Few-Body versus Many-Body Physics

Few-body and many-body physics differ mainly in how we organize a problem. Few-body calculations usually resolve the motion and scattering channels of a small number of constituents. Many-body calculations often organize states by occupations, correlations, collective modes, or thermodynamic variables. There is no universal particle count at which one description must replace the other.

Three resonantly interacting atoms can have intricate bound-state physics. A gas of a hundred thousand noninteracting fermions can have a simple occupation-based solution. Complexity, interactions, and particle number are related, but none alone defines the boundary.

Helpful background. Why Many-Body Physics Is Different introduces collective observables. Scaling of Hilbert Space explains how the number of basis states grows.

Few-body coordinates and many-body occupations

For two particles, center-of-mass and relative coordinates often separate the motion. A three-particle problem requires more relative coordinates and coupled channels, yet can still be treated by resolving all three constituents. The difficulty is already substantial: three identical bosons with short-range interactions and scattering length much larger than the interaction range can exhibit Efimov physics. The universal description additionally needs three-body information, and a trap introduces another length scale. See Braaten and Hammer for the assumptions.

For many particles, listing every coordinate is possible in principle but may obscure the question. A metal’s low-temperature heat capacity, for example, depends on which fermionic modes near the Fermi surface can change occupation. A spin system’s response depends on correlations among local moments. This encourages a change of variables rather than a new microscopic quantum theory.

An ideal Fermi gas is a many-body system despite having no interparticle interaction in its idealized Hamiltonian. Antisymmetry constrains occupations, and observables sum over many occupied modes. Conversely, a single two-level atom can have a thermal density matrix without becoming a many-body system.

Finite systems and bulk behavior

The distinction between few and many particles is separate from the distinction between finite and thermodynamic systems.

Example	Useful description	What the finite calculation establishes
three resonant atoms	relative coordinates and few-body scattering channels	trimer energies and scattering in the specified geometry
four-site Hubbard plaquette	occupation states, spin and charge sectors	exact finite-cluster correlations and excitation energies
24 interacting spins	symmetry sectors, local correlations, entanglement	finite many-body behavior; bulk order needs a size analysis
large ideal Fermi cloud	mode occupations and a density of states	many-body statistics; a continuum approximation needs many resolved modes
one impurity in a Fermi sea	impurity plus a many-particle environment	dressing and collective response of the full impurity–bath system

The impurity example is especially instructive. Counting only the distinguished atom misses the bath degrees of freedom that create the effect. On the other hand, many spins restricted by symmetry to a fully symmetric sector may be represented by one collective spin of dimension $L+1$, much smaller than the unrestricted $2^L$ space. That restriction must be preserved by the dynamics and preparation; it is not valid for arbitrary states of the same sample.

A finite cluster can reveal mechanisms relevant to a bulk phase without itself possessing a thermodynamic phase transition. For an analytic finite-dimensional Hamiltonian, the partition function and free energy are analytic at every positive temperature. Exact ground-state level crossings at zero temperature are a separate possibility. The thermodynamic limit explains how singular bulk behavior can emerge from finite systems.

Two-body interactions in a many-body system

The Hamiltonian

$$H=\sum_{i=1}^{N}h_i+\sum_{i<j}V_{ij}$$

contains only one- and two-particle terms, but its state still describes $N$ particles. Each particle can interact with many others, and repeated pair interactions can build correlations involving the whole sample. The number of particles in an individual term does not limit the number of correlated particles.

Likewise, a two-point correlator such as $\langle n_i n_j\rangle$ is a two-operator observable evaluated in the full many-body state. Its value can depend on all other particles. Locality further distinguishes the number of factors in a term from the spatial distance over which it acts.

When a finite sample behaves like the bulk

There is no single size that makes every observable bulk-like. In a short-range system away from criticality, a local probe is often insensitive to distant boundaries when its distance $d_{\partial}$ from them greatly exceeds the relevant correlation length $\xi$ and interaction range $R$:

$$d_{\partial}\gg \xi,R.$$

A central density can then be close to its bulk value while an edge spectrum or a total fluctuation remains strongly size dependent. Near a continuous transition, $\xi$ grows and the same sample can cease to be large on the relevant scale. Long-range interactions, protected edge modes, constraints, and slow dynamics require additional reasoning.

