Locality in Many-Body Systems

Locality says where an operator acts and how that support is arranged in physical space or on an interaction graph. It does not mean a small Hilbert space, weak coupling, easy computation, relativistic causality, or locality in the Bell-theorem sense.

To specify locality, choose the subsystems, their spatial geometry, and the supports and strengths of the interaction terms. A pair term connecting opposite ends of a chain is two-body but geometrically long-ranged. An onsite Hubbard interaction is quartic in fermion operators but spatially local.

Required background. Review Tensor Products of Hilbert Spaces and Commutators and Anticommutators. The chapter Overview and Scaling of Hilbert Space are helpful but not required.

Support, Range, and Interaction Arity Are Different

Three common meanings of “local” answer different questions.

Property	Definition	Question answered
operator support	the subsystems on which an operator acts nontrivially	where does the operation act?
interaction arity	an upper bound $\lvert X\rvert\le k$ on the number of subsystems in one term	how many factors participate in one term?
geometric range	an upper bound or decay law for $\operatorname{diam}(X)$	how far apart can those factors be?

A complete statement also records the term norms and how many terms touch a site. A Hamiltonian can be two-local yet connect every pair of sites. Dividing its couplings by system size may restore an extensive energy scale, but it does not restore geometric locality.

Operator Support and Physical Geometry

Consider a finite set of sites or subsystems $\Lambda$ with

$$\mathcal H_\Lambda = \bigotimes_{i\in\Lambda} \mathcal h_i.$$

An operator supported on $X\subseteq\Lambda$ has the form

$$O_X = O_X^{(X)} \otimes I_{\Lambda\setminus X}.$$

The support $\operatorname{supp}(O)$ is the smallest such $X$. For ordinary spin or qudit tensor factors, operators with disjoint supports commute:

$$X\cap Y=\varnothing \quad\Longrightarrow\quad [A_X,B_Y]=0.$$

Support alone supplies no distance. The model must also declare a metric $d(i,j)$, usually Euclidean distance, graph distance, or distance inherited from a physical embedding. For sets $X$ and $Y$, define

$$\begin{aligned} d(X,Y) &= \min_{i\in X,\,j\in Y}d(i,j), \\ \operatorname{diam}(X) &= \max_{i,j\in X}d(i,j). \end{aligned}$$

Along a sequence of larger systems, a local observable normally has support size and diameter bounded independently of the total system size. Its location may change with the system—for example, a one-site observable kept near the center—but its physical support does not grow with the volume.

Fermionic locality

Odd fermionic operators on disjoint regions anticommute rather than commute. The physically relevant local observable algebras are normally built from parity-even operators, for which separated regions recover the expected commuting structure. A Jordan–Wigner mapping can turn a local fermionic operator into a long Pauli string; that representation-dependent string does not, by itself, redefine the underlying physical geometry.

The full measurement distinction among local, joint, and global observables belongs to Local and Global Observables.

Interaction Arity Is Not Spatial Range

A lattice Hamiltonian can be decomposed as

$$H_\Lambda = \sum_{X\subseteq\Lambda} h_X,$$

where $h_X$ is supported on $X$.

• It is $k$-local if $|X|\le k$ for every nonzero term.
• It has finite geometric range $R$ if $h_X=0$ whenever $\operatorname{diam}(X)>R$.
• It is short-range or quasi-local more generally when longer terms are allowed but their norms decay sufficiently fast according to a stated condition.

A four-site plaquette term has arity four but can remain geometrically local as the lattice grows. A term $Z_1Z_L$ has arity two but its diameter grows with the length of an open chain. “Few-body interaction” therefore does not mean “few-body system,” and low polynomial degree in creation and annihilation operators does not determine spatial range.

Support, arity, and range are separate data. The upper panel contrasts short-range bonds with a two-site term spanning distant regions. The lower panel depicts a Lieb–Robinson envelope: short-range evolution is quasi-local, but the generic bound permits exponentially small tails outside the effective cone rather than imposing strict zero support.

