Correlation Functions Overview

A many-body correlation function is an expectation value of an ordered product of operators at specified positions, times, or modes. Correlators answer targeted questions that a many-body wavefunction does not answer at a glance: whether distant regions fluctuate together, whether order persists, which momenta dominate, which excitation energies can be reached, how rapidly memory is lost, and how a system responds to a weak probe.

For a state $\rho$,

$$\langle X\rangle_\rho = \operatorname{Tr}(\rho X).$$

The basic two-point function is therefore

$$G_{AB}(1,2) = \left\langle A(1)B(2) \right\rangle_\rho,$$

where each label can include site, position, time, spin, orbital, or other indices. This compact notation is incomplete until the state, operator order, time prescription, normalization, and boundary conditions are stated.

Correlations are central in many-body physics because low-order local observables are often measurable and computationally accessible even when the full state occupies an exponentially large Hilbert space. They are not a magical compression of all information: a small set of low-order correlators generally does not determine an arbitrary interacting state.

Helpful background. Density operators explain the expectation value, and the Heisenberg picture supplies operator time evolution.

Correlation functions and physical questions

This introduction connects spatial correlations to order and time-dependent correlations to excitation spectra. For a quick comparison of conventions, use Correlation Function Definitions. Detailed calculations follow in Equal-Time Correlations, Time-Dependent Correlations, and the Kubo formula.

In the spin examples below, $s^\alpha=\sigma_\alpha/2$ is dimensionless; physical angular momentum is $S^\alpha=\hbar s^\alpha$. Lattice operators are bounded unless stated otherwise. Continuum field products require a regulator or spatial smearing.

Why Correlators Replace a State-Centered View

A pure state of $L$ spin-$1/2$ sites requires $2^L$ amplitudes in a generic basis. A local experiment rarely reconstructs all of them. Instead it may measure quantities such as

$$\left\langle s_i^zs_j^z \right\rangle, \qquad \left\langle n_i n_j \right\rangle, \qquad \left\langle a_i^\dagger a_j \right\rangle,$$

or their Fourier and frequency transforms.

These observables can reveal:

• local density and magnetization profiles;
• correlation lengths and ordering wavevectors;
• superfluid or condensate coherence;
• pairing tendencies;
• collective modes and excitation continua;
• conserved quantities and hydrodynamic tails;
• critical exponents and universality;
• the validity or failure of Gaussian, mean-field, and quasiparticle descriptions.

Correlators also connect theory to experiments. Scattering probes, spectroscopy, microscopy, interferometry, noise measurements, and transport all access particular ordered combinations of operators, often after convolution with form factors, detector resolution, and finite-time windows.

The right question is therefore not merely “What is the correlation function?” It is:

1. Which operators represent the source and detector?
2. Which state or ensemble defines the expectation?
3. Are positions, times, and internal indices fixed?
4. Is the product ordinary, symmetrized, time ordered, normal ordered, or retarded?
5. Is the disconnected contribution subtracted?
6. Which Fourier and normalization convention is used?

The Correlation Hierarchy

Zero-point function

In generating-functional language, the zero-point object is the normalization or partition function. For an ordinary normalized state,

$$\langle I\rangle_\rho = 1.$$

This trivial identity becomes nontrivial in path integrals, perturbation theory, and statistical mechanics because unnormalized vacuum or thermal diagrams must be divided by the appropriate normalization.

One-point functions

A one-point function is an expectation value,

$$\phi_A(1) = \langle A(1)\rangle_\rho.$$

Examples include local density,

$$\bar n_i = \langle \hat n_i\rangle,$$

local magnetization,

$$\mathbf m_i = \langle\mathbf s_i\rangle,$$

and an order parameter selected by a field or boundary condition.

One-point functions reveal inhomogeneity and explicit or spontaneous symmetry breaking. They can vanish for several different reasons:

• the state is genuinely disordered;
• a symmetry forces the expectation to zero;
• a finite system is in a symmetric superposition or mixture of ordered sectors;
• the chosen operator has the wrong quantum numbers;
• spatial averaging cancels a modulated pattern.

Consequently, a vanishing one-point function does not by itself exclude order.

