Many-Body and Quantum Statistical Mechanics

Begin with Identical Particles, Fock Space, and Second Quantization when building the many-particle language. Continue to ensembles, correlations, interacting models, and collective phenomena as the problem requires. Composite Systems and Entanglement owns subsystem structure and reduced states.

Many-body quantum mechanics studies quantum systems with many interacting or statistically distributed degrees of freedom. Quantum statistical mechanics studies ensembles of such systems and explains thermodynamic behavior using quantum states, operators, spectra, and probabilities.

The defining lesson is not merely that the Hilbert space becomes larger. New organizing structures appear: particle statistics, locality, collective excitations, thermodynamic phases, universality, long-range correlations, entanglement scaling, and effective degrees of freedom that may bear little resemblance to the microscopic constituents.

This volume connects the core formalism of quantum mechanics to quantum gases, lattice models, condensed matter, cold atoms, quantum simulation, nonequilibrium dynamics, statistical field theory, and the nonrelativistic route toward quantum field theory.

Two Subjects That Meet

Many-body quantum mechanics and quantum statistical mechanics overlap strongly, but they emphasize different questions.

Many-body quantum mechanics

Many-body theory begins with a microscopic or effective Hamiltonian for many degrees of freedom and asks about:

ground states and low-lying excitations;
correlations and entanglement;
collective modes and quasiparticles;
phases and order;
response to external probes;
real-time dynamics and thermalization;
tractable descriptions of exponentially large state spaces.

The particle number may be fixed or variable. The system may be at zero temperature, finite temperature, or far from equilibrium.

Quantum statistical mechanics

Quantum statistical mechanics asks how incomplete macroscopic information is represented by density operators and how thermodynamic quantities follow from ensembles. Its central objects include thermal states, partition functions, chemical potentials, entropy, free energies, fluctuations, and the thermodynamic limit. Fluctuations and Susceptibilities connects equilibrium widths to heat capacity, compressibility, magnetization response, and the quantum Kubo–Mori correction.

Statistical weights are not a replacement for the Born rule. An ensemble specifies the state or preparation statistics; the Born rule then supplies probabilities for measurements performed on that state. Classical uncertainty about preparation and quantum uncertainty in a measurement can both be present, but they are conceptually distinct.

Why Many-Body Physics Is Different

Several difficulties and opportunities appear together.

Hilbert-space growth

For $L$ distinguishable local degrees of freedom with local dimension $d$ ,

\dim\mathcal H = d^L.

For $L$ spin- $1/2$ sites, the dimension is $2^L$ . Symmetries and conserved quantities divide the space into sectors, but a fixed-magnetization sector can still grow combinatorially:

\dim\mathcal H_{N_\uparrow} = \binom{L}{N_\uparrow}.

This growth is why exact diagonalization is valuable but size limited, and why locality, tensor structure, sparsity, symmetries, low-energy effective theories, and controlled approximations matter.

Identical-particle statistics

Bosons and fermions do not arise by attaching an ordinary label to each particle. Exchange symmetry changes the allowed state space and operator algebra. The canonical definitions live in Identical Particles and Exchange Symmetry, Fock Space and Occupation Number, and Creation, Annihilation, and Second Quantization.

This volume applies that grammar to quantum gases, interacting Hamiltonians, lattice models, and collective phenomena.

Interactions and correlations

Even a Hamiltonian containing only one-body and two-body terms can produce a state whose correlations involve the whole system. A common first-quantized form is

H_N = \sum_{i=1}^{N} h_i + \frac{1}{2} \sum_{\substack{i,j=1\\i\ne j}}^{N} V_{ij}.

In a mode basis, a number-conserving two-body Hamiltonian often takes the form

\begin{aligned} H = {}& \sum_{ij} h_{ij} a_i^\dagger a_j \\ &+ \frac{1}{2} \sum_{ijkl} V_{ijkl} a_i^\dagger a_j^\dagger a_l a_k. \end{aligned}

Writing this Hamiltonian is usually easier than finding its phases, excitations, correlation functions, or long-time dynamics.

Locality

Many physical Hamiltonians are sums of terms acting on nearby sites or fields at nearby points. Locality constrains information propagation and supports effective descriptions even though the global Hilbert space is enormous.

A lattice Hamiltonian may be written schematically as

H = \sum_X h_X,

where each $h_X$ acts only on a finite region $X$ or decays with the size and separation of $X$ . The distinction between short-range and long-range interactions affects extensivity, correlation growth, finite-size scaling, and the existence of conventional thermodynamic limits. Lattice Models Overview develops the graph, local-space, hopping, interaction, constraint, and universality data behind this notation. Tight-Binding Model then solves the canonical quadratic hopping problem.