Finite-Range, Decaying, and Long-Range Hamiltonians

Range is not enough unless interaction strength and coordination remain controlled. The total strength of terms incident on a site is bounded by

$$J_* = \sup_{i\in\Lambda} \sum_{X\ni i} \lVert h_X\rVert.$$

For a regular family of bounded-degree, finite-range spin models, $J_*$ remains bounded as $|\Lambda|$ grows. Rigorous quasi-locality theorems use conditions of this kind, often with distance-dependent weights and more precise assumptions.

For pair interactions on a $d$-dimensional lattice, a power-law model may satisfy

$$\lVert h_{ij}\rVert \le \frac{J} {[1+d(i,j)]^\alpha}.$$

Every term is two-local, but the Hamiltonian is not finite-range for any finite $\alpha$. On a regular lattice, the sum of pair strengths incident on a site behaves schematically like

$$\sum_{r=1}^{L} r^{d-1-\alpha}.$$

This sum converges as $L\to\infty$ when $\alpha>d$. For slower decay, a size-dependent Kac or mean-field normalization is often introduced when an extensive energy is desired. That normalization controls the total interaction strength while leaving direct couplings between distant sites.

Propagation bounds for long-range interactions depend on $\alpha$, dimension, geometry, norm, and theorem hypotheses. One must not apply a finite-range linear light-cone formula unchanged to a Coulomb, dipolar, cavity-mediated, or all-to-all model.

Spatial Locality in Continuum Systems

Particle labels are not spatial regions. For identical particles, spatial locality is expressed naturally through fields or smeared density operators. A smeared observable

$$A(f) = \int d^d x\, f(\mathbf x) \mathcal O(\mathbf x)$$

is associated with the spatial support of the test function $f$.

A two-particle interaction can be written schematically as

$$H_{\mathrm{int}} = \frac{1}{2} \int d^d x\,d^d y\, V(\mathbf x-\mathbf y) \psi^\dagger(\mathbf x) \psi^\dagger(\mathbf y) \psi(\mathbf y) \psi(\mathbf x).$$

Its spatial range is set by $V$, not by the fact that the term contains two creation and two annihilation operators. For bosons or suitable multicomponent fermions, a contact model replaces the potential by an effective delta interaction and yields a local density such as

$$H_{\mathrm{contact}} = \frac{g}{2} \int d^d x\, \psi^\dagger(\mathbf x) \psi^\dagger(\mathbf x) \psi(\mathbf x) \psi(\mathbf x).$$

Continuum fields are operator-valued distributions, local bosonic spaces may be unbounded, and contact interactions can require regularization and renormalization. Standard bounded-spin lattice theorems therefore do not transfer automatically. For identical spinless fermions, the same-component zero-range $s$-wave term vanishes by antisymmetry; derivative interactions, finite range, or additional internal components require their own operator structure. Real-Space Representation explains continuum normalization and discretization.

Local Observables and Global Sums

An onsite density $n_i$ is local. A correlator $n_i n_j$ has support $\{i,j\}$; it is two-local in arity, but its diameter is $d(i,j)$. A block magnetization has support on the block. The total magnetization

$$M = \sum_{i\in\Lambda}m_i$$

is a global observable built from local densities.

A Hamiltonian can therefore be a global sum of local terms. Whether that sum grows proportionally to volume is a further question addressed in Extensive and Intensive Quantities.

Comparing interaction geometries

The same algebraic adjectives can hide different geometries.

Lattice term	Support in spatial sites	Range
$-JZ_jZ_{j+1}$	two neighboring sites	one lattice spacing
$-(J/L)Z_iZ_j$ for all pairs	two sites per term	up to the system diameter
$KP_\square$	four sites on a plaquette	one cell
$Un_{i\uparrow}n_{i\downarrow}$	one site, or two factors if spin modes are separated	onsite

The Hubbard term is a two-particle interaction because it acts on a pair of fermions. Its spatial support nevertheless occupies one site. The choice of local tensor factors determines the arity; the number of creation operators determines the particle-body rank. These are different counts.

