Choosing a Model for Quantum Matter

A useful model keeps the degrees of freedom needed to predict a particular observable in a particular range of conditions. The same semiconductor may need an effective-mass model for low-field conduction, several bands and phonons for optical absorption, or an atomistic description near a single dopant. Choosing between them begins with the measurement, rather than the material’s name.

Helpful background. Quantum Scales and Collective Behavior distinguishes a model, an excitation and a phase. From quantum mechanics to materials explains how a reduced Hamiltonian and its observables arise together.

A model must predict the observable

Begin with a concrete question: the low-temperature heat capacity, an absorption threshold, a spin-wave energy, a conductance, or a change of phase. Specify the temperature, density, fields and frequency or length scale involved. This often identifies degrees of freedom that can be neglected, and others that cannot.

For example, a spin Hamiltonian can describe the magnetic excitations of an insulating magnet below its charge gap. It cannot by itself predict the charge-gap absorption edge: the relevant charged states have been removed. A model with an accurate electron dispersion can still give an incorrect optical intensity if its wavefunctions or light-coupling matrix elements are inaccurate.

This distinction can be seen without a material calculation. Consider two physical two-level Hamiltonians in a fixed laboratory basis,

$$H_A=\frac{\Delta}{2}\sigma_z, \qquad H_B=\frac{\Delta}{2}\sigma_x, \qquad \Delta>0.$$

Both have energies $\pm\Delta/2$. Suppose the probe couples through the same laboratory operator $O=\sigma_z$. In model A, $O$ is diagonal in the energy basis and the transition matrix element between the two levels vanishes. In model B,

$$\left|\langle +x|O|-x\rangle\right|^2=1.$$

Equal energy levels therefore do not imply equal absorption. These are different Hamiltonians relative to a fixed probe, not two coordinate descriptions of the same Hamiltonian: a mere basis change would transform $O$ as well.

The smallest adequate model is one whose omitted processes make an acceptably small difference to the requested prediction. It is not always the model with the fewest terms. Keeping two coupled bands may be simpler and safer than forcing a single-band fit through a crossing.

Which degrees of freedom matter?

Question and regime	Useful starting description	What must still be checked
Carriers close to an isolated band extremum	Effective mass, possibly with several valleys	Energy window, mass anisotropy, intervalley scattering and probe matrix elements
Dispersion across a substantial part of the Brillouin zone	Tight binding or an electronic-structure calculation	Orbital content, hopping range, symmetry and treatment of interactions
Competition between motion and local electronic repulsion	Hubbard or a related interacting lattice model	Relevant bands, filling, screening and neglected interactions
Low-energy magnetism with suppressed charge fluctuations	Heisenberg or a more general spin Hamiltonian	Charge gap, exchange anisotropy, competing interactions and spin magnitude
Small oscillations about a stable crystal structure	Harmonic lattice dynamics	Acoustic sum rule, anharmonicity and the temperature range
Pairing and quasiparticles around a chosen superconducting state	A self-consistent mean-field/Bogoliubov–de Gennes description	Interaction approximation, competing states, fluctuations and electromagnetic response
Long-distance behavior near a continuous transition	A symmetry-based effective field theory	Relevant fields, conservation laws, dimensionality and the range of scaling

These choices are starting points. The model encyclopedia gives concrete Hamiltonians and solvable limits; the following chapters explain their material applications.

Local moments from interacting electrons

For the single-band repulsive Hubbard model at half filling, $U\gg t$ suppresses real double occupancy at energies well below $U$. Virtual hopping nevertheless lowers the energy of suitable spin states and gives antiferromagnetic exchange

$$J=\frac{4t^2}{U}$$

to leading order for a bond with spin-independent hopping $t$. The resulting spin description predicts excitations of order $J$, but does not retain the charge excitations of order $U$. Away from half filling, mobile holes or electrons remain low-energy degrees of freedom; a pure spin model is generally insufficient. A projected hopping-and-exchange model such as the $t$–$J$ model is one possible next description.

The scale separation is the reason the reduction works. The name of the material or a visually narrow band is not enough to justify it. See Hubbard physics and magnetism and spin systems.

Local band crossings and global topology

A Dirac expansion near a band crossing is useful for low-energy dispersion and the response to a small mass term. An unregularized expansion around one point does not specify the wavefunctions over the whole Brillouin zone and cannot, by itself, determine a lattice band’s integer Chern number. The global completion matters. A properly regularized continuum model can carry a well-defined invariant, but its regularization is part of the model.

Model names also need their physical context. The Haldane model of a Chern band and the Haldane phase of an integer-spin chain concern different systems. In the spin-conserving limit, the Kane–Mele model can be understood as two time-reversed Chern sectors. With spin-mixing Rashba coupling, those separate spin-sector Chern numbers no longer provide the general classification; the time-reversal-protected $\mathbb Z_2$ invariant remains the relevant distinction while the required symmetry and bulk gap survive. See topological quantum matter.