Emergence

The useful variables need not be the microscopic particles. Vibrations become phonons, spin deviations become magnons, interacting fermions may support quasiparticles, and broken-symmetry phases support collective modes. An order parameter or hydrodynamic density can summarize behavior that is invisible in a single microscopic basis state.

Emergence does not mean abandoning microscopic quantum mechanics. The canonical Emergence and Effective Degrees of Freedom audit asks which variables are retained, how observables are matched, what controls the error, and what would invalidate the description.

The Thermodynamic Limit

The canonical treatment is Thermodynamic Limit. This overview records only the role the limit plays in organizing the volume.

The thermodynamic limit is central because sharp phases and phase transitions are defined through large-system behavior. For particle number $N$ and volume $V$ ,

\begin{gathered} N\longrightarrow\infty, \qquad V\longrightarrow\infty, \\ \frac{N}{V}=n \quad \text{fixed}. \end{gathered}

The limit must also specify the interaction scaling, geometry, boundary conditions, and sequence of finite systems. A long-range all-to-all interaction may require normalization with system size to keep the energy extensive.

At finite size and positive temperature, a finite-dimensional partition function is a finite sum of smooth positive terms. Sharp nonanalytic thermal phase transitions arise only after an infinite-system limit. Finite systems still show crossovers, avoided crossings, susceptibility peaks, and scaling behavior that anticipate the transition.

The order of limits can matter. For spontaneous symmetry breaking, one often introduces a small symmetry-breaking field $h$ , takes the thermodynamic limit, and only then lets $h\to0$ :

\lim_{h\to0} \lim_{V\to\infty} \langle M\rangle_h

need not equal

\lim_{V\to\infty} \lim_{h\to0} \langle M\rangle_h.

This is one reason finite-volume exact eigenstates and macroscopic broken-symmetry descriptions can look different without being inconsistent.

Thermal States and Ensembles

The density operator is the common language of statistical ensembles and subsystems. For a system in canonical equilibrium at inverse temperature

\beta = \frac{1}{k_{\mathrm B}T},

the thermal density operator is

\rho_\beta = \frac{e^{-\beta H}}{Z}, \qquad Z = \operatorname{Tr} \left( e^{-\beta H} \right).

The partition function $Z$ normalizes the state and generates thermodynamic quantities. For example,

F = -k_{\mathrm B}T\ln Z

is the Helmholtz free energy, and

\langle H\rangle = -\frac{\partial}{\partial\beta} \ln Z

when the Hamiltonian is independent of temperature.

If energy and particle number can be exchanged with a reservoir, the grand-canonical state is

\rho_{\beta,\mu} = \frac{ e^{-\beta(H-\mu\hat N)} }{ \Xi },

with

\Xi = \operatorname{Tr} \left[ e^{-\beta(H-\mu\hat N)} \right].

The chemical potential $\mu$ controls the statistical weight of particle-number sectors. It is not generally the energy of one particle.

The microcanonical, canonical, and grand-canonical ensembles impose different macroscopic constraints. Their predictions may become equivalent for suitable local observables in an appropriate thermodynamic limit, but Ensemble Equivalence is conditional for finite systems, long-range interactions, constrained systems, and phase-coexistence regimes.

Maximum Entropy Principle gives the complementary inference construction: exact support or selected expectation values define a feasible state family, and entropy maximization selects its least-committal representative without supplying a thermalization mechanism.

Equilibrium and Open-System Thermalization

An equilibrium thermal state and an open-system thermalization model are related but not identical objects.

This volume owns equilibrium ensembles, partition functions, and generic many-body statistical mechanics. Measurement and Open Quantum Systems owns reduced dynamics, bath approximations, dissipative generators, and monitored trajectories. A thermal master equation must explain dynamically why a chosen thermal state is stationary and under what weak-coupling and detailed-balance assumptions.

The canonical state can also emerge as the reduced state of a small subsystem of a much larger isolated system under appropriate conditions. Understanding when and why this happens leads to typicality, equilibration, and the eigenstate thermalization hypothesis.

Quantum Statistics

The Quantum Statistics and Ideal Gases gateway routes from exchange symmetry and complete mode labels through occupation laws to Bose and Fermi gas models, degeneracy, traps, and dimensional effects.

For a noninteracting gas, the Hamiltonian is diagonal in mode occupations:

H = \sum_k \epsilon_k n_k.

