Benchmark Problems

A trustworthy benchmark is a falsifiable numerical contract. It specifies the problem before the calculation is run, identifies a trusted answer, declares how agreement will be measured, and explains what a disagreement would diagnose. A familiar Hamiltonian or a plausible plot is not enough.

This page defines a compact benchmark suite for many-body and statistical quantum mechanics. Its nine contracts exercise distinct layers of a computational workflow:

• spin and occupation-number basis construction;
• fermionic signs and bosonic enhancement factors;
• symmetry sectors, degeneracies, and trace moments;
• thermodynamic quadrature and constrained root finding;
• nonlinear self-consistency equations;
• finite-size scaling with known boundary corrections.

The suite is intentionally small enough to run routinely. Passing it does not establish that a method works for every coupling, size, observable, or phase. It establishes only that the implementation reproduces the channels and regimes that the contracts actually exercise.

Purpose and Canonical Scope

This page is the canonical home for the many-body physics benchmark suite and its stable MB-Bxxx contracts. It owns fixed conventions, parameter points, target values, refinement directions, acceptance logic, and failure diagnoses for the listed problems.

It does not rederive each model. The physical definitions and analytic developments remain at their canonical homes:

• Transverse-Field Ising Model;
• Heisenberg Model;
• Hubbard Model;
• Bose–Hubbard Model and Bose–Hubbard Dimer;
• Ideal Bose Gas;
• Ideal Fermi Gas;
• BCS Mean-Field Theory.

The Model-to-Volume Cross-Link Index maps each contract back to its dossier, teaching page, compact card, and application destination. The Reference benchmark taxonomy owns sitewide naming and report fields. Validation Tests owns executable test families and status labels. Exact Diagonalization Preview and Finite-Size Scaling in Numerics explain the methods used here.

What a Contract Contains

Every benchmark in this suite fixes the following data before execution:

Field	Required content
ID	stable identifier of the form MB-Bxxx
model	Hamiltonian or ensemble and canonical page
representation	basis ordering, local convention, or integration variable
sector	conserved quantum numbers, boundary condition, or thermodynamic ensemble
parameters	dimensionless values and the energy or length unit
targets	exact values, trusted digits, identities, or asymptotic coefficients
diagnostics	residuals, moments, degeneracies, observables, and convergence trends
refinement	system size, quadrature order, solver tolerance, or fit window
acceptance	a criterion declared before the result is inspected
horizon	an explicit statement of what the passing result does not validate

These fields prevent a common ambiguity: two codes can claim to solve “the Hubbard dimer” while using different mode orderings, particle sectors, signs, or energy offsets.

Error Measures and Reference Profile

For a scalar observable $O$, record an absolute error and a scale-aware error,

$$\begin{aligned} \epsilon_{\mathrm{abs}} &= \left| O_{\mathrm{num}}-O_{\mathrm{ref}} \right|, \\ \epsilon_{\mathrm{scale}} &= \frac{ \epsilon_{\mathrm{abs}} }{ \max\!\left( |O_{\mathrm{ref}}|, O_{\mathrm{scale}} \right) }. \end{aligned}$$

The explicit $O_{\mathrm{scale}}$ avoids meaningless relative errors when the reference value is zero. For a normalized approximate eigenpair, report a residual such as

$$r_E = \frac{ \left\| H|\widetilde\psi\rangle - \widetilde E|\widetilde\psi\rangle \right\|_2 }{E_{\mathrm{scale}}}.$$

No one tolerance is appropriate for every method. The following reference profile is suitable for the tiny dense matrices below when they are assembled and diagonalized in IEEE double precision:

Quantity	Illustrative pass threshold
relative Hermiticity defect $\|H-H^\dagger\|_F/\max(\|H\|_F,1)$	$10^{-13}$
listed eigenvalue error in units of the stated coupling	$10^{-11}$
normalized trace-moment error	$10^{-12}$
listed ground-state observable error	$10^{-10}$
normalized eigenpair residual	$10^{-11}$

Sparse iterative solvers, stochastic estimates, arbitrary precision, and thermodynamic extrapolations require their own tolerances. A looser threshold can be legitimate, but the reason must be part of the benchmark record rather than chosen after seeing the output.

