Lattice Discretization

A lattice replaces continuous coordinates by a finite set of values and differential operators by discrete ones. The result is a new finite model with its own dispersion, boundary conditions, and cutoff. Recovering continuum physics requires controlled limits; increasing the number of sites is ambiguous unless spacing and physical size are stated separately.

Required background. Finite-difference Schrödinger methods constructs the kinetic operator, and dimensionless variables fixes physical scales.

Derivative symbols reveal the lattice model

On a uniform periodic grid $x_j=ja$, the centered first derivative and three-point second derivative have Fourier symbols

$$D_0(k)=\frac{i\sin(ka)}a, \qquad \Delta_h(k)=-\frac4{a^2}\sin^2(ka/2).$$

They approximate $ik$ and $-k^2$ for $|ka|\ll1$. Their behavior near the Brillouin-zone edge is different: $D_0$ vanishes again at $ka=\pi$, while $-\Delta_h$ is maximal there. Squaring the first-derivative matrix therefore does not produce the usual nearest-neighbor Laplacian.

For a free Schrödinger particle,

$$E_h(k)=\frac{2\hbar^2}{ma^2}\sin^2(ka/2) =\frac{\hbar^2k^2}{2m} \left[1-\frac{(ka)^2}{12}+\cdots\right].$$

This gives a quantitative dispersion benchmark and predicts second-order convergence at fixed physical $k$. Testing only the zero mode cannot detect the spatial error.

Three limits that should not be conflated

The ultraviolet resolution is set by $a$, while the box length is $L=Na$ and the momentum spacing is $2\pi/L$. Refining $a$ at fixed $L$ increases the cutoff but leaves finite-volume effects. Increasing $N$ at fixed $a$ enlarges the box but leaves the lattice dispersion unchanged.

A time step adds another approximation. Hold the physical initial state, observation times, and potential length scales fixed when varying these controls. An initial Gaussian with fixed width in grid points changes its physical width under refinement and is not a valid continuum-convergence experiment.

Periodic, hard-wall, and outgoing boundaries define different problems. Compare a localized observable before reflected signals return, or converge the appropriate boundary treatment.

Real and imaginary time demand different stability checks

Real-time Schrödinger evolution has modal factor $e^{-iE\Delta t/\hbar}$. Forward Euler replaces it by $1-iE\Delta t/\hbar$, whose modulus exceeds one for every nonzero $E$. A stable spatial operator cannot rescue this time integrator.

Imaginary-time projection uses $e^{-\epsilon H}$ with inverse-energy step $\epsilon$. A forward step has factor $1-\epsilon E$; for a nonnegative bounded discrete spectrum, linear stability requires $0\leq\epsilon E\leq2$. The endpoint $\epsilon E=2$ has factor $-1$: it is bounded but does not decay. After shifting the ground energy to zero, every excited mode must satisfy $0<\epsilon(E-E_0)<2$ for repeated forward steps to filter it out. Keeping all modal factors nonnegative is stricter, $\epsilon(E-E_0)\leq1$. A symmetric exponential splitting has a different error structure.

Imaginary time filters energies rather than describing physical unitary trajectories. A Euclidean lattice field integral additionally integrates over field configurations; discretizing a one-particle wavefunction does not implement that many-body quantum field theory.

Gauge transport on links

For charged particles, a discrete covariant difference can use a unitary link $U_j$ transporting the value at $j+1$ to site $j$:

$$(D_U\psi)_j=\frac{U_j\psi_{j+1}-\psi_j}{a}.$$

Under $\psi_j\mapsto g_j\psi_j$ with $|g_j|=1$, require $U_j\mapsto g_jU_jg_{j+1}^{-1}$; then $D_U\psi$ transforms like $\psi_j$. Build the kinetic energy from a compatible adjoint, such as $D_U^\dagger D_U$, with the correct boundary and metric. Link phases encode a specified gauge convention; adding a vector potential to a bare finite difference by an inconsistent formula can break discrete covariance.

Exercises

Why does doubling $N$ not specify a continuum refinement by itself?

Solution

If $L$ is held fixed, $a$ halves and low-momentum discretization errors decrease. If $a$ is held fixed, $L$ doubles and momentum spacing decreases, but the ultraviolet cutoff and dispersion error at fixed $k$ remain. A report must give both $a$ and $L$, together with the time step and boundary conditions.

References

• Gattringer, Christof, and Christian B. Lang, Quantum Chromodynamics on the Lattice: An Introductory Presentation, Lecture Notes in Physics 788, Springer (2010), doi:10.1007/978-3-642-01850-3 — lattice derivatives, link variables, and the field-theory boundary.
• LeVeque, Randall J., Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM (2007), author’s book resources — consistency, dispersion, and stability.