$$

Thus a static Lindhard polarization $\Pi^R\lt0$ and a chemical-potential susceptibility $\chi^R_{nn}\gt0$ can describe the same system. Mixing one convention’s numerator with the other convention’s screening denominator creates false signs and false poles.

Conservation at zero wavevector

At $\mathbf q=0$, the density mode is total particle number:

$$n_{\mathbf q=0} = N.$$

If

$$[H_0,N]=0,$$

then the isolated retarded self-response obeys

$$\chi^R_{NN}(\omega)=0.$$

This does not imply zero compressibility. A thermodynamic chemical-potential change reweights number sectors or compares neighboring fixed-$N$ free energies, whereas isolated unitary evolution preserves $N$.

Continuity constraints

For a conserved number density,

$$\partial_t n + \boldsymbol{\nabla}\cdot\mathbf j =0.$$

In Fourier space,

$$-i\omega\,\delta n + i\mathbf q\cdot\delta\mathbf j =0.$$

Density and longitudinal-current responses must satisfy the corresponding Ward identities. An approximation that violates continuity can produce an incorrect compressibility, spurious spectral weight, or a gauge-dependent result.

Compressibility

For a homogeneous one-component system,

$$n = \frac{\langle N\rangle}{V}.$$

Two common definitions are

$$\kappa_n = \left( \frac{\partial n}{\partial\mu} \right)_{T},$$

and the isothermal thermodynamic compressibility

$$\kappa_T = \frac{1}{n^2} \left( \frac{\partial n}{\partial\mu} \right)_{T}.$$

Thus

$$\kappa_n = n^2\kappa_T.$$

The first has units of density per energy. The second has units of inverse pressure. Calling both quantities $\kappa$ without a definition is a common source of missing factors.

Compressibility sum-rule limit

Under matching equilibrium, normalization, and order-of-limits conventions,

$$\lim_{\mathbf q\to0} \chi_{nn}(\mathbf q,\omega=0) = \kappa_n = n^2\kappa_T.$$

For a charged fluid with long-range Coulomb interactions, distinguish the response to the total local chemical-potential shift from the screened response to an externally imposed potential. The thermodynamic compressibility refers to a neutral equation-of-state variation, or the appropriate irreducible static density response. The fully screened response to an external charge perturbation can vanish as $q\to0$ even when that compressibility is finite; Random Phase Approximation makes the distinction explicit.

This relation is a consistency condition, not permission to replace every $\mathbf q=0$ retarded correlator by a thermodynamic derivative. The finite-wavevector static limit must be taken in a way that permits density redistribution and equilibration.

Local, global, and trap compressibility

On a lattice, define the local response matrix

$$\chi_{ij} = \frac{ \partial\langle n_i\rangle }{ \partial\mu_j }.$$

Three quantities then differ:

$$\chi_{ii} \quad \text{local response to a local source},$$ $$\frac{ \partial\langle n_i\rangle }{ \partial\mu } = \sum_j\chi_{ij} \quad \text{local response to a global source},$$

and

$$\frac{ \partial\langle N\rangle }{ \partial\mu } = \sum_{i,j}\chi_{ij} \quad \text{global number response}.$$

They coincide only in special limits and normalizations. In a trap, a locally incompressible plateau can coexist with a globally compressible cloud whose boundary moves as $\mu$ changes.

Gaps and incompressibility

At zero temperature, a finite interval of chemical potential over which the density is constant implies

$$\kappa_n=0.$$

This is a useful signature of a charge gap or Mott plateau, but finite-size steps, disorder, phase coexistence, and thermal activation must be distinguished from a bulk incompressible phase.

At positive temperature, a gapped phase generally has an exponentially small rather than exactly zero compressibility.

Pairing Susceptibility

Choose a pair annihilation operator with form factor $g_{\mathbf k}$:

$$\Delta_g = \sum_{\mathbf k} g_{\mathbf k} c_{-\mathbf k\downarrow} c_{\mathbf k\uparrow}.$$

Introduce a complex external pair field:

$$H_{\mathrm{pair}}' = -\eta^*(t)\Delta_g - \eta(t)\Delta_g^\dagger.$$

The retarded response of the pair amplitude to the source coupled through $\Delta_g^\dagger$ is

$$\chi^R_{\mathrm{pair}}(t) ={} \frac{i}{\hbar\mathcal V} \theta(t) \times \left\langle \left[ \Delta_g(t), \Delta_g^\dagger(0) \right] \right\rangle,$$

where $\mathcal V$ specifies volume or site normalization (set it to one for a total pair amplitude). In a number-symmetric normal state, the response to $\eta^*$ through $[\Delta_g(t),\Delta_g]$ vanishes by charge selection. In a symmetry-broken state both source components can matter, giving a two-component response matrix.

