Model Encyclopedia

The Model Encyclopedia is the convention-aware routing layer for recurring many-body models. It answers a practical question: what must be specified before a familiar model name becomes a reproducible physical problem?

Each model dossier records the degrees of freedom, Hilbert space, Hamiltonian convention, geometry, boundary conditions, symmetries, control parameters, important limits, observables, solution status, and benchmark route. Detailed derivations remain in the topical teaching chapters. For a task-oriented entry across the volume’s lookup aids, use Many-Body and Quantum Statistical Mechanics Reference. Compact formula lookup remains in Common Many-Body Hamiltonians. The Model-to-Volume Cross-Link Index owns the dossier-to-teaching-to-benchmark-to-application routing map.

Choosing a Model for Quantum Matter owns the inverse, material-facing question: which candidate family is minimally adequate for a declared observable? This encyclopedia begins once the family is named and supplies its reproducible dossier rather than repeating that selection audit.

This separation matters. “The Hubbard model,” “the Heisenberg chain,” or “the ideal Bose gas” names a family until dimension, lattice or continuum spectrum, filling, ensemble, boundaries, and conventions are fixed.

Purpose and Canonical Scope

The encyclopedia owns model dossiers and comparisons. It does not own every result associated with a model.

Layer	Canonical responsibility	Typical use
Model dossier in this chapter	complete model specification, exact-solution status, limits, observables, and benchmark handoff	identify and compare models
Topical teaching article	derivation, physical interpretation, phase structure, and method-specific detail	learn the physics
Common Many-Body Hamiltonians	compact convention-aware formulas	check notation quickly
Reference model cards	short cross-site locator cards	find the canonical volume
Benchmark Problems	stable numerical inputs, outputs, tolerances, and validation rules	test implementations
Reproducible Notebooks	artifact inventory, execution contract, and release status	reproduce published calculations

A dossier may summarize a standard formula, but it links to the page that owns its derivation. For example, the Transverse-Field Ising Model dossier states the convention and solvable regime; the teaching article owns the Jordan–Wigner and Bogoliubov development, while Quantum Phase Transitions owns the general critical-scaling framework.

Until an individual dossier is published, the teaching page listed in the maps below remains the canonical route. This avoids dangling links and keeps the one-canonical-home rule visible during construction.

What Counts as a Model

A useful abstract record is

$$\mathfrak M = \left( \Gamma, \mathcal H, \mathcal A, H(\boldsymbol\lambda), \mathcal B, \mathcal S, \mathcal O \right).$$

Its entries are:

• $\Gamma$: geometry, dimension, lattice or continuum structure, and boundary conditions;
• $\mathcal H$: Hilbert space, including superselection or fixed-number sectors;
• $\mathcal A$: operator algebra and particle statistics or local constraints;
• $H(\boldsymbol\lambda)$: Hamiltonian and a declared parameter convention;
• $\mathcal B$: state, ensemble, or preparation data;
• $\mathcal S$: symmetries and conserved quantities;
• $\mathcal O$: observables used to interrogate the model.

For equilibrium thermodynamics, the ensemble data may replace $H$ by the grand Hamiltonian

$$K = H-\mu N$$

or by a more general constrained generator. Writing $-\mu N$ inside a displayed “Hamiltonian” is common, but a dossier must say whether the object is $H$ or $K$.

A Hamiltonian expression is not enough

Consider the nearest-neighbor hopping expression

$$H_t = -t \sum_{\langle i,j\rangle} \left( c_i^\dagger c_j + c_j^\dagger c_i \right).$$

It does not yet determine:

• whether the operators are bosonic or fermionic;
• whether there is one internal component or several;
• the graph and spatial dimension;
• open, periodic, twisted, or disordered boundaries;
• the particle number or filling;
• whether onsite constraints project the Hilbert space;
• which observables and thermodynamic limit are intended.

The same quadratic expression can therefore be a one-particle band problem, a free Fermi sea, a free lattice Bose gas, or the kinetic term of an interacting model.

