Quenched Disorder and Disorder Averaging

Quenched disorder is sampled once and held fixed while the quantum or thermal degrees of freedom evolve. The correct observable is therefore computed at fixed disorder and averaged only afterward. This ordering distinguishes $\overline{\ln Z_V}$ from $\ln \overline{Z_V}$, fixes which correlation functions are normalized before averaging, and exposes when rare samples invalidate a statement based only on a typical realization.

Required background. Coherent-state path integrals supplies the Euclidean many-body functional integral. Thermal density operators and KMS supplies the fixed-Hamiltonian ensemble whose disorder average is taken.

Helpful background. Connected correlators and cumulants supplies the cumulant expansion used below.

Random fields and the order of averages

Let a static scalar potential $V(\mathbf x)$ couple to the density,

$$S_V=S_0+\int_0^\beta d\tau\int d^d x\,V(\mathbf x)n(\tau,\mathbf x).$$

An ensemble is part of the model. A common Gaussian choice has

$$\overline{V(\mathbf x)}=0, \qquad \overline{V(\mathbf x)V(\mathbf y)}=\Delta C(\mathbf x-\mathbf y),$$

where $C$ is positive as a covariance kernel. White noise means $C(\mathbf r)=\delta^{(d)}(\mathbf r)$ only below a declared ultraviolet cutoff; correlated disorder retains a finite correlation length $\zeta$. The distribution tail, not just $\Delta$, controls rare regions; Lifshits, Gredeskul, and Pastur 1988, ch. 1 develops this ensemble-to-spectrum distinction.

For each realization, $Z_V=\operatorname{Tr}e^{-\beta H[V]}$ and $\langle O\rangle_V=Z_V^{-1}\operatorname{Tr}(Oe^{-\beta H[V]})$. Quenched quantities are

$$F_{\mathrm q}=-T\,\overline{\ln Z_V}, \qquad \overline{\langle O\rangle_V},$$

not $-T\ln\overline{Z_V}$ and not a ratio of separately averaged numerator and denominator. Jensen’s inequality gives $F_{\mathrm q}\ge F_{\mathrm a}$ for the annealed free energy $F_{\mathrm a}=-T\ln\overline{Z_V}$. The distinction is physical: annealing lets the disorder equilibrate with the matter.

The chapter’s dependency structure is summarized in the following original diagram. Inspect the first arrow: the ensemble and order of averages precede every replica, sigma-model, localization, or glass conclusion.

From quenched input to localization observables. Solid arrows add a derivation; dashed annotations mark ensemble, symmetry, dimension, or scale assumptions. The diagram is schematic and not a phase diagram.

Cumulants and disorder-induced interactions

For a general distribution, introduce connected disorder cumulants $K_p$. Averaging a source $J$ gives

$$\ln \overline{e^{\int J V}} =\sum_{p=1}^\infty\frac{1}{p!} \int K_p(x_1,\ldots,x_p)J(x_1)\cdots J(x_p).$$

For the centered Gaussian ensemble only $K_2=\Delta C$ survives. Because the Euclidean coupling supplies $J(\tau,\mathbf x)=-n(\tau,\mathbf x)$, averaging the unnormalized Boltzmann factor produces

$$\overline{e^{-\int Vn}} =\exp\!\left[ \frac{\Delta}{2}\int d^d x\,d^d y\,C(\mathbf x-\mathbf y) \int_0^\beta d\tau\,d\tau'\,n(\tau,\mathbf x)n(\tau',\mathbf y) \right].$$

The two independent imaginary times express static disorder. In an effective action the same term appears with a minus sign because the weight is $e^{-S_{\mathrm{eff}}}$. This sign and the double-time structure are useful checks on replica calculations.

Self-averaging and rare samples

For an extensive sample observable $X_L$, define

$$R_X(L)=\frac{\overline{X_L^2}-(\overline{X_L})^2}{(\overline{X_L})^2}.$$

$R_X\to0$ is self-averaging in the chosen ensemble and limit. It can fail at a random critical point or for observables dominated by broad tails; averages, medians, typical values $X_{\mathrm{typ}}=\exp\overline{\ln X}$, and full distributions then answer different questions. Aharony and Harris 1996 show why critical randomness can prevent strong self-averaging.

Rare-region estimates must retain the tail that creates them. A region of volume $\ell^d$ may occur with probability $p(\ell)\sim e^{-c\ell^d}$ while its relaxation time grows as $\tau(\ell)\sim e^{b\ell^d}$; eliminating $\ell$ produces a power-law distribution of times. Replacing the ensemble by its variance alone can erase this physics.

The canonical disorder and glass claim test matrix records which ensemble, limit, and negative test accompany each later claim.

Exercise

Quenched versus annealed. Let $V$ be a centered Gaussian variable with variance $\Delta$ and let a toy realization have $Z_V=e^{aV}$. Compute $F_{\mathrm q}$ and $F_{\mathrm a}$.

Solution

Since $\ln Z_V=aV$, $\overline{\ln Z_V}=0$ and $F_{\mathrm q}=0$. The Gaussian moment-generating function gives $\overline{Z_V}=e^{a^2\Delta/2}$, so $F_{\mathrm a}=-Ta^2\Delta/2$. Thus $F_{\mathrm q}\ge F_{\mathrm a}$, with strict inequality for $a^2\Delta>0$.

References

• Amnon Aharony and A. Brooks Harris, “Absence of Self-Averaging and Universal Fluctuations in Random Systems near Critical Points,” Physical Review Letters 77 (1996) 3700–3703. DOI
• Ilya M. Lifshits, Sergei A. Gredeskul, and Leonid A. Pastur, Introduction to the Theory of Disordered Systems, Wiley, 1988. WorldCat