From Microscopic Hamiltonians to Continuum Fields

A microscopic Hamiltonian yields a predictive continuum field theory only after one identifies a low-energy sector, projects or integrates out the remaining modes, matches enough observables to determine the retained couplings, and estimates the first omitted effects. Symmetry constrains the operator basis; it does not by itself set the coefficients or prove that discarded bands are irrelevant. This separation-of-scales logic is stated generally in Georgi 1993, pp. 213–218.

Required background. Second-Quantized Bosons and Fermions fixes lattice and continuum normalization. Matching with Amplitudes, Green Functions, and Background Fields supplies the general matching logic.

Helpful background. Degrees of Freedom, Symmetry, and the Local Operator Expansion helps construct a complete operator basis.

Select the low-energy sector

Start from a microscopic Hamiltonian $H_{\rm micro}$ and a target set of observables at characteristic energy and momentum $Q$. The reduction requires four declarations:

1. State: vacuum, dilute gas, Fermi sea, condensate, or another reference state.
2. Retained modes: momenta near specified minima or a Fermi surface, selected bands, spins, valleys, or collective coordinates.
3. Symmetries: exact microscopic symmetries and emergent symmetries expected only through a stated order.
4. Gap or separation: the scale $\Lambda_b$ at which discarded modes, nonlocality, or new inelastic channels become relevant.

Projection and integration are different. Projecting a Hilbert space removes matrix elements to discarded states and can miss virtual processes. Integrating out a gapped sector generates new operators such as longer-range hopping, exchange, and multibody interactions. A controlled construction either retains these effects to the desired order or bounds them.

Expansion near a band minimum

Consider bosons on a hypercubic lattice,

$$H=-t\sum_{\langle ij\rangle}(b_i^\dagger b_j+\text{h.c.}) +\frac U2\sum_i n_i(n_i-1)-\mu_{\rm lat}\sum_i n_i.$$

Near the minimum at $\mathbf k=0$, define $b_i=a^{d/2}\psi(\mathbf x_i)$. Expanding the hopping dispersion and replacing $a^d\sum_i$ by $\int\mathrm d^d x$ gives

$$H_{\rm eff}=\int\mathrm d^d x \left[ \frac{\nabla\psi^\dagger\cdot\nabla\psi}{2m^*} -\mu\,\psi^\dagger\psi +\frac{g_0}{2}\psi^\dagger\psi^\dagger\psi\psi +\mathcal H_{\rm corr} \right],$$

with

$$m^*=\frac1{2ta^2}, \qquad g_0=Ua^d,$$

at tree level. The chemical potential includes the band-bottom shift. The leading kinematic correction contains $-ta^4\sum_ip_i^4/12$, and interaction corrections include derivative and multibody operators.

The equality $g_0=Ua^d$ is a bare matching relation at the lattice scale, not yet a physical scattering amplitude. Loops renormalize it, and at strong lattice coupling the continuum scattering length must be obtained from the lattice two-body problem. This distinction is essential near resonance.

Match observables, not field labels

Choose a set of low-energy quantities sufficient to fix the coefficients at the desired order. Examples include:

• the one-particle pole and curvature, fixing energy offset, residue convention, and effective mass;
• a two-body phase shift or bound-state energy, fixing contact and derivative couplings;
• a matrix element of a density or current, fixing the observable map; and
• a three-body datum when renormalization or the desired precision requires it.

At order $n$, write schematically

$$\mathcal A_{\rm micro}(Q) =\mathcal A_{\rm EFT}(Q;C_0(\Lambda),C_2(\Lambda),\ldots) +O\!\left[(Q/\Lambda_b)^{n+1}\right].$$

Off-shell Green functions can be convenient matching intermediates, but local field redefinitions change them. Pole positions, scattering amplitudes, conserved charges, and properly transformed matrix elements provide invariant checks. Matching one observable and predicting another is stronger than fitting both.