In grand-canonical equilibrium, the mean occupation of a single-particle level with energy $\epsilon$ is

\bar n_{\mathrm B}(\epsilon) = \frac{1}{ e^{\beta(\epsilon-\mu)}-1 }

for bosons and

\bar n_{\mathrm F}(\epsilon) = \frac{1}{ e^{\beta(\epsilon-\mu)}+1 }

for fermions.

These formulas assume noninteracting modes in grand-canonical equilibrium. Their consequences differ dramatically: bosons can accumulate macroscopically in one mode, while fermionic occupation is bounded by the exclusion principle and forms a Fermi sea at low temperature.

In the dilute, nondegenerate regime,

e^{\beta(\epsilon-\mu)} \gg1,

both distributions approach Maxwell–Boltzmann behavior. This is a limiting regime, not a third quantum exchange symmetry. Classical Limit of Quantum Statistics develops the thermal-wavelength criterion, fugacity expansion, and leading exchange corrections.

Correlations and Response

Many-body observables are often densities, currents, order parameters, and correlation functions rather than complete wavefunctions. For operators $A$ and $B$ , the connected correlation is

\langle AB\rangle_c = \langle AB\rangle - \langle A\rangle \langle B\rangle.

It removes the disconnected product of one-point expectations. Spatial decay of connected correlations diagnoses correlation lengths and long-range order. Their Fourier transforms lead to structure factors that can be compared with scattering experiments.

Linear response asks how an observable changes under a weak perturbation. If

H'(t) = -f(t)B,

then a causal response has the schematic form

\delta\langle A(t)\rangle = \int_{-\infty}^{t} dt'\, \chi_{AB}(t-t') f(t').

The Kubo Formula relates $\chi_{AB}$ to a retarded commutator and develops the source convention, equilibrium Lehmann representation, contact terms, conductivity, and order-of-limits cautions. The perturbation, response observable, reference state, switching prescription, and Fourier convention must all be specified.

Correlation Functions and Linear Response routes a physical question and probe protocol to the appropriate correlator, spectrum, susceptibility, sum rule, or transport limit. Correlation functions connect microscopic models to measurable spectra, susceptibilities, transport coefficients, and collective modes; they also form the most direct bridge to field-theoretic language. Correlation Functions Overview organizes the correlation hierarchy, connected subtraction, spatial and temporal decay, long-range order, and ordering conventions. Susceptibilities translates the general kernels into magnetic, density, compressibility, and pairing measurements. Spectral Functions explains how exact transition weight becomes poles, continua, linewidths, and probe-dependent intensity. Fluctuation–Dissipation Theorem shows how thermal detailed balance converts those fluctuations into absorptive response. Sum Rules derives exact spectral moments from equal-time algebra and uses them to test approximations, numerics, and finite-window data. Transport Coefficients Preview connects conserved currents and equilibrium kernels to conductivity, diffusion, viscosity, and hydrodynamic poles.

Three Model Families

Simple Hamiltonians act as laboratories for general many-body ideas.

Spin systems

The isotropic Heisenberg model is

H = J \sum_{\langle i,j\rangle} \mathbf s_i\cdot\mathbf s_j.

It isolates rotationally invariant exchange among localized moments. Its sign distinguishes ferromagnetic and antiferromagnetic bond tendencies, while dimension, spin, geometry, and temperature determine whether the system orders, remains critical, or develops a gapped quantum phase. The model and its exact benchmarks are developed in Heisenberg Model.

The XXZ Spin Chain introduces an axial exchange anisotropy $\Delta$ . It connects ferromagnetic and Néel phases through a gapless Luttinger liquid, maps to interacting spinless fermions, and provides a canonical bridge from Bethe ansatz to low-energy field theory. Jordan–Wigner Transformation derives the spin–fermion dictionary and its parity-dependent ring boundaries.

A transverse-field Ising chain can be written

H = -J \sum_i \sigma_i^z\sigma_{i+1}^z - h \sum_i \sigma_i^x.

The competition between interaction and field makes it a standard model for quantum phase transitions, symmetry breaking, correlations, and exact-solution methods. The Transverse-Field Ising Model develops its phases, exact bulk spectrum, finite-size sectors, and entanglement; Quantum Phase Transitions uses it as a scaling benchmark.

Interacting lattice fermions

The Spinless Fermion Chains $t$ – $V$ model adds a nearest-neighbor density interaction to a one-dimensional hopping band. It gives the minimal fermionic route from a free Fermi sea to a Luttinger liquid, phase separation, and a charge-density-wave insulator, while making boundary twists and Jordan–Wigner parity sectors explicit.