Conventions can move critical numbers

For spin-$1/2$ sites, both Pauli matrices and spin operators are standard:

$$\mathbf S_j = \frac12 \boldsymbol\sigma_j.$$

The two transverse-field Ising conventions

$$\begin{aligned} H_\sigma &= -J_\sigma \sum_j \sigma_j^z\sigma_{j+1}^z - h_\sigma \sum_j \sigma_j^x, \\ H_S &= -J_S \sum_j S_j^zS_{j+1}^z - h_S \sum_j S_j^x \end{aligned}$$

represent the same operator only when

$$J_S=4J_\sigma, \qquad h_S=2h_\sigma.$$

Thus a quoted ratio such as $h/J$ is meaningless until the operator convention is stated. The encyclopedia treats convention conversion as part of the model, not as editorial fine print.

The Dossier Contract

Every model page in this chapter follows the same audit.

Field	Required content	Failure prevented
One-sentence description	degrees of freedom and defining competition	names used as explanations
Hamiltonian	signs, factors, bond counting, and $H$ versus $K$	false parameter comparisons
Hilbert space and algebra	statistics, local basis, constraints, and sectors	wrong basis dimension
Geometry and boundaries	dimension, graph, lattice spacing, and boundary twist	incompatible spectra
Symmetries	exact symmetries, conserved charges, and symmetry-breaking terms	invalid block diagonalization
Control parameters	dimensions and natural dimensionless ratios	unit-dependent phase claims
Important limits	fixed quantities and resulting effective models	uncontrolled model substitutions
Solution status	exactly which quantities are known and in what regime	“exactly solved” overclaim
Observables	definitions and normalization conventions	comparing unlike diagnostics
Minimal example	smallest nontrivial analytic calculation	formulas without operational meaning
Numerical benchmark	basis, parameters, reference values, and tolerances	irreproducible numerics
Cross-links and references	canonical derivations and primary sources	duplicate or unsupported content

The contract is intentionally stricter than a glossary card. A model page should make it possible to reconstruct the finite problem that generated a spectrum or plot.

Exactness Is Question-Dependent

“Exactly solvable” is not a single binary property. A useful status vector is

$$\mathcal X(\mathfrak M) = \left( X_E, X_\Psi, X_Z, X_C, X_D \right),$$

where the components refer respectively to energies, eigenstates, thermodynamics, correlation functions, and dynamics.

A method may determine one component without closing the others. Bethe equations can characterize the spectrum while local form factors remain difficult. A quadratic transformation can diagonalize the Hamiltonian while boundary-sector bookkeeping still matters. A large-$N$ saddle can control disorder-averaged correlators without giving an exact finite-$N$ spectrum.

Solution labels used here

Label	Meaning	What must still be stated
finite exact diagonalization	complete diagonalization of a declared finite Hilbert space	basis, sector, arithmetic precision, and size
quadratic or normal-mode exact	a canonical transformation reduces the model to independent modes	boundary sectors, zero modes, and stability
Bethe-ansatz integrable	eigenstates or thermodynamics follow from coupled rapidity equations	boundary conditions, string assumptions, and observable access
exact mapping	a unitary or spectrum-preserving map relates the model to another one	sectors, boundary terms, and mapped observables
exact special limit	a parameter limit is solved, but the generic model is not	limiting procedure and corrections
controlled asymptotic	an expansion is systematic in a small parameter	expansion parameter and observable-dependent error
numerically converged	truncation and finite-size errors are bounded for the stated quantity	convergence sequence and tolerance
frontier large-N	leading disorder-averaged or saddle-point results are controlled as $N\to\infty$	order of limits and finite-$N$ corrections

None of these labels licenses the phrase “the whole model is solved.” A dossier instead states the object, regime, method, and remaining limitation.

A Verification Workflow

Use the encyclopedia in the following order.

1. Identify the degrees of freedom. Decide whether the local objects are bosons, fermions, spins, oscillators, an impurity plus bath, or random all-to-all modes.
2. Complete the model data. State geometry, boundaries, Hilbert-space constraints, filling or ensemble, and Hamiltonian convention.
3. Choose natural scales. Divide by a nonzero energy $E_\star$ and identify dimensionless controls.
4. Resolve symmetries and sectors. Record conserved charges before diagonalization or approximation.
5. State the solution target. Spectrum, free energy, correlator, response, dynamics, and phase boundary are different tasks.
6. Select a controlled method. Match the method to the target and regime.
7. Run a benchmark. Recover an analytic limit, a small-system reference, and a convergence sequence before making a physical claim.

