Exact Solutions Preview

An exact solution replaces a many-body eigenproblem by a structure that can be characterized without uncontrolled approximation. That structure might be a set of independent normal modes, coupled Bethe equations for rapidities, a commuting family of transfer matrices, or exact integral equations for thermodynamic functions. These outcomes are exact in different senses and offer very different access to observables.

The phrase does not mean that every answer is elementary, that every correlation function is available in closed form, or that a finite computer calculation proves a model integrable. The main skill is therefore diagnostic: identify what structure closes, what quantities it determines, and what remains difficult even after the model is called solvable.

Helpful background. Hilbert-space growth explains the computational obstacle, and matrix diagonalization supplies normal-mode linear algebra.

Exponential state spaces and exact structure

Use $\hbar=k_{\mathrm B}=1$ in the equations below; frequency therefore has energy units and entropy is dimensionless. Lattice positions and wave numbers are in lattice-spacing units.

For $L$ sites of local dimension $d$ , the unconstrained Hilbert-space dimension is

\dim\mathcal H_L=d^L.

A generic Hamiltonian is therefore a matrix whose dimension, not merely its number of entries, grows exponentially. Locality makes the operator sparse and symmetries split it into blocks, but neither fact normally changes the exponential asymptotics of a finite-density sector. Scaling of Hilbert Space develops that obstruction in detail.

Exact solvability occurs when the eigenproblem has additional algebraic or scattering structure. The effective data then scale more gently: one-body eigenvectors, quasiparticle occupations, rapidities, root densities, or a hierarchy of commuting charges replace arbitrary vectors in the full tensor-product space.

This reduction is stronger than knowing a few symmetries. Translation, particle number, parity, or total spin can greatly help a calculation, but a finite list of conserved quantities does not usually determine exponentially many eigenstates.

Several Meanings of Exact

The following uses of exact should not be conflated.

Statement	What is exact	What may remain hard
quadratic or free-mode solution	a canonical transformation diagonalizes the full Hamiltonian	nonlocal observables, determinants, asymptotics, boundary sectors
Bethe-ansatz solution	eigenstates and energies are encoded by coupled algebraic or transcendental equations	root classification, completeness, matrix elements, large-size limits
integrable lattice model	an extensive local or quasilocal commuting hierarchy constrains the dynamics	solving the spectral equations and evaluating observables
exact thermodynamics	free energy and related functions obey exact integral or functional equations	analytic solution of those equations and real-time observables
exact finite-size diagonalization	a chosen finite matrix is diagonalized to numerical precision	the thermodynamic limit and any claim of integrability
rigorous theorem or bound	a stated result follows under explicit hypotheses	construction of all eigenstates or experimentally measurable functions

An implicit equation can be an exact answer. If roots $\{\lambda_j\}$ obey equations derived without approximation and an observable is an exact function of those roots, the result is exact even when the roots require numerical solution. Conversely, a high-precision number for $L=20$ is not automatically an exact solution of the model for arbitrary $L$ .

Exact does not mean closed form

There is no universally useful requirement that an exact result be expressible through elementary functions. Many central exact results take the form of:

a determinant or Pfaffian;
a set of quantization conditions;
a Fredholm determinant;
a nonlinear integral equation;
a convergent form-factor series;
a recursion or functional relation;
an asymptotic formula with rigorously controlled errors.

The mathematical representation and the physical question must be stated together.

Exact does not mean universal

An exact formula can depend on:

open versus periodic boundaries;
a parity, charge, or magnetization sector;
the ordering of limits $L\to\infty$ , $T\to0$ , and $t\to\infty$ ;
a particular normalization of rapidity or spectral parameter;
whether the observable is local in the variables that diagonalize the Hamiltonian.

These choices determine which physical system the result describes.

Route I: Quadratic Hamiltonians

The most transparent exact many-body route is reduction to independent modes. For canonical fermions, a number-conserving quadratic Hamiltonian has the form

H = \sum_{i,j=1}^{M} c_i^\dagger h_{ij}c_j +E_0,

where $h=h^\dagger$ is an $M\times M$ one-body matrix. Choose a unitary matrix $U$ such that

U^\dagger hU = \operatorname{diag} (\varepsilon_1,\ldots,\varepsilon_M).