Nearest-neighbor versus all-to-all support growth

Compare, on a chain of $L$ spins,

$$\begin{aligned} H_{\mathrm{nn}} &= -J\sum_j Z_jZ_{j+1}, \\ H_{\mathrm{all}} &= -\frac{J}{L} \sum_{i<j}Z_iZ_j. \end{aligned}$$

Both Hamiltonians are two-local. For $A=X_0$,

$$\begin{aligned} [H_{\mathrm{nn}},X_0] &= -2iJ \left( Z_{-1}Y_0+Y_0Z_1 \right), \\ [H_{\mathrm{all}},X_0] &= -\frac{2iJ}{L} \sum_{j\ne0} Y_0Z_j. \end{aligned}$$

The nearest-neighbor commutator reaches only adjacent sites at first order. The all-to-all commutator has support involving every other site immediately, although each contribution is suppressed by $1/L$. The normalization makes the characteristic total energy extensive; it does not turn the complete graph into a short-range chain.

How Locality Constrains Operator Spreading

In the Heisenberg picture,

$$A_X(t) = e^{iHt/\hbar} A_X e^{-iHt/\hbar}.$$

Its nested-commutator expansion is

$$A_X(t) = A_X + \frac{it}{\hbar}[H,A_X] + \frac{1}{2!} \left( \frac{it}{\hbar} \right)^2 [H,[H,A_X]] +\cdots.$$

For ordinary tensor-factor operators, a term $h_Z$ disjoint from the current support commutes with it. Successive nested commutators can therefore enlarge support only along chains of overlapping interaction terms. Finite range turns the number of required commutators into a geometric constraint.

The response interpretation is direct. Apply a small local unitary $U_X=e^{-i\epsilon A_X}$ and later measure $B_Y$. To first order,

$$\delta\langle B_Y(t)\rangle = i\epsilon \left\langle [A_X,B_Y(t)] \right\rangle + O(\epsilon^2).$$

The commutator norm therefore bounds how strongly a perturbation near $X$ can influence a measurement near $Y$.

For broad classes of short-range lattice Hamiltonians, a representative Lieb–Robinson bound has the schematic form

$$\left\lVert [A_X(t),B_Y] \right\rVert \le C_{X,Y} \lVert A_X\rVert \lVert B_Y\rVert \exp\!\left[ -\mu \left( d(X,Y)-v_{\mathrm{LR}}|t| \right) \right].$$

Outside the effective cone $d(X,Y)\approx v_{\mathrm{LR}}|t|$, the bound is exponentially small, not exactly zero. It is a state-independent upper envelope. The Lieb–Robinson velocity depends on interaction and norm conventions and is generally not a quasiparticle group velocity, butterfly velocity, measured front velocity, or the relativistic speed of light.

The Lieb–Robinson Bound gives a compact statement. For precise assumptions, decay norms, and extensions to fermions or unbounded interactions, see Lieb–Robinson Bounds.

Why Locality Organizes Phases and Computation

Locality creates structure, but each consequence needs additional assumptions.

Use of locality	What it supports	Necessary qualification
bulk and boundary separation	local observables far from a boundary may become insensitive to it	fails or changes near criticality, with edge modes, long-range forces, or global constraints
phase stability	phases can be compared under paths of local Hamiltonians and local perturbations	locality alone neither opens a gap nor identifies the phase
correlation control	a unique ground state with a size-uniform gap and suitable short-range interactions has exponentially decaying connected correlations	locality alone does not forbid long-range order or critical correlations
entanglement structure	locality motivates area laws and tensor-network representations in important regimes	no universal area law follows for every state, dimension, or time evolution
computation	local terms often enable sparse actions, parallel groupings, local time steps, and finite-time truncations	the Hilbert space can still grow exponentially and local-Hamiltonian problems can remain hard

A local Hamiltonian can have a highly entangled ground state, critical correlations, topological order without a local order parameter, or dynamics that rapidly produce entanglement. Conversely, a geometrically nonlocal Hamiltonian can have a simple product eigenstate. Hamiltonian locality and state correlations are different properties.

For the separate consequences, see Area Laws, Operator Entanglement and Scrambling, and Simulation of Lattice Models.

Exercises

Exercise 1: Support and Diameter

On an open six-spin chain with graph distance $d(i,j)=|i-j|$, find the support size and diameter of

$$A=Z_2, \qquad B=X_4X_5, \qquad C=Y_2Z_3, \qquad D=Z_1Z_6.$$

Which pairs commute solely because their supports are disjoint?