The Hubbard model is

\begin{aligned} H = {}& -t \sum_{\langle i,j\rangle,\sigma} \left( c_{i\sigma}^\dagger c_{j\sigma} + c_{j\sigma}^\dagger c_{i\sigma} \right) \\ &+ U \sum_i n_{i\uparrow}n_{i\downarrow}. \end{aligned}

It balances kinetic delocalization against onsite interaction. The generic model is simple to state and difficult to solve, making it a central benchmark for exact, mean-field, perturbative, and numerical methods.

The model, its conventions, exact limits, and elementary diagnostics are developed in Hubbard Model.

Interacting lattice bosons

The Bose–Hubbard Model replaces fermionic operators with bosonic operators and includes hopping plus onsite repulsion. It connects superfluid order, number fluctuations, Mott-like insulating behavior, optical lattices, and quantum simulation. Its superfluid–Mott boundary provides a second benchmark in Quantum Phase Transitions.

These models belong here as generic Hamiltonians and methods laboratories. Detailed material realizations and experimental hardware belong in their dedicated volumes.

Phases, Order, and Criticality

Enter Phases, Order, and Criticality to decide whether a claim needs a phase definition, an order diagnostic, symmetry-breaking logic, a thermal or quantum transition, critical scaling, or a beyond-Landau branch. The gateway records dependencies and evidence tests; the specialist pages own detailed definitions and derivations.

A phase of matter is a stable regime of macroscopic behavior, usually characterized by symmetry, order, topology, excitations, and response. A local order parameter may distinguish phases associated with spontaneous symmetry breaking, but not every phase is characterized by a local order parameter.

At a continuous transition, a correlation length may diverge:

\xi \sim |g-g_c|^{-\nu},

where $g$ is a control parameter and $\nu$ is a critical exponent. Long-distance behavior can become insensitive to many microscopic details, leading to universality.

A thermal phase transition is driven by temperature or entropy–energy competition. A Quantum Phase Transition occurs at zero temperature as a nonthermal Hamiltonian parameter is varied. Finite temperature can still reveal a quantum-critical crossover region, but the zero-temperature transition and finite-temperature crossover should not be conflated.

The symmetry foundations live in Symmetry, Angular Momentum, and Spin. This volume owns the many-body use of order parameters, thermodynamic limits, collective modes, and critical scaling.

Quasiparticles and Collective Modes

Enter Quasiparticles and Collective Modes when the task is to decide whether observed spectral structure supports a particle-like excitation, a collective mode, a branch-specific example, or the breakdown of those descriptions.

A quasiparticle is a long-lived excitation that behaves approximately like a particle within a specified energy and momentum range. It is defined by the poles or peaks of response and Green functions, not by being one microscopic constituent.

Phonons, magnons, Bogoliubov excitations, particle–hole modes, polarons, and Fermi-liquid quasiparticles are examples. Their lifetime and spectral weight determine whether the quasiparticle description is useful.

Not all interacting systems possess sharp quasiparticles. One-dimensional liquids, quantum-critical systems, strongly incoherent regimes, and some topological phases require other organizing principles.

Many-Body Entanglement

Entanglement is already defined in Composite Systems and Entanglement. Enter Many-Body Entanglement and Information to choose the appropriate partition, state-class, diagnostic, scaling, representation, or operator-growth branch. Many-Body Entanglement Overview owns the detailed subsystem, measure, and scaling taxonomy. This volume studies how entanglement scales with subsystem size and how that scaling organizes phases, criticality, dynamics, and numerical representations.

For a pure state divided into region $A$ and its complement,

S_A = -\operatorname{Tr} \left( \rho_A\ln\rho_A \right)

is the entanglement entropy. Its canonical finite-system definition is reviewed in Entanglement Entropy. Spatial cuts, area and volume laws, critical scaling, and tensor-network consequences are developed in Entanglement Entropy in Many-Body Systems.

Ground states of many local gapped systems often obey an area-law pattern, while generic highly excited states can show volume-law scaling. These statements require assumptions and admit important qualifications. Thermal entropy and entanglement entropy are not interchangeable, even when their leading scaling resembles one another.

Entanglement structure also explains why tensor-network methods can efficiently represent some physically important states despite the exponential dimension of the full Hilbert space.