For equilibrium pairing at fixed chemical potential, use $\mathcal K=H-\mu N$ in the Gibbs operator and pair evolution. Since $[N,\Delta_g]=-2\Delta_g$, this rotating-frame pair frequency differs from the physical-$H$ frequency by $2\mu/\hbar$. State which frame is used when comparing a static pair source with spectroscopy.

Number symmetry

The source explicitly breaks particle-number symmetry while it is present. It is a generating field, not a claim that the exact source-free finite system fails to conserve particle number.

In a number eigenstate,

$$\langle\Delta_g\rangle=0$$

at zero source, yet

$$\left. \frac{ \partial\langle\Delta_g\rangle_\eta }{ \partial\eta } \right|_{\eta=0}$$

can be large. A susceptibility measures readiness to respond; it need not equal a pre-existing order parameter.

Pairing instability

In a normal-state approximation, an attractive interaction can produce a criterion schematically of the form

$$1 = g\chi_{\mathrm{pair}}^{(0)}(T_{\mathrm c}).$$

The precise sign, volume factor, and form-factor normalization depend on how the interaction and pair operator are defined. BCS Mean-Field Theory derives the reduced-model calculation.

In a mean-field or ladder calculation, a pole of the normal-state pair susceptibility marks failure of that normal-state approximation. More generally, a diverging susceptibility can accompany critical or algebraic pair correlations without true long-range order. It does not by itself determine a superconducting phase, gap, stiffness, or fluctuation-corrected critical behavior.

Distinct superconducting diagnostics

Do not identify the pair susceptibility with:

• the anomalous expectation value $\langle cc\rangle$;
• the single-particle excitation gap;
• phase stiffness or superfluid density;
• the Meissner response;
• pair-binding energy;
• off-diagonal long-range order.

These quantities can be related in specific theories, but they are not interchangeable definitions.

Longitudinal and Transverse Projectors

For a vector response at nonzero wavevector, define

$$P^L_{ij}(\mathbf q) = \frac{q_iq_j}{q^2},$$

and

$$P^T_{ij}(\mathbf q) = \delta_{ij} - \frac{q_iq_j}{q^2}.$$

Then

$$\chi_L = P^L_{ij}\chi_{ij},$$

while the transverse part is

$$\chi^T_{ij} = P^T_{ik} \chi_{k\ell} P^T_{\ell j}.$$

Longitudinal versus transverse relative to $\mathbf q$ is distinct from longitudinal versus transverse relative to an ordered moment. State which direction defines the decomposition.

At $\mathbf q=0$, the projector is undefined. The zero-wavevector tensor must be obtained by a directional limit or from symmetry, not by dividing by $q^2$ after setting $q=0$.

Cross-Susceptibilities

A cross-susceptibility has different source and detector channels:

$$\chi_{AB} = \frac{\delta\langle A\rangle}{\delta f_B}.$$

Its imaginary part need not have a definite sign. Passivity constrains the absorptive quadratic form of the full response matrix, not each off-diagonal component separately.

Examples include:

• spin response to an electric field in spin–orbit-coupled systems;
• density response to a pairing source in a symmetry-broken phase;
• magnetization response to strain;
• coupled charge and orbital channels;
• thermoelectric response between particle and energy currents.

Transport cross-coefficients require additional care because currents, gradients, and contact terms enter. Their thermodynamic-force and Onsager–Casimir conventions are developed in Transport Coefficients Preview rather than this static channel dictionary.

Susceptibility and Symmetry

An unbroken symmetry can force a response to vanish. For a symmetry-preserving state and geometry, a correlator of $A$ and $B$ can be nonzero only if the product of their symmetry representations contains an invariant scalar. If it does not, then

$$\chi_{AB}=0$$

For a correlator written $\langle A B^\dagger\rangle$, the second representation is conjugated. This matters for complex channels: raising and lowering operators have opposite charges and can have a nonzero cross response even though their individual charge representations differ.

Selection rules can therefore hide a low-energy excitation from one probe while leaving it visible to another. A missing peak need not mean a missing state.

Under equilibrium microscopic reversibility, Onsager–Casimir relations can connect transposed channels after reversing magnetic fields and other time-reversal-odd controls. Retarded and Advanced Response gives the bounded reciprocity statement and its assumptions.

