Universal Relations and Tan Contact

Tan’s contact is the coefficient of a local opposite-spin pair operator after renormalization. It connects observables that live at very different scales: the large-momentum tail, the derivative of the energy with scattering length, the pressure relation, and high-frequency probe tails. These are independent measurements of one quantity only when their normalization, dimensional window, and range corrections agree.

Required background. Short-Range Scattering Data as Many-Body Inputs fixes zero-range matching, and Spectral Moments and Many-Body Sum Rules supplies short-time constraints.

Helpful background. Free-Field OPE Preview and Canonical Treatment introduces operator-product logic used in the interacting extension.

Contact normalization and momentum tail

Consider a two-component gas of equal-mass fermions in three dimensions with

N_\sigma=\int\!\frac{\mathrm d^3k}{(2\pi)^3}n_\sigma(\mathbf k).

Define the extensive contact $C$ by

n_\sigma(k)\longrightarrow\frac{C}{k^4} \quad\text{for}\quad k_{\rm many}\ll k\ll R^{-1}.

Both spin components have the same leading tail. The contact density is $\mathcal C=C/V$ . Alternative Fourier conventions move volume and $(2\pi)^3$ factors; a quoted contact must state which tail it multiplies.

The window is essential. $k$ must exceed Fermi, thermal, pairing, and trap-variation scales so that the two particles approach locally, yet remain below the inverse interaction range. At still larger $k$ , microscopic potential structure replaces the zero-range tail.

Adiabatic relation

After matching the bare coupling to $a$ , the Hellmann–Feynman theorem gives

\frac{\partial E}{\partial(-a^{-1})}\bigg|_{S,N,V} =\frac{C}{4\pi m},

or equivalently $\partial E/\partial a^{-1}=-C/(4\pi m)$ , with $\hbar=1$ . At finite temperature the same relation holds for the free energy at fixed $T,N,V$ . The derivative acts on the physical inverse scattering length, not on a regulator-dependent bare coupling.

To see the structure, write the renormalized interaction energy as $C_0\langle O_4\rangle$ . Differentiating $C_0^{-1}=m/(4\pi a)+$ regulator terms converts $\partial C_0/\partial(-a^{-1})$ into $C_0^2m/(4\pi)$ . The finite combination $m^2C_0^2\langle O_4\rangle$ is precisely $C$ in this convention.

Pressure relation from scaling

For a homogeneous zero-range gas, extensivity and dimensional scaling imply

P=\frac{2E}{3V}+\frac{C}{12\pi maV}.

At unitarity the second term vanishes and $P=2E/(3V)$ . Away from unitarity, varying volume changes the dimensionless combination $a^{-1}V^{1/3}$ ; inserting the adiabatic derivative yields the contact term. Finite effective range adds another derivative contribution proportional to $\partial E/\partial r_e$ , so the zero-range pressure identity cannot be used as exact when $k_Fr_e$ is resolved.

Short-distance and high-frequency relations

The operator-product expansion states that the leading nonanalytic short-distance contribution to a product of fields is the contact operator. Fourier transformation produces the $k^{-4}$ tail. The same Wilson coefficient controls high-frequency tails in radio-frequency spectroscopy and dynamic response, but probe-specific final-state interactions and matrix elements multiply or modify the asymptote. The general OPE derivation is given by Braaten and Platter 2008, while Tan’s original energy, momentum, and pressure relations appear in Tan 2008a and Tan 2008b.

For identical bosons, an analogous two-body contact exists, but Efimov physics introduces a three-body contact and log-periodic subleading structures. Fermionic two-component formulas must not be transferred without those terms.

Cross-validating the contact

Extract $C$ from at least two observables with the same normalization. A momentum-tail fit should vary its lower and upper fit boundaries; an adiabatic extraction should propagate uncertainty in $a$ and entropy or temperature; a pressure extraction should include trap and range corrections. Agreement supports a common short-distance description. Disagreement locates missing range, probe, final-state, or nonequilibrium effects rather than permitting separate redefinitions of $C$ .

Common pitfalls

Fitting outside the contact window. Low momentum contains many-body structure; momenta near $R^{-1}$ resolve microscopic physics.

Mixing contact density and total contact. $\mathcal C=C/V$ has a different dimension and volume scaling.

Ignoring probe final states. An rf tail is not simply the momentum tail with frequency relabeled.

Exercises

Derive the unitary pressure relation

Use dimensional analysis for a homogeneous zero-range gas at $a^{-1}=0$ to relate $E$ and $P$ .

Solution

With no interaction length, scaling all lengths by $\lambda$ sends $V\to\lambda^3V$ and $E\to\lambda^{-2}E$ at fixed entropy and particle number. Thus $-3VP=\mathrm dE/\mathrm d\ln\lambda=-2E$ , giving $P=2E/(3V)$ .

Compare contact conventions

If a paper defines $\tilde n_\sigma(\mathbf k)=n_\sigma(\mathbf k)/(2\pi)^3$ , what coefficient multiplies $k^{-4}$ in $\tilde n$ ?