Nonequilibrium Dynamics

Enter Nonequilibrium Many-Body Dynamics to choose the protocol, regime branch, diagnostic package, and bounded claim. Nonequilibrium Overview owns the detailed preparation, generator, probe, timescale, and order-of-limits ledger needed before making an equilibration or thermalization claim.

An isolated many-body system can evolve unitarily while local observables relax toward values described by statistical mechanics. This raises several distinct questions:

Does equilibration occur for the chosen initial state and observable?
Are late-time values described by a thermal ensemble?
Which conserved quantities constrain the state?
How do correlations and entanglement spread?
What roles do integrability, disorder, chaos, and driving play?

The eigenstate thermalization hypothesis is a powerful framework for generic nonintegrable systems, not a universal theorem covering every Hamiltonian. Integrable systems, constrained models, finite systems, and localization phenomena require separate analysis.

Nonequilibrium many-body dynamics also connects to quantum quenches, Floquet systems, prethermalization, scrambling, Loschmidt echoes, and hydrodynamics.

From Many-Body QM to Field Theory

Second-quantized many-body Hamiltonians already use operator-valued fields in nonrelativistic settings. Correlation functions, generating functionals, imaginary time, coherent-state path integrals, collective fields, and renormalization ideas then lead naturally toward statistical and quantum field theory. Enter Bridges to QFT and Statistical Field Theory to choose the branch appropriate to the question. Why Many-Body QM Leads to QFT distinguishes exact operator reformulations, effective collective fields, and the additional principles required by relativistic QFT.

The boundary is deliberate:

this volume develops nonrelativistic many-body field language and finite-temperature entry points;
the Relativistic QM bridge develops relativistic wave equations and the conceptual transition to quantum fields;
QFT.org owns full relativistic field theory, thermal QFT, renormalization-group machinery, and Schwinger–Keldysh field theory.

The goal here is to make that transition motivated and technically prepared without duplicating the later canonical treatment.

Map of the Volume

The dedicated Map of the Volume organizes the subject as a dependency graph:

\text{microscopic specification} \longrightarrow \text{state or preparation} \longrightarrow \text{diagnostics} \longrightarrow \text{emergent organization}.

Its chapter-by-chapter map shows where ensembles, models, correlations, phases, entanglement, dynamics, computation, and field-theory bridges enter. Use the Many-Body Hilbert Spaces and Operators gateway when the missing step is choosing a state space, operator language, conserved sector, or real- versus momentum-space representation. Enter Lattice Models and Spin Systems when the next decision is the model family, geometry, boundary convention, or exact-method route. Enter Interacting Systems and Approximation Methods when the model is fixed but the approximation family, control parameter, retained correlations, or validation test is not. The detailed Interacting Many-Body Systems Overview remains the direct route for a regime-level audit. Enter Finite-Temperature Methods when a thermal question requires a choice among Gibbs traces, imaginary time, Matsubara functions, spectral reconstruction, equilibrium path integrals, KMS structure, or real-time evolution. Enter Computational Many-Body QM when the problem and observable are fixed but the numerical branch, resource model, or required validation evidence is not; use the Computational Many-Body Overview for the detailed method and error audit. The map also keeps three controls attached to every route: the limiting procedure, the method and its error, and the evidence status of the conclusion.

Reading Paths

The Reading Paths page gives complete routes for a first course, equilibrium statistical mechanics, condensed matter, AMO and cold atoms, quantum information, nonequilibrium dynamics, computation, QFT preparation, and researcher lookup. Each route states entry capabilities, a recommended sequence, and an exit checkpoint.

Readers new to the subject should first use the Overview and Orientation portal. Readers who already know their goal can go directly to Reading Paths; readers diagnosing dependencies should use the map.

What This Volume Owns

This volume is the canonical home for:

equilibrium quantum statistical ensembles and thermal density operators;
Bose–Einstein and Fermi–Dirac statistics in many-body applications;
ideal quantum gases and generic weakly interacting gas theory;
generic lattice and spin Hamiltonians;
many-body mean-field and response methods;
correlation functions, spectral functions, and the Kubo formula at a many-body level;
generic phases, order parameters, quantum criticality, and universality;
quasiparticles and collective modes as many-body structures;
many-body entanglement scaling and thermalization entry points;
nonrelativistic bridges to statistical field theory and QFT;
model-centered benchmarks and physics requirements for many-body computation.