The transformed operators

\gamma_\alpha = \sum_i U_{i\alpha}^*c_i

obey the same canonical anticommutation relations, and

H = \sum_{\alpha=1}^{M} \varepsilon_\alpha \gamma_\alpha^\dagger\gamma_\alpha +E_0.

Every many-body eigenstate is labeled by occupations $n_\alpha\in\{0,1\}$ :

|n_1,\ldots,n_M\rangle = \prod_{\alpha=1}^{M} (\gamma_\alpha^\dagger)^{n_\alpha}|0\rangle,

E_{\{n\}} = E_0 + \sum_\alpha n_\alpha\varepsilon_\alpha.

The exponential list of energies has not disappeared, but it is generated from only $M$ one-body numbers. That compression is the exact structure.

Translation-invariant example

For a periodic chain with hopping $t$ and chemical potential $\mu$ ,

H = -t\sum_j (c_j^\dagger c_{j+1}+c_{j+1}^\dagger c_j) - \mu\sum_j n_j.

The discrete Fourier transform gives

c_j = \frac{1}{\sqrt L} \sum_k e^{ikj}c_k,

H = \sum_k \varepsilon(k)c_k^\dagger c_k, \qquad \varepsilon(k) = -2t\cos k-\mu.

The allowed $k$ values still depend on the boundary twist. Tight-Binding Model gives the one-body derivation and band interpretation; the tight-binding chain gives finite-system examples.

Paired quadratic Hamiltonians

Number conservation is not required. A fermionic quadratic Hamiltonian may contain pairing terms,

H = \sum_{i,j} c_i^\dagger A_{ij}c_j + \frac12\sum_{i,j} \left( c_i^\dagger B_{ij}c_j^\dagger + c_j B_{ij}^*c_i \right) +C,

with $A=A^\dagger$ and $B=-B^{\mathsf T}$ . Introduce the Nambu vector

\Psi = \begin{pmatrix} c_1&\cdots&c_M&c_1^\dagger&\cdots&c_M^\dagger \end{pmatrix}^{\mathsf T}.

Then

H = \frac12\Psi^\dagger \mathcal H_{\mathrm{BdG}} \Psi +C'.

A Bogoliubov transformation preserving the canonical anticommutation relations brings the finite fermionic problem to

H = E_{\mathrm{vac}} + \sum_{a=1}^{M} E_a\eta_a^\dagger\eta_a, \qquad E_a\ge0.

This is how the one-dimensional transverse-field Ising and anisotropic XY chains become free quasiparticle theories after the Jordan–Wigner map. The exact spin–fermion dictionary and parity-dependent momentum grids are developed on Jordan–Wigner Transformation.

For bosons, a quadratic Hamiltonian can also be treated by a Bogoliubov transformation, but the transformation is paraunitary rather than unitary in Nambu space. Positivity and dynamical stability must be checked; an algebraic diagonal form with complex frequencies does not describe stable independent oscillators.

Gaussian states and Wick reduction

A nondegenerate fermionic quadratic ground state and a thermal exponential of a quadratic operator over the full Fock space are Gaussian. Individual quasiparticle occupation eigenstates are also Gaussian. An arbitrary superposition in a degenerate ground space need not be Gaussian, and projection of a thermal state onto fixed number or parity generally destroys Wick factorization. Their equal-time information is encoded by two-point functions such as

C_{ij} = \langle c_i^\dagger c_j\rangle, \qquad F_{ij} = \langle c_i c_j\rangle.

Higher products reduce to sums of pairings. For example, in a number-conserving Gaussian state,

\langle c_i^\dagger c_j^\dagger c_k c_l \rangle = C_{il}C_{jk} - C_{ik}C_{jl}.

This reduction is exact, but a nonlocal spin observable can become a long fermion string and hence a determinant or Pfaffian. A free Hamiltonian does not make every observable a one-body quantity. Wick’s theorem derives the contraction rule.

What quadratic solvability buys

Once the canonical transform is known, one can usually obtain:

the complete finite-size spectrum;
ground-state and thermal occupations;
unrestricted thermal traces as products over modes, with fixed-number traces obtained by coefficient extraction;
Gaussian correlation matrices;
unitary dynamics of creation and annihilation operators;
many entanglement measures from restricted correlation matrices.