Two-point functions

The ordered two-point function is

$$G_{AB}(1,2) = \langle A(1)B(2)\rangle_\rho.$$

It compares two insertions and is the first level that can diagnose spatial or temporal relationships. If $A(1)$ and $B(2)$ do not commute, reversing the order produces a different object:

$$\langle A(1)B(2)\rangle_\rho \ne \langle B(2)A(1)\rangle_\rho$$

in general.

Two-point functions are especially important because:

• quadratic or Gaussian theories are largely controlled by them;
• linear response is built from a two-point commutator;
• one-particle spectra are encoded in field two-point functions;
• static and dynamic structure factors are their Fourier transforms;
• correlation length and long-range order are usually diagnosed from their asymptotics.

Higher-point functions

An ordered $n$-point function is

$$G^{(n)}(1,\ldots,n) = \left\langle O_1(1)\cdots O_n(n) \right\rangle_\rho.$$

Higher-point functions diagnose nonlinear response, non-Gaussian fluctuations, scattering among excitations, multi-particle coherence, and constraints invisible to two-point data.

For canonical fields in a Gaussian state with a specified ordering convention, Wick factorization reduces higher moments to sums of products of one- and two-point functions. Equivalently, their connected cumulants above second order vanish. Nonlinear observables of Gaussian fields can still have higher cumulants, and in an interacting non-Gaussian state higher connected functions contain independent information.

The phrase “the two-point function determines the theory” is therefore valid only under substantial assumptions, such as Gaussianity or a reconstruction framework using the complete hierarchy rather than one correlator.

Connected Correlations

Define centered operators

$$\delta A = A-\langle A\rangle_\rho I, \qquad \delta B = B-\langle B\rangle_\rho I.$$

For a fixed operator order, the connected two-point function is

$$\begin{aligned} G^c_{AB}(1,2) &= \left\langle \delta A(1)\delta B(2) \right\rangle_\rho \\ &= \langle A(1)B(2)\rangle_\rho - \langle A(1)\rangle_\rho \langle B(2)\rangle_\rho. \end{aligned}$$

The disconnected product is what would remain if the two insertions factorized at the level of this moment. Subtracting it isolates correlated fluctuations.

For three ordered operators,

$$\langle ABC\rangle_c ={} \langle ABC\rangle - \langle A\rangle\langle BC\rangle - \langle B\rangle\langle AC\rangle - \langle C\rangle\langle AB\rangle + 2\langle A\rangle \langle B\rangle \langle C\rangle.$$

The ordering in every moment must remain consistent. General connected functions are quantum analogues of cumulants and can be generated by the logarithm of a source-dependent generating functional.

The dedicated Connected Correlation Functions page develops the full partition hierarchy, cluster decomposition, phase selection, correlation lengths, finite-size constraints, and practical subtraction workflow.

Connected is not the same as symmetrized

For noncommuting Hermitian observables, $G^c_{AB}$ can be complex. The real symmetrized covariance is

$$\operatorname{Cov}_\rho(A,B) = \frac{1}{2} \left\langle \delta A\,\delta B + \delta B\,\delta A \right\rangle_\rho.$$

Connectedness subtracts means. Symmetrization changes operator order. They solve different problems and should not be conflated.

Connected is not the same as entangled

If a state factorizes across two regions,

$$\rho_{AB} = \rho_A\otimes\rho_B,$$

then every connected correlation between an operator on $A$ and an operator on $B$ vanishes.

If every pair of local observables has zero connected correlation, the state does factorize (for finite-dimensional subsystems, expand it in complete operator bases). A single vanishing correlator is insufficient. Conversely, a nonzero correlator need not imply entanglement: a separable mixture can correlate local outcomes. For example,

$$\rho_{\mathrm{cl}} = \frac{1}{2} \left( \lvert\uparrow\uparrow\rangle \langle\uparrow\uparrow\rvert + \lvert\downarrow\downarrow\rangle \langle\downarrow\downarrow\rvert \right)$$

is not entangled, yet

$$\langle s_1^z\rangle = \langle s_2^z\rangle = 0, \qquad \langle s_1^zs_2^z\rangle_c = \frac{1}{4}.$$

A nonzero connected correlator diagnoses failure of product factorization for the chosen operators, not entanglement by itself.