Large Response, Divergence, and Instability

A susceptibility becomes large when a weak source can move the state strongly. Possible reasons include:

• a small excitation gap with a nonzero matrix element;
• many nearly degenerate transitions;
• a long correlation length;
• proximity to a continuous transition;
• phase coexistence or switching;
• an approximate denominator nearing zero;
• a finite-size level crossing;
• population inversion or active gain.

These mechanisms are physically distinct.

Finite systems

At finite temperature, a finite-dimensional system has an analytic partition function for finite sources. Its equilibrium susceptibility is finite. Sharp peaks can grow with size, but a true divergence requires a thermodynamic or continuum limit.

At zero temperature, a finite system can have nonanalytic level crossings. A level crossing is not automatically the finite-size version of a bulk continuous transition.

Critical scaling

For an extensive order parameter $M$ that commutes with the equilibrium Hamiltonian, the bulk isothermal susceptibility is

$$\chi_M = \frac{\beta}{V} \operatorname{Var}(M),$$

For a noncommuting quantum order parameter, replace this equal-time variance by its imaginary-time Kubo–Mori integral. In the commuting case, critical correlations can make $\operatorname{Var}(M)$ grow faster than $V$. The peak height, width, and location then obey finite-size scaling rather than ordinary central-limit scaling.

The correct analysis compares several sizes and a specified scaling ansatz. One large finite peak is not proof of criticality. Critical Exponents and Scaling explains how peak height, width, crossing drift, and corrections constrain a scaling fit.

Inverse susceptibility

In an effective quadratic theory, an instability often appears when an eigenvalue of the inverse static susceptibility approaches zero:

$$\lambda_{\min} \left[ \boldsymbol{\chi}^{-1} \right] \longrightarrow0.$$

This criterion is basis meaningful only after channel normalization is fixed. An approximation can also generate a false zero by violating conservation laws, double counting interactions, or using inconsistent self-energy and vertex corrections.

Measurement Dictionary

Different experiments extract susceptibilities in different ways.

Method	Source	Detector	Important caveat
magnetometry	applied magnetic field	total moment	internal field and demagnetization
magnetic resonance	oscillating magnetic field	transverse magnetization	phase calibration and linewidth origin
equation of state	chemical potential or pressure	density	held-fixed variables and trap inversion
lattice modulation	local potential	density mode	chemical-potential versus potential sign
scattering	probe momentum and energy	structure factor	requires fluctuation–dissipation conversion
pair-field calculation	complex source $\eta$	pair amplitude	source breaks number symmetry
numerical source derivative	explicit Hamiltonian parameter	chosen observable	finite difference and equilibration

Scattering commonly measures an ordered correlation spectrum rather than $\chi^R$ directly. Converting it to the absorptive response requires equilibrium, temperature, operator-order, and normalization information.

Numerical Evaluation

Hubbard-chain spin susceptibility

For an equilibrium calculation in a Hubbard chain, first fix hopping $t$, interaction $U$, filling, length, and boundary conditions. Choose the canonical or grand-canonical ensemble to match the controlled variables. Fixing total magnetization suppresses its fluctuations, so that constraint cannot be imposed indiscriminately when calculating its equilibrium response.

State the field coupling, magnetization operator, and whether the reported susceptibility is total, per site, or per volume. Compare a short chain with exact diagonalization before extending the calculation. Vary temperature and physical size separately from numerical tolerances. An equilibrium field derivative and an isolated dynamic response can probe different protocols; state the order of static, uniform, and large-system limits before comparing them.

Finite differences

For an equilibrium source,

$$\chi \approx \frac{ \langle A\rangle_{+\delta f} - \langle A\rangle_{-\delta f} }{ 2\delta f }.$$

Repeat at several $\delta f$. A field that is too large leaves linear response; one that is too small amplifies solver and sampling noise.

Correlation estimators

When an exact fluctuation or Kubo–Mori identity applies, a zero-source estimator can be more efficient than separate source simulations. Autocorrelation time, ensemble constraints, disconnected subtraction, and operator ordering must still be controlled.

Lehmann and real-time methods

Exact diagonalization can evaluate matrix elements and transition denominators directly. Real-time evolution can extract $\chi^R(t)$ after an impulse. Finite size gives discrete lines, and finite duration limits frequency resolution.

Imaginary-time methods

Quantum Monte Carlo and thermal tensor-network methods often produce imaginary-time correlations. Static susceptibilities may be accessible by integration without reconstructing a full real-frequency spectrum. Dynamic susceptibilities require analytic continuation, which is ill conditioned.