Solution

$A$ has support $\{2\}$, size one, and diameter zero. $B$ has support $\{4,5\}$, size two, and diameter one. $C$ has support $\{2,3\}$, size two, and diameter one. $D$ has support $\{1,6\}$, size two, and diameter five. Thus $D$ is two-local but geometrically long-ranged.

The pairs $(A,B)$, $(A,D)$, $(B,C)$, $(B,D)$, and $(C,D)$ have disjoint supports and therefore commute by tensor-factor locality. Only $(A,C)$ overlaps, so disjoint support alone gives no conclusion for that pair; its Pauli factors must be checked. For fermionic odd operators, replace ordinary commutation by the appropriate graded-locality statement.

Exercise 2: Normalization Does Not Restore Range

Compare the nearest-neighbor and complete-graph Ising Hamiltonians on $L$ sites. Count the number of pair terms and the number touching one site. Explain the role of the factor $1/L$ in the complete-graph model.

Solution

An open nearest-neighbor chain has $L-1$ pair terms, and at most two touch any site. The complete graph has $L(L-1)/2$ pair terms, and $L-1$ touch every site. With coefficients of order $J$, its total interaction energy can scale as $L^2$. Multiplying each pair by $1/L$ reduces the characteristic total scale to order $L$. Every pair still couples directly, so the normalized model remains geometrically all-to-all.

Exercise 3: Nested-Commutator Growth

Let $H=\sum_j h_{j,j+1}$ be a nearest-neighbor chain and let $A_1$ be supported on site 1. Show that the $n$th nested commutator

$$\operatorname{ad}_H^n(A_1) = [H,[H,\ldots,[H,A_1]\ldots]]$$

is supported within sites $1,\ldots,n+1$.

Solution

For $n=0$, the claim is the assumed support of $A_1$. Suppose the $n$th nested commutator is supported within $1,\ldots,n+1$. In the next commutator, every bond term disjoint from that region commutes with it. The only term that can extend the right boundary is $h_{n+1,n+2}$, so the new support is contained in $1,\ldots,n+2$. Induction proves the claim. This support counting explains the path structure behind a Lieb–Robinson bound, but it is not the theorem’s norm estimate.

Exercise 4: A Lieb–Robinson Tolerance

Suppose a schematic bound is

$$\lVert[A_X(t),B_Y]\rVert \le C\lVert A_X\rVert\lVert B_Y\rVert e^{-\mu(d-v|t|)}.$$

Find a sufficient separation $d$ that makes the right-hand side no larger than a tolerance $\varepsilon>0$. Why does the result not define an exact causal cone?

Solution

Solving the inequality gives the sufficient condition

$$d \ge v|t| + \frac{1}{\mu} \ln\!\left( \frac{C\lVert A_X\rVert\lVert B_Y\rVert} {\varepsilon} \right).$$

The location depends on the chosen tolerance and on nonuniversal constants. At any finite separation the exponential upper bound is generally nonzero, so it supplies an approximate influence cone rather than strict relativistic vanishing outside a light cone.

References

• S. Bravyi, M. B. Hastings, and F. Verstraete, “Lieb–Robinson Bounds and the Generation of Correlations and Topological Quantum Order,” Physical Review Letters 97, 050401 (2006), doi:10.1103/PhysRevLett.97.050401.
• M. B. Hastings and T. Koma, “Spectral Gap and Exponential Decay of Correlations,” Communications in Mathematical Physics 265, 781–804 (2006), doi:10.1007/s00220-006-0030-4.
• E. H. Lieb and D. W. Robinson, “The Finite Group Velocity of Quantum Spin Systems,” Communications in Mathematical Physics 28, 251–257 (1972), doi:10.1007/BF01645779.
• B. Nachtergaele and R. Sims, “Lieb–Robinson Bounds in Quantum Many-Body Physics,” Contemporary Mathematics 529, 141–176 (2010), doi:10.1090/conm/529/10429.
• M. C. Tran, A. Y. Guo, C. L. Baldwin, A. Ehrenberg, A. V. Gorshkov, and A. Lucas, “Lieb–Robinson Light Cone for Power-Law Interactions,” Physical Review Letters 127, 160401 (2021), doi:10.1103/PhysRevLett.127.160401.