Equal-Time Correlations

Equal-time correlators compare operators on one spatial slice. Common lattice examples are:

Spin correlation

$$C_s(i,j) = \left\langle \mathbf s_i\cdot\mathbf s_j \right\rangle.$$

Connected density correlation

$$C_n(i,j) = \langle n_i n_j\rangle - \langle n_i\rangle \langle n_j\rangle.$$

One-body density matrix

$$G^{(1)}(i,j) = \left\langle a_i^\dagger a_j \right\rangle.$$

Pair correlation

$$P(i,j) = \left\langle \Delta_i^\dagger\Delta_j \right\rangle,$$

where $\Delta_i$ is a specified pair operator.

These objects answer different questions. Density correlations probe number fluctuations, $G^{(1)}$ probes single-particle coherence, and $P$ probes pair coherence. No one formula should be labeled simply “the correlation” without naming its operators.

At coincident points, operator algebra and regularization matter. For lattice fermions, $n_i^2=n_i$ for one mode. In the continuum, products at the same position can contain contact terms or require a regulator. Coincident-point formulas should not be obtained by blindly setting two separated coordinates equal.

The dedicated Equal-Time Correlations page develops these four operator channels, their exact checks, spatial averaging, measurement maps, and finite-size cautions.

Spatial Translation Symmetry

For a translation-invariant state and translation-covariant local operators,

$$G_{AB}(i,j) = G_{AB}(\mathbf r_i-\mathbf r_j).$$

The two-position problem becomes a function of separation. If the state breaks translation symmetry, contains a boundary, trap, impurity, or disorder realization, the dependence on the center coordinate generally remains.

For modulated correlations, a useful asymptotic form is

$$G^c(r) \sim A \cos(Qr+\varphi) \frac{e^{-r/\xi}}{r^p},$$

where:

• $Q$ is a characteristic ordering or oscillation wavevector;
• $\xi$ is a correlation length;
• $p$ is an algebraic prefactor exponent;
• $A$ and $\varphi$ depend on normalization and operator choice.

This expression is schematic rather than universal. Several correlation lengths or oscillation wavevectors can coexist, and long-range interactions can alter the asymptotic form.

Schematic spatial signatures. A short-range connected correlator can decay exponentially with correlation length $\xi$; a critical correlator can decay algebraically; and a full order-parameter correlator can approach a nonzero plateau $m^2$. Oscillations and algebraic prefactors may accompany any of these envelopes.

Exponential Clustering

In a short-range correlated phase,

$$\left\lvert G^c_{AB}(r) \right\rvert \lesssim C e^{-r/\xi}$$

for large separation, possibly multiplied by a power of $r$. The length $\xi$ sets the scale over which local fluctuations remain appreciably correlated.

For a finite-range quantum spin Hamiltonian with uniformly bounded local terms, a unique ground state and a gap that stays nonzero as the system grows give exponential clustering of bounded local observables. The uniform gap is essential: a finite-size gap that closes with size is insufficient. Degenerate ground spaces and long-range interactions require additional hypotheses; see Hastings and Koma (2006).

Exponential decay does not mean the correlation is exactly zero beyond $\xi$. It means the asymptotic envelope is exponentially suppressed.

Algebraic Correlations

At many continuous critical points and in stable gapless phases,

$$G^c_{OO}(r) \sim \frac{ \cos(\mathbf Q\cdot\mathbf r+\varphi) }{ r^{2\Delta_O} },$$

where $\Delta_O$ is the scaling dimension of the operator in the long-distance theory.

Algebraic decay has no finite exponential correlation length. It does not imply a nonzero long-range plateau because the correlator still tends to zero. It is often called quasi-long-range order in appropriate low-dimensional phases. It signals fluctuations on all scales and often leads to nonanalytic momentum dependence near $\mathbf Q$.

Finite-size data can make a large but finite correlation length resemble a power law over a restricted range. Reliable identification requires multiple sizes, distances well separated from the lattice spacing and boundaries, and comparison of competing functional forms.

Long-Range Order

Long-Range Order develops the finite-size and modulated-order versions of the following criterion.