A two-band elimination

Let low and high orbitals be separated by $\Delta$ and mixed by $V$. In block form,

$$H=\begin{pmatrix}H_L&V\\V^\dagger&H_H\end{pmatrix}.$$

For $E\ll\Delta$, eliminating the high component gives the energy-dependent low-space operator

$$H_{\rm eff}(E)=H_L+V(E-H_H)^{-1}V^\dagger.$$

Expanding the resolvent yields

$$H_{\rm eff}=H_L-VH_H^{-1}V^\dagger+O(EV^2/\Delta^2).$$

The second term is the virtual correction missed by naive projection. Its size $V^2/\Delta$ and the next ratio $E/\Delta$ provide independent magnitude and truncation checks. This elementary Schrieffer–Wolff structure recurs in superexchange, resonant two-channel models, and band projections.

Observable maps and double counting

An effective Hamiltonian is incomplete without effective observables. If a unitary transformation $e^S$ block-diagonalizes $H$, then a microscopic operator $O$ maps to

$$O_{\rm eff}=P e^S O e^{-S}P.$$

Using $PH P$ together with the untransformed $POP$ can violate sum rules at the same order as the Hamiltonian correction. Similarly, adding a phenomenological interaction already generated by integrating out a mode double counts it. Every retained coupling should state which modes have already contributed.

Validity and stopping rule

A continuum reduction is accepted only when:

• all operators allowed at the claimed order are included or shown redundant;
• matched observables are cutoff stable after coefficients are refitted;
• at least one unfit observable agrees through the expected error;
• the error changes with $Q/\Lambda_b$ as predicted; and
• no discarded gap closes in the state or parameter range under study.

Stop using the theory when the expansion parameter ceases to be small, an omitted channel goes on shell, a symmetry changes, or regulator variation cannot be absorbed by the retained coefficients.

Common pitfalls

Projecting instead of integrating out. Virtual high-energy states generate finite low-energy operators. Setting the high component to zero misses them.

Matching a bare coefficient once. A coefficient depends on regulator and operator basis. The observable must remain stable under allowed changes after rematching.

Transforming the Hamiltonian but not the probe. Effective currents and densities inherit corrections. Ignoring them can preserve energies while corrupting response.

Exercises

Derive the leading virtual correction

For $H_H=\Delta$ and a scalar mixing $V=v$, find the low eigenvalue through $O(v^2/\Delta)$.

Solution

The exact lower eigenvalue of $\left(\begin{smallmatrix}E_L&v\\v^*&\Delta\end{smallmatrix}\right)$ is

$$E_-=E_L-\frac{|v|^2}{\Delta-E_L}+O(|v|^4/\Delta^3) =E_L-\frac{|v|^2}{\Delta}+O(E_L|v|^2/\Delta^2,|v|^4/\Delta^3).$$

It agrees with $-VH_H^{-1}V^\dagger$ at leading order.

Identify the first anisotropic correction

Expand the hypercubic nearest-neighbor dispersion through fourth order and explain why it is not simply proportional to $(\mathbf p^2)^2$.

Solution

The expansion is $ta^2\mathbf p^2-(ta^4/12)\sum_ip_i^4+\cdots$. The quartic invariant $\sum_ip_i^4$ respects the hypercubic point group but is not fully rotationally invariant; a separate $(\mathbf p^2)^2$ operator may occur. Their difference measures emergent rotational breaking.

Continue

Nonrelativistic Power Counting and Universality ranks the generated operators and turns $Q/\Lambda_b$ into an error estimate. Short-Range Scattering Data as Many-Body Inputs replaces the tree-level contact coefficient by a matched amplitude.

References

• Georgi, Howard. “Effective Field Theory.” Annual Review of Nuclear and Particle Science 43 (1993): 209–252. DOI.

Further reading

• Braaten, Eric, and Hans-Werner Hammer. “Universality in Few-Body Systems with Large Scattering Length.” Physics Reports 428 (2006): 259–390. DOI. Open PDF.
• Schrieffer, J. Robert, and Peter A. Wolff. “Relation between the Anderson and Kondo Hamiltonians.” Physical Review 149 (1966): 491–492. DOI.