It does not duplicate:

basic tensor products, exchange symmetry, Fock space, and creation–annihilation algebra from Composite Systems and Entanglement;
basic density-operator definitions from Core Formalism;
open-system bath reductions, channels, and Lindblad dynamics from Measurement and Open Quantum Systems;
material-specific bands, superconductors, quantum Hall systems, topology, and mesoscopic devices from Quantum Matter;
experimental cold-atom, molecular, and cavity implementations from Atomic, Molecular, and Optical Physics;
reusable numerical algorithms and software infrastructure from Computational QM;
full relativistic, thermal, and nonequilibrium field theory from QFT.org.

Conventions to State

Every many-body calculation should make the following visible.

System size: Give $N$ , number of sites, volume, or mode cutoff.
Statistics: State whether degrees of freedom are bosonic, fermionic, distinguishable, spin, or constrained.
Ensemble: State microcanonical, canonical, grand canonical, pure state, or nonequilibrium preparation.
Boundary conditions: State open, periodic, twisted, or continuum boundaries.
Thermodynamic limit: Specify what grows and which densities or couplings remain fixed.
Hamiltonian convention: Define signs, double-counting factors, and energy zero.
Units: State whether $k_{\mathrm B}=1$ , $\hbar=1$ , or lattice spacing is set to one.
Fourier convention: Define normalization, momentum grid, and reciprocal-volume factors.
Correlation convention: State time ordering, connected subtraction, and operator ordering.
Symmetry sector: State conserved particle number, magnetization, momentum, parity, or other restrictions.
Approximation: Identify mean-field, weak-coupling, low-density, low-temperature, continuum, or large-system assumptions.
Evidence status: Distinguish exact result, controlled approximation, numerical observation, experimental fact, and active conjecture.

The site-wide conventions for units and constants, tensor-product ordering, density matrices, and Fourier transforms apply throughout.

Common Mistakes

Treating many-body quantum mechanics as one-particle mechanics with more indices.
Confusing indistinguishability with ignorance of persistent particle labels.
Confusing an occupation number with a probability.
Forgetting fermionic signs when reordering operators.
Using grand-canonical formulas without checking the conserved quantities and reservoir interpretation.
Treating the chemical potential as the energy of one particle in every context.
Calling a finite-size crossover a sharp phase transition without a scaling analysis.
Discussing spontaneous symmetry breaking without specifying the order of limits.
Assuming every interacting system has sharp quasiparticles.
Interpreting a disconnected correlation as evidence of genuine correlated fluctuations.
Confusing thermal entropy with entanglement entropy.
Treating mean-field theory as exact because it is self-consistent.
Applying continuum density-of-states formulas unchanged to a lattice band.
Ignoring boundary conditions and symmetry sectors in finite-size numerics.
Presenting a frontier diagnosis such as many-body localization as a settled universal classification.

Exercises

Two-level canonical ensemble

A system has energies $0$ and $\Delta$ . Find its canonical partition function, excited-state probability, and mean energy.

Solution

The partition function is

Z = 1+e^{-\beta\Delta}.

The excited-state probability is

p_e = \frac{e^{-\beta\Delta}} {1+e^{-\beta\Delta}} = \frac{1} {e^{\beta\Delta}+1}.

Therefore

\langle H\rangle = \Delta p_e = \frac{\Delta} {e^{\beta\Delta}+1}.

The appearance of a Fermi-like algebraic form here does not make the two-level system a fermion gas; it follows from having one ground state and one excited state.

Extensivity of all-to-all interactions

Suppose every pair among $N$ degrees of freedom contributes an interaction of order $J$ . How does the interaction energy scale with $N$ ? What normalization makes the energy extensive?

Solution

The pair count grows quadratically, so order-one pair couplings produce a superextensive aligned energy. A coupling proportional to $1/N$ restores an order- $N$ leading scale. It does not restore locality or additivity. The full calculation and the required scaling ledger are in Extensive and Intensive Quantities.

Connected correlations

Let the state factorize across sites $i$ and $j$ , and let $A_i$ and $B_j$ act on the two sites. Show that the connected correlation vanishes.

Solution

For a product state,

\langle A_iB_j\rangle = \langle A_i\rangle \langle B_j\rangle.

Therefore

\langle A_iB_j\rangle_c = \langle A_iB_j\rangle - \langle A_i\rangle \langle B_j\rangle =0.

A nonzero connected correlation signals failure of this product factorization for the chosen observables, though it need not by itself imply entanglement in a mixed state.

Cross-Links

References

R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).
M. Kardar, Statistical Physics of Particles, Cambridge University Press (2007).
K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987).
A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).
A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).
P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015).
G. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid, Cambridge University Press (2005).
S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011).
L. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity, Oxford University Press (2016).