For a Hermitian order-parameter operator $O_i$, long-range order is indicated by a nonzero asymptotic full correlator,

$$\lim_{\lvert\mathbf r_i-\mathbf r_j\rvert\to\infty} \langle O_i O_j\rangle = m^2 \ne 0,$$

with an appropriate phase factor for modulated order.

In a symmetry-broken pure phase with $\langle O_i\rangle=m$, cluster decomposition gives

$$\langle O_iO_j\rangle \longrightarrow \langle O_i\rangle \langle O_j\rangle,$$

so the connected correlator tends to zero even though the full correlator tends to $m^2$.

In a finite symmetric state, one often has

$$\langle O_i\rangle = 0$$

while

$$\langle O_iO_j\rangle \longrightarrow m^2$$

over distances short compared with the system size. The full two-point function then detects incipient order that the one-point function misses. This is why finite-size studies often use squared order parameters or structure factors rather than inserting symmetry breaking by hand.

For ordering wavevector $\mathbf Q$, define

$$m_{\mathbf Q}^2(L) = \frac{1}{L^2} \sum_{i,j} e^{i\mathbf Q\cdot(\mathbf r_i-\mathbf r_j)} \langle O_iO_j\rangle.$$

If

$$\lim_{L\to\infty} m_{\mathbf Q}^2(L) \gt 0,$$

the scaling is consistent with long-range order. Boundary conditions, aspect ratio, operator normalization, and order of limits must accompany the claim.

Off-diagonal long-range order

Order need not appear in a density-density correlator. For bosons, the one-body density matrix

$$\rho^{(1)}(\mathbf x,\mathbf y) = \left\langle \psi^\dagger(\mathbf x) \psi(\mathbf y) \right\rangle$$

diagnoses one-particle coherence. In an inhomogeneous system, the basis-independent Penrose–Onsager criterion examines whether its largest eigenvalue scales extensively with particle number.

Pair condensation is instead diagnosed through an appropriate two-particle density matrix or pair correlator. The operator and reduced-density-matrix rank must match the type of order being tested. The normalization and contraction rules for $\Gamma^{(2)}$ are fixed in Two-Body Operators.

Time-Dependent Correlations

For a time-independent Hamiltonian, Heisenberg operators are

$$A_H(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}.$$

An ordinary ordered correlator is

$$C_{AB}(t,t') = \left\langle A_H(t)B_H(t') \right\rangle_\rho.$$

If the state is stationary,

$$[\rho,H] = 0,$$

then simultaneous time translation changes nothing:

$$C_{AB}(t,t') = C_{AB}(t-t',0).$$

This reduction to one time difference does not hold for a generic quench state, driven system, aging state, or explicitly time-dependent Hamiltonian.

Lehmann representation

Let

$$H\lvert n\rangle = E_n\lvert n\rangle, \qquad \rho = \sum_n p_n \lvert n\rangle\langle n\rvert$$

for a stationary state diagonal in energy. Then

$$C_{AB}(t) ={} \sum_{n,m} p_n e^{i(E_n-E_m)t/\hbar} \times \langle n\lvert A\rvert m\rangle \langle m\lvert B\rvert n\rangle.$$

The correlator is a weighted sum of transition frequencies. Its Fourier transform,

$$S_{AB}(\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t} C_{AB}(t),$$

has the spectral form

$$S_{AB}(\omega) ={} 2\pi \sum_{n,m} p_n \langle n\lvert A\rvert m\rangle \langle m\lvert B\rvert n\rangle \times \delta\!\left( \omega-\frac{E_m-E_n}{\hbar} \right).$$

For $B=A^\dagger$, the weights are nonnegative. The locations encode energy differences, while the weights encode matrix elements and state populations.

Temporal decay and recurrence

In a finite isolated system with a discrete spectrum, $C_{AB}(t)$ is generally quasiperiodic rather than irreversibly decaying to zero. Apparent relaxation can arise from dephasing among many frequencies, and recurrences remain possible.

A smooth decay law usually presupposes a thermodynamic or continuum limit, coarse graining, disorder averaging, coupling to an environment, or a restricted observational time window. A conserved component can produce a nondecaying contribution even in a large system.

The phrase “correlation time” is therefore operational. It may describe an exponential envelope, an integral of a normalized correlator, or the width of a spectral feature. The definition must be stated.