The main remaining work lies in boundaries, zero modes, degeneracies, asymptotic evaluation, and observables nonlocal in the diagonal variables.

What makes the route fail

Add a density interaction,

V = \sum_{i<j} U_{ij}n_i n_j.

A generic nonzero interaction between distinct modes remains quartic under a linear canonical transformation. The same-mode fermionic term $n_i^2=n_i$ would instead be only a one-body shift. The Heisenberg equation for $c_i$ now involves cubic operators, whose equations involve higher products. The linear operator algebra no longer closes. Special interacting models may still be Bethe-ansatz integrable, but they are not rendered free by the same one-body diagonalization.

Route II: Coordinate Bethe Ansatz

Bethe ansatz addresses certain interacting one-dimensional systems by exploiting highly constrained scattering. Consider a sector with $M$ particles or spin deviations at ordered positions

x_1<x_2<\cdots<x_M.

Away from collisions, a coordinate Bethe wavefunction is a superposition of plane waves,

\psi(x_1,\ldots,x_M) = \sum_{P\in S_M} A(P) \exp\!\left( i\sum_{a=1}^{M}k_{P_a}x_a \right).

The interaction appears in the relations among amplitudes when neighboring particles exchange order. In the simplest scalar case,

A(P\circ\tau_a) = S(k_{P_a},k_{P_{a+1}})A(P),

where $\tau_a$ swaps adjacent momenta and $S$ is the two-body scattering phase.

Periodic quantization

Move particle $j$ once around a ring of length $L$ . It accumulates a propagation phase and scatters through all other particles. Periodicity gives the schematic Bethe equations

e^{ik_jL} \prod_{\ell\ne j} S(k_j,k_\ell) = 1, \qquad j=1,\ldots,M.

The energy and momentum often remain additive in the rapidity data,

E = \sum_{j=1}^{M}\varepsilon(k_j)+E_{\mathrm{ref}},

P = \sum_{j=1}^{M}p(k_j) \pmod{2\pi}.

The model is interacting because the allowed $k_j$ are coupled through $S$ . Independent modes would have separate one-body quantization conditions.

Rapidities and Bethe roots

Momentum is not always the natural variable. A rapidity $\lambda$ can rationalize the scattering phase or expose the symmetry of the equations. A common logarithmic structure is

L p(\lambda_j) + \sum_{\ell\ne j} \theta(\lambda_j-\lambda_\ell) = 2\pi I_j,

where $I_j$ are integers or half-integers fixed by the sector and boundary convention. The solutions $\lambda_j$ are Bethe roots.

Roots need not all remain real. Complex roots can form patterns associated with bound states. The familiar string hypothesis is often an asymptotically powerful classification, but finite-size string deviations, exceptional solutions, and completeness questions must be treated rather than hidden.

Why one dimension helps

In one dimension, particle order can change only through encounters. Exact Bethe structure requires those encounters to factor consistently into two-body scattering, with no diffractive production of new independent momenta. For particles carrying internal labels, different sequences of pairwise exchanges must agree. That consistency leads to the Yang–Baxter equation discussed below.

What the ansatz actually supplies

For a successful model and sector, Bethe ansatz can provide:

quantization equations for allowed roots;
exact energy and momentum formulas in terms of those roots;
eigenvectors through coordinate amplitudes or algebraic creation operators;
a path to thermodynamic root densities;
in favorable cases, norms, form factors, determinants, and correlation functions.

Each item requires additional work. Writing down Bethe equations is not the same as classifying every admissible solution, proving completeness, or evaluating a dynamical correlator.

The XXZ chain as the canonical example

The uniform nearest-neighbor spin- $1/2$ XXZ chain is interacting for anisotropy $\Delta\ne0$ and nevertheless Bethe-ansatz integrable. Its page derives the coordinate ansatz, gives a rapidity convention, explains the exact phase boundaries, and previews thermodynamic Bethe ansatz. See XXZ Spin Chain for those model-specific statements.

The free XX point, $\Delta=0$ , sits at the intersection of the two routes: Jordan–Wigner makes it quadratic, and its Bethe scattering reduces to fermionic exchange. This overlap is a special limit, not evidence that every Bethe-integrable chain is free.

Route III: Quantum Integrability

There is no single context-free definition of quantum integrability accepted for every finite-dimensional, continuum, lattice, and field-theoretic problem. Naively counting commuting operators is too weak: any nondegenerate finite Hamiltonian has spectral projectors that commute with it, but those projectors do not constitute useful local conservation laws.