For a finite Hamiltonian, a useful nondimensionalization is

$$\begin{aligned} \widehat H &= \frac{H-E_{\mathrm{ref}}}{E_\star}, \\ \widehat\beta &= \beta E_\star, \\ \widehat t &= \frac{E_\star t}{\hbar}. \end{aligned}$$

This makes comparisons portable across unit conventions. It does not remove the need to report the original physical scales.

Model Map

The planned encyclopedia contains eighteen model dossiers. The tables below state their canonical teaching and benchmark routes without duplicating those pages.

Gases and collective chains

Model	Defining competition	Controlled or exact scope	Current canonical routes
Ideal Bose Gas	Bose occupation with a specified one-particle spectrum and fixed density or chemical potential	mode factorization is exact; condensation depends on dimension, spectrum, and thermodynamic limit	Teaching article; MB-B006
Ideal Fermi Gas	Pauli filling of a specified one-particle spectrum	mode factorization is exact; low-temperature asymptotics require a smooth density of states	Teaching article; MB-B007
Harmonic Chain	inertia versus quadratic intersite restoring forces	stable quadratic chain is exactly reducible to normal modes; anharmonicity is outside that solution	Teaching derivation
Lieb–Liniger Model Preview	one-dimensional kinetic energy versus contact repulsion	coordinate Bethe ansatz gives the spectrum and thermodynamics; local dynamics remain a separate problem	Model dossier; Low-Dimensional Quantum Gases; Nonrelativistic Field Theory
Tonks–Girardeau Gas Preview	bosonic symmetry with an impenetrability constraint	hard-core limit maps spectra and density observables to free fermions; off-diagonal observables require the mapping strings	Model dossier; Low-Dimensional Quantum Gases; Luttinger Liquid Preview

The ideal gases are exactly noninteracting, but that does not make every thermodynamic limit trivial. For the Bose gas, the existence of an ordinary condensate is controlled by the infrared density of states. For the Fermi gas, a Sommerfeld expansion is controlled by $k_{\mathrm B}T/E_{\mathrm F}$ only when the relevant density of states and chemical potential behave smoothly.

For the harmonic chain, translational invariance produces an acoustic zero mode. It must be treated by fixing the center of mass, selecting boundary conditions, or retaining the zero-mode degree of freedom explicitly. Simply deleting it changes the model.

Spin and lattice families

Model	Defining competition	Controlled or exact scope	Current canonical routes
Spin-$1/2$ Chain	tensor-product spins with local fields and finite-range couplings	umbrella family, not one solvability class	Lattice Models Overview; Boundary Conditions on Lattices
Transverse-Field Ising Model	Ising alignment versus noncommuting transverse field	standard nearest-neighbor chain without longitudinal field maps to quadratic fermions	Teaching article; MB-B001
Heisenberg Chain	isotropic exchange and possible field or boundary twist	uniform spin-$1/2$ chain is Bethe-ansatz integrable; generic dimensions and couplings are not	Teaching article; MB-B002
XXZ Chain	exchange anisotropy $\Delta$ versus field and filling	standard one-dimensional chain is Bethe-ansatz integrable; phases and root structures depend on $\Delta$	Teaching article; Exact Solutions Preview
Tight-Binding Chain	hopping, onsite energies, filling, and boundary phase	one-body or quadratic translation-invariant problem is exactly diagonal in momentum; interactions change the class	Tight-Binding Model; Tight-Binding Chain card
Hubbard Dimer	hopping versus onsite fermion interaction on two sites	finite sectors are exactly diagonalizable and expose singlet, triplet, charge, and strong-coupling structure	Exact Hubbard Dimer Spectrum; MB-B003
Hubbard Chain	fermion hopping versus onsite interaction and filling	uniform one-dimensional model has a Bethe-ansatz solution; generic lattices require approximation or numerics	Hubbard Model; MB-B004
Bose–Hubbard Dimer	boson tunneling versus onsite interaction at fixed total number	each fixed-$N$ sector is a finite exact problem of dimension $N+1$	Bose–Hubbard Model; MB-B005
Bose–Hubbard Chain	boson hopping versus onsite repulsion and chemical potential	atomic, free, hard-core, and infrared limits are controlled; the generic soft-core chain is not integrable	Bose–Hubbard Model; Finite-Size Scaling in Numerics