The dedicated Time-Dependent Correlations page develops the ordinary two-time function, its Fourier convention, detailed balance, finite-system recurrence, numerical estimation, and nonequilibrium generalization.

Ordering Prescriptions

Several two-point objects use the same operators but answer different questions.

Object	Definition	Main role
ordinary ordered	$\langle A(t)B(0)\rangle$	fluctuations and transition spectra
reversed ordered	$\langle B(0)A(t)\rangle$	complementary transition ordering
symmetrized	$\frac12\langle\{\delta A(t),\delta B(0)\}\rangle$	real noise-like fluctuations
commutator	$\langle[A(t),B(0)]\rangle$	quantum order sensitivity and response kernel
retarded susceptibility	$\frac{i}{\hbar}\theta(t)\langle[A(t),B(0)]\rangle$	causal response to $H'=-fB$
time ordered	$\langle\mathcal T A(t)B(0)\rangle$	perturbation theory and propagators
imaginary time	$\langle\mathcal T_\tau A(\tau)B(0)\rangle_\beta$	thermal equilibrium methods

Green-function conventions often add factors of $-i$, $1/\hbar$, and statistics-dependent signs. The table gives structural roles, not a universal notation standard.

An ordinary correlator is not automatically a response function. Fluctuations and Susceptibilities explains when an integrated equilibrium correlator gives a static thermodynamic response and when Kubo–Mori ordering is required. For a perturbation $H'(t)=-f(t)B$, the retarded susceptibility of $A$ is

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

Causality enters through $\theta(t)$, and the commutator compares the two operator orders. Retarded and Advanced Response develops the paired support, analyticity, and spectral discontinuity. The Kubo Formula develops the source derivation, sign conventions, and contact-term subtleties.

Momentum and Frequency Space

For a translation-invariant lattice, the static structure factor associated with an operator $O_i$ is commonly

$$S_O(\mathbf q) = \frac{1}{L} \sum_{i,j} e^{-i\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} \langle O_iO_j\rangle.$$

Some authors use the connected correlation instead, subtract elastic or disconnected pieces, or normalize by particle number rather than site number.

A dynamic structure factor can be defined as

$$S_O(\mathbf q,\omega) ={} \frac{1}{2\pi L} \int_{-\infty}^{\infty} dt\, e^{i\omega t} \times \sum_{i,j} e^{-i\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} \langle O_i(t)O_j(0)\rangle.$$

The momentum dependence reveals spatial patterns; the frequency dependence resolves transition energies and lifetimes. A sharp dispersing peak can indicate a quasiparticle, while a broad continuum can indicate decay, fractionalization, disorder, or unresolved multi-particle states. Instrumental resolution must be separated from intrinsic width.

Fourier transformation does not remove convention choices. Signs, factors of $2\pi$, volume factors, connected subtraction, and whether $\omega$ is angular frequency or energy all matter.

The dedicated Structure Factors page develops these transforms, elastic and inelastic weight, detailed balance, sum-rule checks, and probe-specific forward models.

Correlations and Conservation Laws

Exact constraints produce sum rules among correlators. If total particle number

$$\hat N = \sum_j n_j$$

is fixed with no fluctuation, then

$$\begin{aligned} \sum_j \langle \delta n_i\delta n_j \rangle &= \left\langle \delta n_i\delta N \right\rangle \\ &= 0. \end{aligned}$$

Positive onsite number variance must be balanced by offsite anticorrelations. Such correlations can arise from a global constraint even without a local interaction.

Likewise, in a total-spin singlet,

$$\mathbf s_{\mathrm{tot}} \lvert\psi\rangle = 0,$$

so

$$\sum_j \left\langle \mathbf s_i\cdot\mathbf s_j \right\rangle = 0.$$

The positive onsite term $s(s+1)$ is balanced by negative correlations with other sites. Sum rules are powerful checks on both analytic derivations and numerical data.

Choosing a correlator for a model

In the Heisenberg model, spin correlations identify magnetic patterns and excitation spectra. In the Hubbard model, charge, spin, one-body, and pair correlators answer different questions about the same state. Enhancement of one short-distance pair correlation is not sufficient to establish a superconducting phase.