Quasiparticle spectra and stability

A fermionic Bogoliubov–de Gennes Hamiltonian is Hermitian and has real eigenvalues for any specified Hermitian mean-field Hamiltonian. That fact alone does not establish that the assumed order parameter solves the self-consistency equation, minimizes the appropriate thermodynamic potential, or remains stable against fluctuations. A plausible quasiparticle spectrum is one part of a superconducting calculation. It is not a substitute for testing the state around which the spectrum was constructed.

Choose a transport description by scale

Transport requires both states and a mechanism for carrying, relaxing or redistributing a conserved quantity. Several commonly used methods answer different versions of this problem.

• Boltzmann transport follows a distribution of sufficiently well-defined quasiparticles. Its collision term must include the processes that relax the measured current; a single relaxation time is an additional approximation.
• Landauer transport relates conductance to transmission between specified leads. It is especially useful when contacts, coherence and a finite device geometry are central. Extensions can include interactions or inelastic processes, but those are not supplied by the simplest elastic formula.
• Kubo response computes a linear-response coefficient from equilibrium correlation functions with a stated perturbation and limiting procedure. Choosing an approximate Hamiltonian or evaluating the correlator approximately still requires justification.
• Hydrodynamics describes slowly varying densities after local equilibration. Momentum relaxation, boundaries and constitutive coefficients must be specified. A clean electronic fluid can have a large collision rate while retaining a long momentum-relaxation time.

These are not successive rungs of an accuracy ladder. A nanostructure with a few transmitting channels and a bulk fluid with frequent momentum-conserving collisions may need different calculations even when made from the same material. Compare the sample dimensions with the mean free path, phase-coherence length and equilibration length before choosing. The transport chapter develops the corresponding regimes.

Test the reduction before interpreting the result

Three comparisons are particularly informative.

Recover a known limit. A lattice model should reproduce its noninteracting, atomic or long-wavelength limit where applicable. A magnetic calculation should respect the symmetries and conserved quantities of its stated Hamiltonian. These checks can expose an incorrect sign or a missing term before comparison with data.

Vary what was omitted. Increase the number of retained bands, hopping range, cluster size or numerical resolution. Alternatively, estimate the leading neglected process using the separation between its energy scale and the one being studied. Numerical convergence within one Hamiltonian does not test whether that Hamiltonian contains the necessary physics.

Compare another observable or condition. Parameters fitted to one spectrum should, where possible, predict a different temperature, field, momentum or response without an independent refit of every point. Distinct models can agree with the same data; a useful follow-up measurement is one on which their predictions differ.

For a finite system, distinguish a level crossing or an enhanced correlation from a thermodynamic phase transition. Establishing a phase may require size dependence, an appropriate order parameter or invariant, and control of the order of limits. A small exactly solved cluster provides a valuable benchmark without automatically representing the macroscopic phase diagram.

Exercises

1. A probe distinguishes equal spectra. For the two Hamiltonians above, replace the probe by $O=\sigma_x$. Which model now has a nonzero transition matrix element? What would happen if both the Hamiltonian and probe were transformed by the same unitary operator?

Solution

The transition strength is one in model A and zero in model B. Under a common unitary transformation, $H\to UHU^\dagger$, $O\to UOU^\dagger$ and $|n\rangle\to U|n\rangle$, every matrix element is unchanged. Physical disagreement comes from different Hamiltonians relative to the laboratory probe, not from a change of basis.

2. Two energy windows in a Mott insulator. Take $t=0.10\,\mathrm{eV}$ and $U=4.0\,\mathrm{eV}$. Estimate $J$. Would the resulting spin model be a suitable starting point for a $5\,\mathrm{meV}$ magnetic excitation or for optical transitions near $4\,\mathrm{eV}$?

Solution

$J=0.010\,\mathrm{eV}=10\,\mathrm{meV}$. The spin model can describe the magnetic excitation if other low-energy terms are negligible or included. It cannot describe the optical charge excitation near $U$, because the states with changed charge occupancy were eliminated. The numerical ratio $t/U=0.025$ supports the strong-coupling expansion, but does not rule out additional material-specific interactions.

3. An apparently converged gap. Increasing a calculation from 100 to 1000 momentum points changes its gap by less than $0.1\%$. Does this establish that the material’s gap is predicted to that accuracy?

Solution

It tests momentum discretization for the chosen model and method. Errors from omitted bands, interactions, structural uncertainty and the approximate treatment of the retained Hamiltonian can remain much larger. A convergence claim should name the quantity being varied and the uncertainty it addresses.

References

• Coleman, P. Introduction to Many-Body Physics. Cambridge University Press, 2015. doi:10.1017/CBO9781139020916.
• Kane, C. L., and E. J. Mele. “$Z_2$ Topological Order and the Quantum Spin Hall Effect.” Physical Review Letters 95, 146802 (2005). doi:10.1103/PhysRevLett.95.146802.
• Martin, R. M. Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press, 2004.