The generic “spin-$1/2$ chain” entry is deliberately a family card. Adding anisotropy, a longitudinal field, next-neighbor exchange, disorder, long-range coupling, or a boundary twist can destroy an exact mapping while preserving the same local Hilbert space.

For Jordan–Wigner-solvable chains, periodic spin boundaries do not become naive periodic fermion boundaries. The fermion parity sector controls the boundary term. Jordan–Wigner Transformation owns that bookkeeping.

Pairing, impurity, and frontier models

Model	Defining competition	Controlled or exact scope	Current canonical routes
Reduced BCS Model	pair scattering among time-reversed levels	Richardson equations give an exact finite-level pairing solution; BCS mean field controls a different thermodynamic description	Exact dossier; BCS Mean-Field Theory; MB-B008
Kondo Model	impurity spin exchange with a conduction bath	perturbative RG, special integrable formulations, and numerical renormalization group control complementary regimes	Model dossier; Kondo Model Preview
Anderson Impurity Model	impurity charge energy, onsite repulsion, and bath hybridization	resonant-level limit is quadratic; selected formulations are integrable; correlated spectra are commonly treated by controlled numerics	Model dossier; teaching preview
SYK Model Preview	random all-to-all few-body Majorana interactions	disorder-averaged large-$N$ saddle and infrared reductions are controlled; finite-$N$ spectra and fluctuations are distinct problems	Model dossier; Scrambling and OTOCs Preview; Large-N Methods Preview

These entries need especially careful language. The reduced BCS Hamiltonian has an exact Richardson solution, while the familiar gap equation belongs to a symmetry-breaking mean-field treatment. The Kondo temperature is a generated crossover scale whose numerical prefactor depends on the bandwidth and coupling convention. The SYK saddle describes a declared disorder ensemble and order of limits, not every finite random Hamiltonian realization.

Model Relationships and Controlled Limits

Model names become most useful when the map between them is explicit.

Hubbard to Heisenberg

At half filling and large repulsive $U/t$, virtual doublon–holon fluctuations in the Hubbard model generate antiferromagnetic exchange,

$$J_{\mathrm{eff}} = \frac{4t^2}{U} + O\!\left( \frac{t^3}{U^2} \right).$$

The low-energy spin model is valid only below the charge gap and within the projected singly occupied subspace. It does not reproduce Hubbard charge dynamics.

Anderson to Kondo

In the local-moment regime

$$\epsilon_d<0, \qquad \epsilon_d+U>0,$$

eliminating virtual empty and doubly occupied impurity states gives an antiferromagnetic exchange of schematic form

$$J_K \simeq 2\lvert V\rvert^2 \left( \frac{1}{-\epsilon_d} + \frac{1}{\epsilon_d+U} \right).$$

The reduction changes the Hilbert space: impurity charge fluctuations are integrated out. Effective Hamiltonians in Many-Body Systems owns the validity audit.

Lieb–Liniger to Tonks–Girardeau

For line density $n$ and repulsive contact coupling $g_{1\mathrm D}$, a common dimensionless interaction is

$$\gamma = \frac{m g_{1\mathrm D}} {\hbar^2 n}.$$

The limit $\gamma\to\infty$ at fixed density produces impenetrable bosons. The energy spectrum approaches that of free spinless fermions, but the bosonic one-body density matrix is not the fermionic one because the mapping contains a nonlocal sign structure.

Lieb–Liniger Model Preview owns the finite-coupling conventions, Bethe equations, thermodynamic integral equations, and two-boson ring benchmark. Tonks–Girardeau Gas Preview owns the limiting Bose–Fermi map, ring parity twist, observable-specific equivalences, and four-boson benchmark.