Response matrices

For multiple channels:

1. use the same units and normalization across sizes;
2. symmetrize only when a justified equilibrium identity requires it;
3. report eigenvectors as well as eigenvalues;
4. test stability against basis enlargement;
5. separate symmetry-enforced zeros from numerical noise.

Common pitfalls

Equal-time variance is not always the static quantum susceptibility. The simple $\beta\operatorname{Var}(B)$ expression requires a conserved $B$. Noncommuting operators need the imaginary-time integral, as the transverse-spin example in the Kubo derivation shows.

Local and uniform sources ask different questions. The diagonal entry $\chi_{ii}$ measures a local perturbation; a uniform chemical-potential change uses the entire row sum. Correlations between sites can change that sum substantially.

A large pair response is not a complete superconducting diagnosis. Pairing amplitude, phase stiffness, and magnetic screening are distinct measurements, particularly in low dimensions and near strong fluctuations.

Exercises

Exercise 1: Curie susceptibility

For one spin-$1/2$ moment with

$$H = -g\mu_{\mathrm B}B \frac{\sigma_z}{2},$$

derive the zero-field susceptibility to $B$.

Solution

The two energies are

$$E_\pm = \mp \frac{g\mu_{\mathrm B}B}{2}.$$

The partition function is

$$Z = 2\cosh \left( \frac{\beta g\mu_{\mathrm B}B}{2} \right).$$

The magnetic moment is

$$\langle\mu_z\rangle = \frac{1}{\beta} \frac{\partial\ln Z}{\partial B} = \frac{g\mu_{\mathrm B}}{2} \tanh \left( \frac{\beta g\mu_{\mathrm B}B}{2} \right).$$

Differentiating,

$$\chi_B(B) = \frac{\beta(g\mu_{\mathrm B})^2}{4} \operatorname{sech}^2 \left( \frac{\beta g\mu_{\mathrm B}B}{2} \right).$$

Thus

$$\chi_B(0) = \frac{(g\mu_{\mathrm B})^2} {4k_{\mathrm B}T}.$$

For a number density $n_s$ of independent moments, multiply by $n_s$ to obtain the bulk response to $B$.

Exercise 2: Chemical potential versus potential energy

A calculation gives

$$\delta n = \Pi\,\delta U$$

with $\Pi(\mathbf q,0)\lt0$. Show how to express the same result using a chemical-potential source.

Solution

The potential-energy perturbation is

$$H' = +\int d^d r\, U n.$$

The chemical-potential convention is

$$H' = -\int d^d r\, \delta\mu\,n.$$

Therefore

$$U=-\delta\mu.$$

Substitution gives

$$\delta n = -\Pi\,\delta\mu.$$

Hence

$$\chi_{nn} = \frac{\delta n}{\delta\mu} = -\Pi.$$

A negative static polarization and positive static chemical-potential susceptibility are consistent.

Exercise 3: Conserved uniform density

Show why a particle-number-conserving isolated system can have

$$\chi^R_{NN}(\omega)=0$$

and still have nonzero isothermal compressibility.

Solution

If

$$[H_0,N]=0,$$

then

$$N(t)=N.$$

The retarded commutator is therefore

$$\chi^R_{NN}(t) = \frac{i}{\hbar} \theta(t) \langle[N,N]\rangle =0.$$

The isothermal compressibility instead compares equilibrium states at nearby chemical potentials:

$$\kappa_n = \left( \frac{\partial n}{\partial\mu} \right)_T.$$

Changing $\mu$ reweights number sectors in the grand-canonical ensemble or compares neighboring fixed-$N$ free energies. Isolated unitary evolution does neither. The two responses use different protocols and limit orders.

Exercise 4: Local and global compressibility

Given

$$\chi_{ij} = \frac{ \partial\langle n_i\rangle }{ \partial\mu_j },$$

derive the response of site $i$ and of total number $N$ to a uniform chemical-potential change.

Solution

A uniform change satisfies

$$\delta\mu_j = \delta\mu$$

for every $j$. Therefore

$$\delta\langle n_i\rangle = \sum_j \chi_{ij} \delta\mu,$$

so

$$\frac{ \partial\langle n_i\rangle }{ \partial\mu } = \sum_j\chi_{ij}.$$

Since

$$N=\sum_i n_i,$$

the total response is

$$\frac{ \partial\langle N\rangle }{ \partial\mu } = \sum_{i,j}\chi_{ij}.$$

The diagonal element $\chi_{ii}$ is only the response at $i$ to a source applied at $i$. It is not generally either of these uniform-source responses.