For translation-invariant quantum lattice models, a productive working criterion is the existence of an extensive family of independent local or quasilocal charges,

[H,Q_n]=0, \qquad [Q_m,Q_n]=0,

with the number of relevant charges growing with system size. Locality or quasilocality prevents arbitrary spectral projectors from trivializing the definition.

This is a structural criterion, not a slogan. One must specify the family of systems, the boundaries, the locality notion, and the independence of the charges.

A commuting transfer matrix

In many integrable chains, the central object is a transfer matrix $T(u)$ depending on a spectral parameter $u$ and satisfying

[T(u),T(v)]=0

for all $u$ and $v$ . Expanding around a regular point $u_0$ generates commuting charges schematically as

Q_{n+1} \propto \left. \frac{d^n}{du^n} \log T(u) \right|_{u=u_0}.

For an appropriate normalization, one member of this hierarchy is the Hamiltonian. With a regular local $R$ matrix and compatible boundaries, higher derivatives generate increasingly extended local charges. Commutativity alone does not prove locality or completeness; extra transfer-matrix families can be needed for quasilocal charges.

The phrase transfer matrix originates in classical statistical mechanics. A two-dimensional classical lattice model and a one-dimensional quantum chain can share the same algebraic object; commuting transfer matrices then organize both exact partition functions and quantum conserved charges.

Yang–Baxter consistency

Let $R_{ab}(u)$ act on a pair of auxiliary or local spaces. The Yang–Baxter equation has the schematic form

R_{12}(u-v) R_{13}(u-w) R_{23}(v-w) = R_{23}(v-w) R_{13}(u-w) R_{12}(u-v).

It states that two different sequences of pairwise rearrangements give the same many-body result. This is the algebraic expression of factorized scattering. Building monodromy and transfer matrices from such an $R$ matrix yields the commuting family above under suitable boundary conditions.

For matrix-valued scattering, this condition is essential: scalar phases commute automatically, whereas internal spin or species labels make exchange order nontrivial.

Integrability is more than symmetry

A global symmetry supplies charges, multiplets, and selection rules. Integrability supplies an extensive hierarchy sufficient to constrain scattering and dynamics far beyond a finite-dimensional Lie algebra. Thus:

translation invariance does not by itself imply integrability;
conservation of particle number does not imply integrability;
global $SU(2)$ symmetry does not imply integrability;
absence of obvious symmetry does not rule out hidden integrable structure;
a single extra conserved operator is not normally enough.

Symmetry and integrability can coexist. In the Heisenberg and XXZ chains, familiar spin symmetries organize states while transfer-matrix charges provide the stronger integrable structure.

Integrable does not mean noninteracting

An interacting integrable model can have nontrivial phase shifts, bound states, dressed quasiparticles, and strongly correlated eigenstates. What is absent is generic diffractive many-body scattering, not all scattering. The outgoing rapidity set is constrained by the incoming one, while pairwise phases encode interactions.

Free models are integrable in a broad sense because each mode occupation is conserved. Their charge hierarchy is especially transparent. Interacting Bethe models form a richer class whose quasiparticles are dressed by the state and ensemble.

Thermodynamic Bethe ansatz

At finite density, the number of Bethe roots grows with system size. Their thermodynamic limit is described by particle and hole densities, $\rho(\lambda)$ and $\rho^{\mathrm h}(\lambda)$ . To keep signs explicit, consider a single species with an even kernel and choose

\rho_{\mathrm t}=\rho+\rho^{\mathrm h}=a+K*\rho,\qquad (K*f)(\lambda)=\int d\mu\,K(\lambda-\mu)f(\mu).

The kernel comes from the derivative of a scattering phase, and $a$ is the bare density of allowed roots. Many Bethe states share the same smooth densities. Their counting gives the entropy density

s=\int d\lambda\, \left[\rho_{\mathrm t}\ln\rho_{\mathrm t} -\rho\ln\rho-\rho^{\mathrm h}\ln\rho^{\mathrm h}\right].