Quadratic and interacting boundaries

Several encyclopedia entries meet at a quadratic boundary:

$$\begin{aligned} H_{\mathrm{Hub}}\big|_{U=0} &= H_{\mathrm{free\ f}}, \\ H_{\mathrm{BH}}\big|_{U=0} &= H_{\mathrm{free\ b}}, \\ H_{\mathrm{AIM}}\big|_{U=0} &= H_{\mathrm{res}}, \\ H_{\mathrm{TFIM}}\big|_{h_z=0} &\xrightarrow{\mathrm{JW}} H_{\mathrm{quad\ f}}. \end{aligned}$$

Here “free f,” “free b,” “res,” and “quad f” denote free fermions, free bosons, the resonant-level model, and the quadratic Jordan–Wigner fermion problem. A numerical implementation should recover the corresponding spectra before being trusted in the interacting regime.

Symmetry and Sector Audit

The same Hamiltonian can have different numerical and physical content in different sectors.

Structure	Typical conserved quantity	Consequence
number-conserving bosons or fermions	$[H,N]=0$	work at fixed $N$ or state the grand ensemble
spin-rotation invariance	$[H,\mathbf S_{\mathrm{tot}}]=0$	classify by total spin as well as $S^z_{\mathrm{tot}}$
axial spin symmetry	$[H,S^z_{\mathrm{tot}}]=0$	block by magnetization, but not generally by total spin
Ising parity	$[H,\prod_j\sigma_j^x]=0$ in the standard convention	resolve even and odd sectors
translation invariance	$[H,T]=0$	label momentum only when boundaries preserve translation
inversion	$[H,P]=0$ for a symmetric geometry	separate parity sectors
particle–hole symmetry	parameter- and lattice-dependent	compare spectra only at the matching symmetric point

Degeneracies should be compared only after the same symmetry resolution. A level crossing between different sectors can be exact; a same-symmetry crossing is often avoided unless an additional conserved quantity is present.

Observables Define the Question

Two calculations can solve the same Hamiltonian and still answer different questions.

Target	Representative quantity	Essential convention
equilibrium energy	$E_0$, gap, or free energy	energy zero and sector
occupation	$\langle n_i\rangle$ or $n_{\mathbf k}$	mode and Fourier normalization
order	$\langle O\rangle$ or $\langle O_iO_j\rangle$	finite-size symmetry treatment
response	$\chi^R_{AB}(\omega)$	Fourier sign, retarded prescription, and units
spectrum	$A(\mathbf k,\omega)$	operator, broadening, and sum rule
entanglement	$S_A$ or entanglement spectrum	subsystem and logarithm base
dynamics	$\langle O(t)\rangle$ after a quench	initial state and time convention
transport	conductivity or Drude weight	current operator, limits, and boundary protocol

An exact energy spectrum does not automatically determine local matrix elements. A correct ground-state energy does not validate a spectral function. Each dossier therefore names observables separately from solution methods.

Benchmarks and Reproducibility

The first numerical validation suite covers nine stable contracts:

• transverse-field Ising spectrum and finite-size gap scaling;
• spin-$1/2$ Heisenberg-ring spectrum;
• Hubbard dimer and four-site Hubbard chain;
• two-site Bose–Hubbard spectrum;
• ideal Bose- and Fermi-gas thermodynamics;
• the mean-field BCS gap equation.

The canonical parameters, expected outputs, tolerances, and failure diagnostics live in Benchmark Problems. A dossier links to those contracts instead of restating their numbers.

A model-specific computation should pass three levels of evidence:

$$\begin{gathered} \text{identity or limiting check} \\ \Downarrow \\ \text{finite benchmark} \\ \Downarrow \\ \text{convergence study} \end{gathered}$$

Agreement at one system size is not a convergence study. Agreement with a plot is not a numerical tolerance. A calculation becomes reproducible only when its model record, code artifact, environment, parameters, and validation outputs are all identifiable; Reproducible Notebooks owns that artifact-level contract.