Exercise 5: Static ground-state susceptibility

For a nondegenerate ground state of $H_0$ and perturbation $H(f)=H_0-fB$, derive

$$\chi_B(0) = 2 \sum_{m\ne0} \frac{ \lvert B_{m0}\rvert^2 }{ E_m-E_0 }.$$

Solution

Second-order nondegenerate perturbation theory gives

$$E_0(f) ={} E_0 - fB_{00} - f^2 \sum_{m\ne0} \frac{ \lvert B_{m0}\rvert^2 }{ E_m-E_0 } + O(f^3).$$

The expectation conjugate to $f$ is

$$\langle B\rangle_f = -\frac{dE_0(f)}{df}.$$

Hence

$$\chi_B(0) = \left. \frac{ d\langle B\rangle_f }{ df } \right|_{f=0} = -E_0''(0),$$

which yields the stated sum. Every denominator is positive, so the susceptibility is nonnegative.

Degenerate ground states require degenerate perturbation theory and a specified symmetry-breaking or ensemble prescription.

Exercise 6: Pair-field source

For

$$H_{\mathrm{pair}}' = -\eta^*\Delta - \eta\Delta^\dagger,$$

identify the retarded correlator that gives the response of $\langle\Delta\rangle$ to $\eta$.

Solution

The source $\eta$ multiplies $\Delta^\dagger$ with a minus sign. Therefore the Kubo detector is $A=\Delta$ and the source-coupled operator is $B=\Delta^\dagger$:

$$\chi^R_{\Delta\Delta^\dagger}(t) = \frac{i}{\hbar} \theta(t) \left\langle \left[ \Delta(t), \Delta^\dagger(0) \right] \right\rangle.$$

If an intensive pair amplitude is desired, divide by the chosen volume or site normalization.

The source explicitly changes particle number by two while present. The derivative is evaluated at zero source, so this construction is compatible with an exactly number-conserving source-free Hamiltonian.

Exercise 7: Positivity of a static matrix

Let

$$\chi^T_{ab} = -\frac{ \partial^2\Omega }{ \partial f_a\partial f_b }.$$

Explain why $\mathbf v^{\mathsf T}\boldsymbol{\chi}^T\mathbf v\ge0$ for every real source vector $\mathbf v$ in a Hermitian equilibrium channel basis.

Solution

Choose one combined Hermitian operator using real coefficients:

$$B_{\mathbf v} = \sum_a v_aB_a$$

with source amplitude $\lambda$. The quadratic form is the second response of the conjugate expectation:

$$\mathbf v^{\mathsf T} \boldsymbol{\chi}^T \mathbf v = -\left. \frac{d^2\Omega(\lambda)}{d\lambda^2} \right|_{\lambda=0}.$$

The equilibrium grand potential is concave in a linearly coupled source. Equivalently, the second derivative is a Kubo–Mori covariance of $B_{\mathbf v}$ with itself and is nonnegative after the minus sign. Thus the susceptibility matrix is positive semidefinite.

A negative eigenvalue signals an inconsistent convention, an unstable constrained state, or a calculation that is not the equilibrium Hessian being claimed.

Exercise 8: Finite-size peak

A sequence of finite lattices shows a growing magnetic-susceptibility peak. List the checks needed before claiming a continuous phase transition.

Solution

One should:

1. verify that the source, moment normalization, and volume factor are identical across sizes;
2. estimate statistical and truncation errors;
3. test several sizes rather than two;
4. fit the peak height, width, and location to a finite-size scaling ansatz;
5. compare with correlation-length and order-parameter observables;
6. rule out a level crossing, first-order coexistence, Schottky peak, or crossover;
7. control boundary conditions and aspect ratio;
8. state the order of thermodynamic and zero-field limits.

A finite system has no finite-temperature divergence. The claim must come from consistent scaling toward a thermodynamic singularity.

Related reading

• Random Phase Approximation distinguishes irreducible polarization and screened density response.
• Magnetic Susceptibility explains experimental fields, demagnetization, and unit conversion.
• Fermi Liquid Theory Preview relates compressibility and spin response to Landau parameters.
• BCS Mean-Field Theory derives a specific pair-response instability criterion.
• Critical Exponents and Scaling develops finite-size tests for growing response peaks.

References

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• H. B. Callen and T. A. Welton, “Irreversibility and Generalized Noise”, Physical Review 83, 34–40 (1951).
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• R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems”, Journal of the Physical Society of Japan 12, 570–586 (1957).
• R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, 2nd ed., Springer, 1991.
• J. Lindhard, “On the Properties of a Gas of Charged Particles”, Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser 28, no. 8 (1954).
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