Even a noninteracting gas has exchange correlations. Gaussian-state Wick factorization and fixed-particle-number constraints explain different contributions; neither should be mistaken for a dynamical interaction.

Correlations, Reduced States, and Information

For two regions $A$ and $B$, every correlator of local operator bases can be computed from the reduced state $\rho_{AB}$. Conversely, a complete operator basis of sufficiently many expectation values reconstructs that reduced state.

In practice, one usually measures only a small subset. Two states can share all one- and selected two-point functions while differing in higher correlations, entanglement, topology, or global constraints.

Correlation functions and information measures answer complementary questions:

• a correlator is operator specific and can retain sign, momentum, and dynamical information;
• mutual information is basis independent and bounds total correlation between regions;
• entanglement measures separate quantum correlation under specified assumptions;
• reduced density matrices contain the complete local information but grow rapidly with subsystem size.

No single scalar measure replaces the full diagnostic set.

Comparing a calculation with a measurement

A finite isolated system has discrete transition lines. Replacing each delta function by a Lorentzian of width $\eta$ makes a readable plot, but $\eta$ is not a physical decay rate. Test how conclusions change with size and broadening. The spectral-functions discussion distinguishes intrinsic linewidths from resolution.

An experimental spectrum also includes the probe’s matrix elements, spatial form factor, temperature, and detector resolution. Compare normalization and integrated weight as well as peak positions. Imaginary-time simulations introduce a further difficulty: recovering real-frequency spectra requires an ill-conditioned analytic continuation.

Common pitfalls

A zero mean need not mean disorder. A finite symmetric state can have $\langle O_i\rangle=0$ and a nondecaying two-point function. Compare the full and connected correlators and specify whether a symmetry-breaking phase has been selected.

A plateau and a slow decay are different. Algebraic decay tends to zero, however slowly. Multiple sizes and separations are needed to distinguish it from long-range order or exponential decay with a large correlation length.

Ordering and subtraction are independent choices. Centering removes one-point means; symmetrization averages two operator orders; a retarded response adds causality and a commutator. Changing one choice does not silently change the others.

Exercises

Product-state factorization

Let $\rho=\rho_A\otimes\rho_B$, with $X_A$ acting only on $A$ and $Y_B$ only on $B$. Show that their connected correlation vanishes.

Solution

Using the tensor-product trace,

$$\begin{aligned} \langle X_AY_B\rangle_\rho &= \operatorname{Tr}_{AB} \left[ (\rho_A\otimes\rho_B) (X_A\otimes Y_B) \right] \\ &= \operatorname{Tr}_A(\rho_AX_A) \operatorname{Tr}_B(\rho_BY_B) \\ &= \langle X_A\rangle_\rho \langle Y_B\rangle_\rho. \end{aligned}$$

Therefore

$$\langle X_AY_B\rangle_c = 0.$$

The converse does not follow from one operator pair: vanishing of one selected connected correlator does not prove that the state factorizes.

Classical correlation without entanglement

For

$$\rho_{\mathrm{cl}} = \frac{1}{2} \left( \lvert\uparrow\uparrow\rangle \langle\uparrow\uparrow\rvert + \lvert\downarrow\downarrow\rangle \langle\downarrow\downarrow\rvert \right),$$

compute $\langle s_1^z\rangle$, $\langle s_2^z\rangle$, and $\langle s_1^zs_2^z\rangle_c$. Why does the answer not prove entanglement?

Solution

Each local spin is equally likely to be up or down, so

$$\langle s_1^z\rangle = \langle s_2^z\rangle = 0.$$

Both components have aligned outcomes:

$$\langle s_1^zs_2^z\rangle = \frac{1}{2} \left( \frac{1}{4} + \frac{1}{4} \right) = \frac{1}{4}.$$

Hence

$$\langle s_1^zs_2^z\rangle_c = \frac{1}{4}.$$

The density operator is explicitly a convex mixture of product states, so it is separable. The connected correlation comes from classical uncertainty about which aligned product state was prepared.

Fixed-number sum rule

Suppose $\hat N=\sum_jn_j$ has the exact value $N$ in the state. Prove

$$\sum_j \langle\delta n_i\delta n_j\rangle = 0.$$

What does this imply if $\langle(\delta n_i)^2\rangle\gt0$?