Reading Routes

Quantum statistics

Begin with the Ideal Bose Gas dossier or Ideal Fermi Gas dossier to fix conventions, then use the corresponding teaching article for the full thermodynamics. Continue to low-dimensional gases only after the density of states and thermodynamic limit are clear.

Lattice computation

Start with Lattice Models Overview, then choose the Ising, Heisenberg, Hubbard, or Bose–Hubbard family. Use Exact Diagonalization Preview before the matching benchmark contract.

Exact and integrable systems

Use Exact Solutions Preview to distinguish free-mode diagonalization, Jordan–Wigner mappings, and Bethe ansatz. Then read the model dossier with its boundary and observable limitations.

Pairing and impurities

For pairing, separate the exact reduced Hamiltonian from BCS Mean-Field Theory. For impurities, begin with the Anderson Impurity Model dossier and use its teaching preview for spectra, thermodynamics, transport, and DMFT. Then read the Kondo Model dossier and Kondo Model Preview for the projected spin problem. This order keeps the local-moment reduction from being mistaken for an identity.

Frontier models

The SYK dossier is a research bridge. Its large-$N$ statements require explicit disorder averaging and order-of-limits labels. Scrambling and OTOCs Preview owns the operational chaos diagnostics and their limitations.

Common Mistakes

• Treating a model name as a complete Hamiltonian and state specification.
• Quoting $h/J$, $U/t$, or $J_K\rho$ without defining operator and density-of-states conventions.
• Confusing $H$ with $H-\mu N$.
• Calling a finite matrix diagonalization an exact solution of the thermodynamic model.
• Calling an integrable spectrum a closed solution for every correlation function.
• Presenting mean-field BCS results as the exact reduced-BCS solution.
• Ignoring Jordan–Wigner parity sectors under periodic boundaries.
• Comparing spectra with different energy zeros or unresolved symmetry sectors.
• Using a bosonic occupation cutoff without a convergence sequence.
• Replacing the Anderson impurity by a Kondo spin outside the local-moment regime.
• Treating the Tonks–Girardeau mapping as equality of all bosonic and fermionic observables.
• Applying a one-dimensional exact result to higher dimensions.
• Treating a leading large-$N$ saddle as an exact statement at finite $N$.
• Inferring a thermodynamic phase transition from one small-system level crossing.
• Citing a compact dossier where the canonical derivation or primary result should be cited.

Exercises

1. Convert spin conventions. Consider

$$H = -J_\sigma \sum_j \sigma_j^z\sigma_{j+1}^z - h_\sigma \sum_j \sigma_j^x.$$

Rewrite it using $S_j^\alpha=\sigma_j^\alpha/2$. If the Pauli convention is critical at $h_\sigma/J_\sigma=1$, what is the corresponding ratio $h_S/J_S$?

Solution

Substituting $\sigma_j^\alpha=2S_j^\alpha$ gives

$$H = -4J_\sigma \sum_j S_j^zS_{j+1}^z - 2h_\sigma \sum_j S_j^x.$$

Therefore

$$J_S=4J_\sigma, \qquad h_S=2h_\sigma,$$

and the equivalent critical ratio is

$$\frac{h_S}{J_S} = \frac12.$$

The physics is unchanged; only the parameter convention moved the quoted number.

2. Classify three exactness claims. Classify each statement using the solution labels on this page.

   1. A Hubbard dimer is completely diagonalized in its six-dimensional two-particle sector.
   2. The thermodynamic one-dimensional Hubbard chain is described by coupled Bethe equations.
   3. A two-dimensional Hubbard ground-state energy is stable under increasing tensor-network bond dimension within a reported tolerance.

Solution

The first is finite exact diagonalization: it is complete for the declared finite sector, not for arbitrary lattices.

The second is Bethe-ansatz integrable for the standard one-dimensional model and compatible boundaries. The claim should still distinguish spectrum and thermodynamics from local real-time correlators.

The third is numerically converged for the stated observable and convergence protocol. It is not an analytic exact solution, and the evidence must include finite-size, geometry, and algorithmic errors as applicable.

