Hamiltonian Truncation

Hamiltonian truncation replaces an infinite or very large state space by a finite subspace selected by energy, occupation, or another physically motivated cutoff. The finite matrix can be diagonalized accurately while omitted states still alter its low-energy physics. This page treats the numerical projection and its error; field-theory renormalization requires additional model-specific analysis.

Helpful background. Rayleigh–Ritz supplies variational projection, and basis-set convergence supplies refinement tests.

Projected and effective Hamiltonians

Let $P$ project onto retained states and $Q=I-P$. Writing the full eigenvector as $\psi_P+\psi_Q$ gives

$$\begin{aligned} PHP\psi_P+PHQ\psi_Q&=E\psi_P,\\ QHP\psi_P+QHQ\psi_Q&=E\psi_Q. \end{aligned}$$

When $E-QHQ$ is invertible on the relevant omitted space,

$$\psi_Q=(E-QHQ)^{-1}QHP\psi_P,$$

and the exact retained-space equation contains

$$H_{\rm eff}(E)=PHP+PHQ(E-QHQ)^{-1}QHP.$$

Bare truncation keeps only $PHP$. The second term describes virtual excursions into discarded states. If omitted energies are all well above $E$, the denominator is negative and these excursions usually lower low-energy levels. A nearby omitted state destroys the assumption that its influence is small.

Approximating this correction can improve cutoff convergence, but the resulting effective Hamiltonian need not retain the naive Ritz upper-bound property. Avoid counting a correction twice if it is already absorbed into fitted parameters.

Operator products do not commute with truncation

In general,

$$PA^2P=(PAP)^2+PAQAP.$$

Therefore squaring a truncated matrix can differ from truncating the exact square. For dimensionless oscillator position $x=(a+a^\dagger)/\sqrt2$ and the first $N$ oscillator states,

$$Px^2P-(PxP)^2 =\frac{N}{2}|N-1\rangle\langle N-1|.$$

Only the highest retained state can leave the space by one application of $x$ and return by the next. This identity gives an exact diagnostic for potential-matrix construction. Computing $x^4$ by repeatedly multiplying a too-small $x$ matrix can contaminate several boundary rows. Assemble analytic matrix elements or use sufficient temporary padding before projection.

The same issue affects commutators. No finite-dimensional matrices satisfy $[x,p]=iI$ exactly because the trace of a commutator is zero. Test the canonical relation on low-energy states, where truncation effects are small, rather than demanding an impossible full-matrix identity.

A two-space error estimate

Consider

$$H=\begin{pmatrix}0&v\\v&\Delta\end{pmatrix}, \qquad \Delta>0,$$

with real $v$, retaining only the first state. Bare truncation gives $E_P=0$. The exact effective equation is $E=v^2/(E-\Delta)$, and the lower root is $E_-=(\Delta-\sqrt{\Delta^2+4v^2})/2$. For $|v|\ll\Delta$, $E_-\approx-v^2/\Delta$.

This benchmark tests the sign and scale of omitted-state effects. Small omitted probability, of order $(v/\Delta)^2$, does not mean the corresponding energy shift can be ignored at the desired precision.

Cutoff studies and observables

Use nested retained spaces when possible, preserve exact symmetries, and track states by sectors or projectors. Monitor weight near the cutoff boundary, but do not treat it as a universal error bound. High-energy intermediate states can matter in response functions even when the ground-state boundary weight is tiny.

Vary cutoff and any basis scale separately. For bosonic occupation cutoffs, compare number distributions and high moments. For a low-energy effective theory, operators as well as the Hamiltonian may require matching corrections. A converged energy with an uncorrected observable can remain inconsistent.

Exercises

For $N=1$, compare the exact oscillator ground-state expectation $\langle0|x^2|0\rangle$ with the square of the one-state matrix $PxP$.

Solution

Parity gives $PxP=0$, whose square is zero. The exact expectation is $1/2$, arising from the omitted intermediate state $|1\rangle$. The correction formula gives $N/2=1/2$ on the only retained state.

References

• Bai, Z., J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, 2000, online text — projected eigenproblems.
• Feshbach, H., “Unified theory of nuclear reactions,” Annals of Physics 5, 357–390 (1958), doi:10.1016/0003-4916(58)90007-1 — projection and energy-dependent effective operators.