Solution

Exact particle number means

$$\delta N = \hat N-\langle\hat N\rangle$$

annihilates the support of the state. Since

$$\delta N = \sum_j\delta n_j,$$

we have

$$\begin{aligned} \sum_j \langle\delta n_i\delta n_j\rangle &= \left\langle \delta n_i \sum_j\delta n_j \right\rangle \\ &= \langle\delta n_i\delta N\rangle = 0. \end{aligned}$$

The $j=i$ term is the nonnegative onsite variance. If it is positive, the sum of offsite connected correlations must be negative. The global number constraint therefore produces anticorrelation somewhere away from the site.

Lehmann representation

For a stationary state $\rho=\sum_np_n\lvert n\rangle\langle n\rvert$, derive the spectral representation of $\langle A(t)B(0)\rangle$.

Solution

Insert one energy-basis resolution of the identity:

$$\begin{aligned} \langle A(t)B\rangle &= \sum_n p_n \langle n\lvert e^{iHt/\hbar} A e^{-iHt/\hbar} B \rvert n\rangle \\ &= \sum_{n,m} p_n e^{i(E_n-E_m)t/\hbar} A_{nm}B_{mn}. \end{aligned}$$

Fourier transformation uses

$$\int_{-\infty}^{\infty} dt\, e^{ixt} = 2\pi\delta(x)$$

and gives

$$S_{AB}(\omega) = 2\pi \sum_{n,m} p_nA_{nm}B_{mn} \delta\!\left( \omega-\frac{E_m-E_n}{\hbar} \right).$$

Symmetric finite-volume order

Consider the spin state

$$\lvert\mathrm{cat}\rangle = \frac{ \lvert\uparrow\uparrow\cdots\uparrow\rangle + \lvert\downarrow\downarrow\cdots\downarrow\rangle }{\sqrt2}.$$

Compute $\langle s_i^z\rangle$ and $\langle s_i^zs_j^z\rangle$ for distinct sites. What does the result teach about finite-size order diagnostics?

Solution

The two branches contribute opposite local magnetizations, and the cross terms vanish for $L\gt1$. Therefore

$$\langle s_i^z\rangle = 0.$$

In both branches, spins at distinct sites are aligned, so

$$\langle s_i^zs_j^z\rangle = \frac{1}{4}.$$

The one-point function vanishes because the state preserves the spin-flip symmetry, but the two-point function does not decay. Squared order parameters and long-distance correlations can reveal finite-size precursors of symmetry breaking even when a symmetry eigenstate has zero order-parameter expectation.

Structure-factor scaling

Suppose a translation-invariant scalar correlator approaches

$$\langle O_iO_j\rangle \longrightarrow m^2$$

at large separation and define

$$S_O(0) = \frac{1}{L} \sum_{i,j} \langle O_iO_j\rangle.$$

How does $S_O(0)$ scale with $L$?

Solution

There are order $L^2$ distant pairs, and each contributes asymptotically $m^2$. The prefactor is $1/L$, so

$$S_O(0) \sim Lm^2$$

up to subleading short-distance and finite-size corrections. Equivalently,

$$\frac{S_O(0)}{L} \longrightarrow m^2.$$

A peak that grows only as order one under this normalization does not by itself demonstrate long-range order.

Where to go next

• Connected Correlation Functions develops cumulants and clustering.
• Structure Factors converts spatial patterns into momentum-space observables.
• Green Functions in Many-Body QM treats particle addition and removal.
• Fluctuation–Dissipation Theorem relates equilibrium fluctuations to absorptive response.
• Many-Body Entanglement Overview explains what correlation functions do and do not reveal about entanglement.

References

• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).
• H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Matter Physics, Oxford University Press (2004).
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015).
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
• M. B. Hastings and T. Koma, “Spectral Gap and Exponential Decay of Correlations”, Communications in Mathematical Physics 265, 781–804 (2006).
• 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).
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).
• O. Penrose and L. Onsager, “Bose–Einstein Condensation and Liquid Helium”, Physical Review 104, 576–584 (1956).
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011).
• C. N. Yang, “Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of Superconductors”, Reviews of Modern Physics 34, 694–704 (1962).