The logarithms combine into occupation ratios, so the arbitrary common density unit cancels. Minimize $f=e-Ts-\mu q$ subject to the density relation. Defining a pseudoenergy by $\rho^{\mathrm h}/\rho=e^{\epsilon/T}$ gives

\epsilon=e^{\mathrm{bare}}-\mu q^{\mathrm{bare}} -T K*\ln(1+e^{-\epsilon/T}).

The same sign convention gives a dressed charge

q^{\mathrm{dr}}=q^{\mathrm{bare}} +K*(\vartheta q^{\mathrm{dr}}),\qquad \vartheta=\frac{\rho}{\rho_{\mathrm t}}.

Adding one root rearranges the other roots; dressing measures that collective adjustment. These three equations use the same kernel convention. Several species, strings, different scattering-phase conventions or asymmetric kernels require the appropriate matrix signs and transposes, not substitution into this scalar example.

Once the equations and their admissible species are established for a model, they determine its exact bulk free energy and equilibrium derivatives. They still require numerical solution in many cases. Finite-size effects and real-time matrix elements are additional problems. The XXZ treatment connects this construction to a specific lattice Hamiltonian.

A Hierarchy of Exact Information

Solving one layer does not guarantee the next.

Layer	Typical exact object	Additional challenge
energies	mode sums or Bethe-root formulas	enumerate admissible roots and degeneracies
eigenvectors	Fock states, coordinate waves, algebraic Bethe states	normalization and completeness
thermodynamics	product partition function or TBA equations	solve coupled integral equations and limits
static correlators	Wick determinant, form factors, multiple integrals	sum intermediate states and extract asymptotics
dynamical correlators	spectral or form-factor expansions	analytic continuation and oscillatory sums
finite boundaries	reflection equations or boundary modes	boundary bound states and altered quantization
nonequilibrium evolution	Gaussian propagation or integrable quasiparticle data	initial-state overlaps and long-time limits

This hierarchy explains why an article may correctly call a Hamiltonian exactly solvable while quoting numerical correlation functions. The spectrum can be exact even when the observable requires a difficult resummation. For complete charge data and Bethe root densities after a quench, see Integrability and Generalized Gibbs Ensembles Preview.

Spectrum versus observables

For an operator $O$ , a zero-temperature dynamical structure factor contains matrix elements as well as energies:

S_O(q,\omega) = 2\pi \sum_n |\langle n|O_q|0\rangle|^2 \delta(\omega-E_n+E_0).

Exact eigenvalues determine the delta-function locations. They do not determine the weights $|\langle n|O_q|0\rangle|^2$ . Those form factors can be the harder part of the problem.

Finite volume versus thermodynamic limit

Bethe equations may be exact for every finite $L$ , while root-density equations describe only the $L\to\infty$ limit. Conversely, universal finite-size corrections can reveal conformal data not visible in the leading bulk energy. Always label which limit an exact statement concerns.

Why Exact Solvability Is Exceptional

The exact routes above require closure conditions that a generic local perturbation does not respect.

Quadratic closure is nongeneric

For the paired quadratic Hamiltonian above, with the same matrices $A$ and $B$ , commutators with linear fermion operators remain linear:

[H,c_i] = -\sum_j (A_{ij}c_j+B_{ij}c_j^\dagger).

For one mode, $H=\varepsilon c^\dagger c$ gives $[H,c]=-\varepsilon c$ , fixing the sign. The pairing contribution follows from $[c_j^\dagger c_k^\dagger,c_i]=\delta_{ki}c_j^\dagger-\delta_{ji}c_k^\dagger$ and $B_{jk}=-B_{kj}$ . Quartic interactions generate cubic operators. Their commutators generate still higher products, producing an operator hierarchy rather than a finite-dimensional linear problem.

Factorized scattering is restrictive

In a generic interacting many-body collision, outgoing momenta can explore configurations not obtainable by merely permuting incoming rapidities. This diffraction prevents a wavefunction from being assembled consistently from a fixed set of plane-wave momenta and two-body phases.

For matrix-valued scattering, even nondiffractive two-body data must satisfy Yang–Baxter consistency. That is a set of functional constraints on couplings, not an automatic property of one-dimensionality.

Conserved-charge hierarchies are fragile

Let

H(\lambda) = H_0+\lambda V,

where $H_0$ has a local conserved charge $Q$ . To continue it as

Q(\lambda) = Q+\lambda\,\delta Q+O(\lambda^2),

conservation at first order requires

[H_0,\delta Q] + [V,Q] = 0.

