From Few-Body to Many-Body Physics

Adding degrees of freedom changes more than the dimension of a Hamiltonian. Boundaries become less important for some observables, collective modes become useful, and a sequence of finite systems can approach a sharp phase transition. These changes need different arguments: a large Hilbert space, a local interaction, and a thermodynamic limit are not the same property.

Helpful background. Tensor products explain how local state spaces combine. Why Many-Body Physics Is Different introduces the physical questions developed here.

Enter this chapter

Read the following sequence for a first introduction, or choose the question you need.

Question	Page	Main idea
Why do collective questions need new ideas?	Why Many-Body Physics Is Different	Correlations, emergent variables, and thermodynamic reasoning complement state counting
When does a problem become many-body?	Few-Body versus Many-Body Physics	The useful description depends on the observables and physical regime, not a universal particle-count threshold
How many states are available?	Scaling of Hilbert Space	Tensor products, exchange symmetry, conserved sectors, and truncations give different state counts
Which degrees of freedom interact directly?	Locality	Spatial support and interaction range differ from the number of particles in an interaction term
How should a quantity change with size?	Extensive and Intensive Quantities	Totals, densities, boundary corrections, and long-range interactions scale differently
What does an infinite-system statement mean?	Thermodynamic Limit	Define a sequence with specified density, geometry, couplings, and boundary conditions
How does a finite system differ from that limit?	Finite-Size Effects	Level spacing, gaps, boundaries, recurrences, and resolution can compete
When are collective variables useful?	Emergence and Effective Degrees of Freedom	Match a reduced description to observables in a stated regime

A finite chain and its large-system limit

Consider an open chain of $L$ spin-$1/2$ sites with

$$H_L=-J\sum_{j=1}^{L-1}\sigma_j^z\sigma_{j+1}^z -h\sum_{j=1}^{L}\sigma_j^x.$$

The Pauli matrices are dimensionless, and $J$ and $h$ have units of energy. Before symmetry reduction, the Hilbert-space dimension is $2^L$. Each term acts on one site or two neighbors, so the interaction is spatially local even though the state may contain correlations throughout the chain.

Open boundaries give $L-1$ bonds. Closing the chain adds one bond, an order-one change in the Hamiltonian for fixed $J$. Its contribution to the energy per site is bounded by $|J|/L$, although boundary conditions can have more pronounced effects on gaps, edge states, or the selection of a symmetry sector.

A thermodynamic statement concerns a sequence $L\to\infty$, with couplings and the intended state or ensemble specified. A calculation at $L=20$ remains a finite system, however difficult it is to diagonalize. At positive temperature its finite-dimensional partition function is smooth. At zero temperature, even a finite system can have a level crossing, so a ground-state cusp alone does not establish a bulk transition.

Compare the same physical quantity at different sizes

Before extrapolating, state what the size label means. For a square lattice, $L$ might be the linear extent and the site count $L^2$. Keep the shape and aspect ratio fixed unless changing them is part of the question. State whether density or particle number is held fixed, and define the observable’s normalization.

The relevant length or time scale also matters. A correlation length comparable to the system size can obscure bulk decay. A finite simulation time can miss a narrow spectral feature. A disturbance reflected from a boundary can return before the intended relaxation has finished. Increasing numerical precision does not remove any of these physical effects.

For computational examples, follow Finite-Size Scaling Numerics and the Computational QM volume. For rigorous restrictions on propagation, see Lieb–Robinson Bounds.

From microscopic variables to collective ones

An effective description keeps the variables needed for a chosen question and estimates the effect of those it omits. A few long-wavelength density modes may describe sound without resolving every particle coordinate. A low-energy spin Hamiltonian may describe an electronic system when charge excitations cost much more energy.

This reduction is justified by its regime and predictions, not by the simplicity of its notation. Ask which observables are reproduced, which energy or length scales are excluded, and what correction grows as the approximation fails. Continue to effective Hamiltonians, quasiparticles, or field-theory bridges according to the variables that remain.

Exercises

Classify three problems. Compare (a) three interacting bosons in a trap, (b) an exact calculation for a 24-site spin chain, and (c) a nonanalytic ground-state energy density obtained as $L\to\infty$.

Solution

(a) is a few-body problem, even if interactions make it difficult. (b) is finite many-body physics. (c) explicitly concerns an infinite-system limit. A sequence of finite calculations may support the conclusion in (c), but each member of the sequence remains finite.

A boundary correction. For the chain above, let $E_0^{\rm open}$ and $E_0^{\rm periodic}$ be the ground-state energies. Bound their difference without solving either model.

Solution

The Hamiltonians differ by $V=-J\sigma_L^z\sigma_1^z$, whose operator norm is $|J|$. Applying the variational principle to either ground state gives $|E_0^{\rm periodic}-E_0^{\rm open}|\leq|J|$. Dividing by $L$ bounds the energy-density difference by $|J|/L$. This does not give a corresponding relative bound for a gap that may itself approach zero.

References

• Anderson, P. W., “More Is Different,” Science 177, 393–396 (1972), doi:10.1126/science.177.4047.393.
• Campostrini, M., Pelissetto, A., and Vicari, E., “Finite-size scaling at quantum transitions,” Physical Review B 89, 094516 (2014), doi:10.1103/PhysRevB.89.094516.
• Coleman, P., Introduction to Many-Body Physics, Cambridge University Press (2015), doi:10.1017/CBO9781139020916.
• Lieb, E. H., and Robinson, D. W., “The finite group velocity of quantum spin systems,” Communications in Mathematical Physics 28, 251–257 (1972), doi:10.1007/BF01645779.