For example, a 100-site chain with $\xi=3$ can have a nearly bulk-like center while its first few sites retain order-one boundary corrections. Averaging over every site includes those boundary regions and typically converges more slowly than the central value. Finite-Size Effects works out this comparison and explains shell filling, momentum quantization, and recurrences.

The practical question is therefore what degrees of freedom and scales control the observable. Emergence and Effective Degrees of Freedom develops this next step through phonons, projected spins, and diffusion.

Exercises

Exercise 1: Classify the Claim, Not Just the Sample

For each case, state the problem regime, physical scale, and claim type.

1. The exact binding energies of three resonant atoms in a harmonic trap.
2. A correlation function in the ground state of a 20-site spin chain.
3. The measured central density of a trapped cloud containing $10^5$ atoms.
4. The statement that the free-energy density converges as $N,V\to\infty$ at fixed $N/V$.

Solution

1. This is a few-body problem, microscopic in physical scale, and an exact or approximate claim about one finite system.
2. This is a finite many-body problem. Whether it is macroscopic for that correlator depends on its support, the correlation length, boundaries, and the requested accuracy. The stated result is finite-system evidence only.
3. The cloud is finite many-body and may be macroscopic for a coarse central-density observable. The measurement is still a finite-system result, not an infinite-system theorem.
4. This is a thermodynamic-limit claim about a specified sequence. It does not describe the size of any single sample.

Exercise 2: A peak without a phase transition

A single two-level system has energies $0$ and $\Delta>0$. Its heat capacity is

$$\frac{C}{k_{\mathrm B}}= \frac{x^2 e^{-x}}{(1+e^{-x})^2},\qquad x=\frac{\Delta}{k_{\mathrm B}T}.$$

Explain why it has a peak at an intermediate temperature even though there is no thermal phase transition. What happens to $C/N$ for $N$ independent copies?

Solution

At low temperature the excited state is almost empty, so changing temperature barely changes the energy. At high temperature the two populations approach equality and again change slowly. The positive heat capacity tends to zero at both ends and therefore reaches a maximum between them. Its expression is analytic for every $T>0$.

For independent copies, $Z_N=(1+e^{-x})^N$ and $C_N=NC$. Thus $C_N/N=C$ has exactly the same smooth peak for every size, including the thermodynamic limit. A peak alone does not establish a phase transition.

Exercise 3: Three Meanings of Two-Body

Distinguish the following statements:

1. $H=\sum_{i<j}V_{ij}$ is a two-body Hamiltonian.
2. $C_{ij}=\langle S_i^zS_j^z\rangle$ is a two-body observable.
3. $H$ acts on exactly two particles.

Solution

Statement 1 says that each interaction term has support on at most two constituents; the full system may contain arbitrarily many constituents. Statement 2 says that the measured operator has support on two sites, even though its expectation value is evaluated in a many-body state. Statement 3 alone specifies a two-particle system. None of the first two statements implies the third.

Exercise 4: One Sample, Two Scale Judgments

An open chain has length $L=100$ and a correlation length $\xi=3$. Compare a one-site observable at the center with an observable localized on the first site. Can either be called bulk-like from $L/\xi$ alone?

Solution

The center is about 50 sites from either boundary, so $d_{\partial}/\xi\gg1$ supports—but does not prove—the expectation that a local central observable is insensitive to the edges. One should still compare sizes and boundary conditions and check for long-range interactions, conserved global constraints, or other slow scales. The first-site observable is an edge observable by construction; a large $L/\xi$ does not turn it into a bulk observable. Thus macroscopicity is relative to support and target, not only to the global ratio $L/\xi$.

References

• P. W. Anderson, “More Is Different,” Science 177, 393–396 (1972), doi:10.1126/science.177.4047.393.
• E. Braaten and H.-W. Hammer, “Universality in Few-Body Systems with Large Scattering Length,” Physics Reports 428, 259–390 (2006), doi:10.1016/j.physrep.2006.03.001.
• M. Campostrini, A. Pelissetto, and E. Vicari, “Finite-Size Scaling at Quantum Transitions,” Physical Review B 89, 094516 (2014), doi:10.1103/PhysRevB.89.094516.
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015), doi:10.1017/CBO9781139020916.
• R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011), doi:10.1016/C2009-0-62310-2.