For a generic $V$ , there is no local or quasilocal $\delta Q$ solving this equation for every charge in the hierarchy. Special deformations can preserve integrability, but they occupy constrained families in coupling space.

Exceptional does not mean physically irrelevant

Exact models are valuable because they:

provide nonperturbative benchmarks for numerical methods;
identify phases and transitions that survive nearby perturbations;
supply controlled starting points for perturbation theory;
reveal quasiparticles, anomalies, and selection rules hidden in microscopic variables;
test universal low-energy descriptions;
expose failures of thermalization and unusual transport constraints;
calibrate experiments in tunable one-dimensional systems.

The exact point may be finely tuned while its qualitative lessons remain robust over a finite region.

Integrability Breaking and Prethermal Structure

Suppose $H=H_{\mathrm{int}}+\lambda V$ with $|\lambda|\ll1$ . The charges of $H_{\mathrm{int}}$ are no longer exactly conserved, but their time derivatives begin at order $\lambda$ :

\frac{dQ_n}{dt} = i[H,Q_n] = i\lambda[V,Q_n].

Expectation values can therefore remain close to integrable behavior for a long intermediate window before generic thermalization mechanisms dominate. The lifetime depends on the perturbation, state, dimensionality, and resonances; it is not universally $1/|\lambda|$ .

This prethermal use of an exact model is conceptually different from claiming the perturbed model remains integrable. Approximate charges can organize dynamics without commuting with the full Hamiltonian exactly.

Diagnostics and Their Limits

No single small-system diagnostic proves integrability. Useful evidence includes:

construction of a Yang–Baxter $R$ matrix or commuting transfer matrix;
an explicit extensive family of local or quasilocal charges;
a complete Bethe-ansatz construction with matching state counts;
factorized scattering without diffraction;
exact functional relations or solvable transfer-matrix spectra;
spectral statistics after every ordinary symmetry has been resolved.

Level statistics

Integrable spectra often show approximately Poissonian level-spacing statistics, while generic chaotic systems in an appropriate symmetry class show level repulsion. This comparison is meaningful only after separating particle number, momentum, parity, total spin, and other exact symmetries. Mixing independent symmetry blocks can manufacture apparent Poisson statistics.

Level statistics remain a diagnostic, not a definition. Free models, localized systems, finite-size crossovers, accidental degeneracies, and unresolved symmetries can all complicate the inference. Many-body quantum chaos explains symmetry classes and level-statistics diagnostics.

Counting conservation laws

Every spectral projector $|n\rangle\langle n|$ commutes with a finite nondegenerate Hamiltonian. Declaring these projectors conserved would make every finite model integrable and erase the physical distinction. The useful question is whether charges have controlled locality, extensivity, independence, and a construction that persists as $L$ grows.

Boundaries, Twists, and Sectors

Exact structure is inseparable from boundary data.

For free particles, a twist $\phi$ changes momenta to

k_m = \frac{2\pi m+\phi}{L}.

For Jordan–Wigner fermions obtained from a periodic spin chain, the effective twist depends on fermion parity. One must diagonalize in the compatible sector and project onto states of the matching parity.

For Bethe systems, periodic boundaries give bulk Bethe equations, while integrable open boundaries require reflection matrices satisfying boundary consistency equations. A generic boundary term may break the commuting transfer-matrix family even when the bulk couplings remain integrable.

Boundary contributions are subextensive in the leading bulk free energy, but they can control:

exact finite-size levels and momenta;
edge or boundary bound states;
ground-state degeneracy;
persistent currents under twists;
boundary critical exponents;
entanglement cuts and topological zero modes.

Boundary conditions on lattices compares these closures and their shell effects.

Exact Diagonalization Is a Different Phrase

In computational many-body physics, exact diagonalization means constructing and diagonalizing the Hamiltonian in a finite basis, often after resolving symmetries. If the basis is complete for that finite system and arithmetic error is controlled, its eigenpairs are numerically exact for that chosen $L$ .

This does not imply:

a formula valid for arbitrary system size;
polynomial computational cost;
an extensive conserved-charge hierarchy;
factorized scattering;
exact thermodynamic-limit behavior.