3. Count the Bose–Hubbard dimer sector. Two bosonic modes contain a fixed total of $N$ particles. Show that the sector dimension is $N+1$, and state why an unrestricted local occupation cutoff is unnecessary for this finite problem.

Solution

A basis state has occupations

$$\lvert n_1,n_2\rangle, \qquad n_1+n_2=N.$$

Choosing $n_1=0,1,\ldots,N$ fixes $n_2=N-n_1$, so there are

$$\dim\mathcal H_N=N+1$$

basis states. Number conservation already bounds each local occupation by $N$. An additional cutoff below $N$ would change the sector, while one above $N$ adds no states.

4. Audit an incomplete tight-binding claim. A calculation reports “the tight-binding ground-state energy is $-2t$.” List at least five pieces of model data needed before the statement can be checked.

Solution

One must specify at least the graph or dimension, number of sites, boundary conditions, hopping convention and bond counting, particle statistics, particle number or filling, internal degeneracy, onsite terms, and whether the reported quantity is a one-particle energy or a many-body ground-state energy. For a periodic nearest-neighbor one-dimensional one-particle model with $t>0$ and allowed momentum $k=0$, the band minimum is $-2t$. Other geometries, twists, sizes, fillings, or sign conventions need not share that value.

5. Check the sign of the Kondo exchange. In the Anderson local-moment window, use the displayed expression for $J_K$ to determine its sign for $U>0$ and nonzero $V$. Why does that not prove the Anderson and Kondo models are identical?

Solution

The local-moment inequalities imply

$$-\epsilon_d>0, \qquad \epsilon_d+U>0.$$

Both denominators in

$$J_K \simeq 2\lvert V\rvert^2 \left( \frac{1}{-\epsilon_d} + \frac{1}{\epsilon_d+U} \right)$$

are positive, so $J_K>0$ in the displayed convention: the exchange is antiferromagnetic.

The models are not identical because the reduction projects out empty and doubly occupied impurity states and is controlled only below their charge-excitation scales. The Anderson model retains charge fluctuations, mixed valence, and charge-transfer spectral features that the spin-only Kondo model cannot represent.

References

1. A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
2. A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).
3. S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011).
4. T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press (2004).
5. H. Bethe, “Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette”, Zeitschrift für Physik 71, 205–226 (1931).
6. E. Lieb, T. Schultz, and D. Mattis, “Two Soluble Models of an Antiferromagnetic Chain”, Annals of Physics 16, 407–466 (1961).
7. J. Hubbard, “Electron Correlations in Narrow Energy Bands”, Proceedings of the Royal Society A 276, 238–257 (1963).
8. E. H. Lieb and F. Y. Wu, “Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension”, Physical Review Letters 20, 1445–1448 (1968).
9. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, “Boson Localization and the Superfluid-Insulator Transition”, Physical Review B 40, 546–570 (1989).
10. R. W. Richardson, “A Restricted Class of Exact Eigenstates of the Pairing-Force Hamiltonian”, Physics Letters 3, 277–279 (1963).
11. P. W. Anderson, “Localized Magnetic States in Metals”, Physical Review 124, 41–53 (1961).
12. J. Kondo, “Resistance Minimum in Dilute Magnetic Alloys”, Progress of Theoretical Physics 32, 37–49 (1964).
13. E. H. Lieb and W. Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State”, Physical Review 130, 1605–1616 (1963).
14. M. Girardeau, “Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension”, Journal of Mathematical Physics 1, 516–523 (1960).
15. J. Maldacena and D. Stanford, “Remarks on the Sachdev–Ye–Kitaev Model”, Physical Review D 94, 106002 (2016).

Further Study

• Common Many-Body Hamiltonians for compact formulas and convention checks.
• Exact Solutions Preview for a careful comparison of free modes, mappings, and Bethe ansatz.
• Effective Hamiltonians in Many-Body Systems for controlled low-energy reductions.
• Benchmark Problems for stable numerical validation contracts.
• Reproducible Notebooks for artifact status and execution requirements.
• Reference Model Encyclopedia for cross-volume lookup cards.
• SYK Model Preview for the Majorana ensemble, large-$N$ saddle, infrared regime, and finite-$N$ benchmark.