Exact diagonalization is indispensable precisely because most models are not analytically solvable. It also tests exact formulas at small $L$ . See Matrix Diagonalization for the numerical linear-algebra layer.

Exercises

Build a free many-body spectrum

Two fermionic normal modes have Hamiltonian

H = \varepsilon_1 n_1 + \varepsilon_2 n_2 +E_0, \qquad n_a\in\{0,1\}.

List the complete spectrum and compute both the unrestricted Fock-space thermal trace and the fixed- $N=1$ canonical partition function. Explain where the exponential many-body state count is hidden.

Solution

The four occupation states and their energies are

\begin{array}{c|c} (n_1,n_2)&E\\ \hline (0,0)&E_0\\ (1,0)&E_0+\varepsilon_1\\ (0,1)&E_0+\varepsilon_2\\ (1,1)&E_0+\varepsilon_1+\varepsilon_2 \end{array}

Therefore

Z = \sum_{n_1,n_2=0}^{1} e^{-\beta(E_0+n_1\varepsilon_1+n_2\varepsilon_2)}

factorizes as

Z = e^{-\beta E_0} (1+e^{-\beta\varepsilon_1}) (1+e^{-\beta\varepsilon_2}).

For $M$ modes there are still $2^M$ many-body occupation patterns, but every energy and thermal weight is generated from only $M$ one-mode energies. The product is an unrestricted trace (equivalently a grand partition function at chemical potential zero). At fixed $N=1$ , instead

Z_1=e^{-\beta E_0}(e^{-\beta\varepsilon_1}+e^{-\beta\varepsilon_2}).

The exact solution compresses the description; it does not remove the number constraint or reduce the literal number of eigenstates.

Generate commuting charges

Assume $T(u)$ is invertible near $u_0$ and

[T(u),T(v)]=0

for all nearby $u,v$ . Explain why coefficients in the expansion

\log T(u) = \sum_{n=0}^{\infty} (u-u_0)^n C_n

commute with one another.

Solution

Because the entire family $T(u)$ commutes, analytic functions of different members also commute in the common domain of definition. In particular,

[\log T(u),\log T(v)]=0.

Insert both power-series expansions:

0 = \sum_{m,n\ge0} (u-u_0)^m(v-u_0)^n [C_m,C_n].

The variables $u-u_0$ and $v-u_0$ are independent. Every coefficient must therefore vanish:

[C_m,C_n]=0.

Derivatives of $\log T(u)$ at $u_0$ are proportional to these $C_n$ , so they form a commuting hierarchy. Showing that the resulting operators are local or quasilocal requires additional properties of the transfer-matrix construction.

Separate exact outputs

Suppose a Bethe-ansatz solution gives every finite-volume energy $E_n$ and momentum $P_n$ but no matrix elements of a local operator $O$ . Which of the following are immediately determined?

the density of states;
the partition function at finite $L$ ;
the locations of spectral lines in $S_O(q,\omega)$ ;
the spectral weights;
the equal-time correlator $\langle O_jO_0\rangle$ .

Solution

The spectrum determines the density of states and, in principle, the finite-volume partition function

Z(\beta)=\sum_n e^{-\beta E_n}.

Energy and momentum differences determine where transitions are kinematically allowed, so they determine the possible locations of spectral lines.

The weights require matrix elements such as

|\langle n|O_q|0\rangle|^2.

Without them, neither the spectral weights nor the equal-time correlation function is fixed. Equal-time correlations can be obtained by integrating a known dynamical structure factor, but the energies alone are insufficient.

Degeneracy does not force a Gaussian state

For four fermionic modes with $H=0$ , consider $|\psi\rangle=(|0000\rangle+|1111\rangle)/\sqrt2$ . Find $C_{ij}=\langle c_i^\dagger c_j\rangle$ , the anomalous two-point means, and $\langle n_1n_2\rangle$ . Does Wick factorization hold?

Solution

Every normal occupation is $1/2$ , off-diagonal normal means vanish, and all anomalous two-point means vanish: two operators cannot connect vacuum to the four-particle component. Hence $C=I/2$ . But $\langle n_1n_2\rangle=1/2$ , whereas Gaussian Wick factorization would give $C_{11}C_{22}-|C_{12}|^2=1/4$ . The state is a ground state of this degenerate quadratic Hamiltonian and is not Gaussian.

References

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