Suite at a Glance

ID	Problem	Primary implementation layer	Trusted anchor
MB-B001	transverse-field Ising chain	spin-bit assembly and boundary bonds	exact $L=2$ spectrum and $L=4$ trace moments
MB-B002	spin-$1/2$ Heisenberg ring	spin normalization and multiplets	complete $L=4$ spectrum
MB-B003	Hubbard dimer	fermionic signs and spin sectors	analytic six-state spectrum
MB-B004	four-site Hubbard chain	many-fermion basis and observables	independently reproducible 36-state block
MB-B005	two-site Bose–Hubbard model	bosonic ladder factors	analytic three-state spectrum
MB-B006	ideal Bose gas	Bose functions and condensation branch	polylogarithm root and condensate fraction
MB-B007	ideal Fermi gas	Fermi integrals and Sommerfeld limit	zero-temperature values and low-$T$ coefficients
MB-B008	BCS gap equation	nonlinear self-consistency	exact zero-temperature gap
MB-B009	critical Ising gap scaling	size sequence and extrapolation	exact open-chain gap for every $L$

The order is useful. Run algebraic finite-cluster tests before interpreting a large simulation. A sophisticated extrapolation cannot repair a basis or sign error.

MB-B001: Transverse-Field Ising Chain

Contract

Use Pauli matrices with eigenvalues $\pm1$ and

$$H_{mathrm{OBC}} = -J\sum_{i=1}^{L-1} \sigma_i^z\sigma_{i+1}^z -h\sum_{i=1}^{L}\sigma_i^x.$$

The computational basis obeys

$$\sigma^z|0\rangle=|0\rangle, \qquad \sigma^z|1\rangle=-|1\rangle.$$

For periodic boundaries, add the bond $-J\sigma_L^z\sigma_1^z$ exactly once. No constant shift is applied.

The primary analytic case is $L=2$ with open boundaries. Its four eigenvalues are

$$\left\{ -\sqrt{J^2+4h^2}, -J, +J, +\sqrt{J^2+4h^2} \right\}.$$

At $J=1$ and $h=0.7$, the sorted spectrum is

$$\begin{aligned} E/J = \{&-1.720465053409, -1, \\ &1, 1.720465053409\}. \end{aligned}$$

Stronger assembly check

Use $L=4$, periodic boundaries, $J=1$, and $h=0.7$. The distinct energy levels and degeneracies are

$E/J$	degeneracy
$-4.563856065203$	1
$-4.441311123147$	1
$-1.735286090565$	1
$-1.400000000000$	2
$-0.441311123147$	1
$0$	4
$0.441311123147$	1
$1.400000000000$	2
$1.735286090565$	1
$4.441311123147$	1
$4.563856065203$	1

The full Hilbert-space dimension is $16$, and two useful basis-independent moments are

$$\begin{aligned} \operatorname{Tr}H &=0, \\ \frac{1}{16}\operatorname{Tr}H^2 &= 4(J^2+h^2) \\ &=5.96. \end{aligned}$$

Required checks

1. Confirm the dimension $2^L$ and Hermiticity.
2. Reproduce the $L=2$ spectrum before adding periodic boundaries.
3. For $L=4$, reproduce the degeneracy sum, ground energy, trace, and second moment.
4. Verify the global spin-flip symmetry $P=\prod_i\sigma_i^x$ through $[H,P]=0$.
5. If symmetry blocks are used, reconstruct the complete spectrum from all blocks.

What failures diagnose

• A factor-of-two discrepancy usually means spin operators $S^\alpha=\sigma^\alpha/2$ were substituted for Pauli matrices.
• An incorrect $L=2$ periodic result often comes from counting the same undirected bond twice.
• A correct ground energy with a wrong trace moment points to incomplete basis enumeration or a hidden energy shift.
• Missing degeneracies can indicate accidental symmetry breaking in the matrix assembly.

Passing MB-B001 validates these finite-chain conventions. It does not validate the thermodynamic critical point, long-distance correlations, or a time-evolution routine.

MB-B002: Spin-1/2 Heisenberg Ring

Contract

Use dimensionless spin operators $\mathbf S_i=\boldsymbol\sigma_i/2$ and the four-site periodic Hamiltonian

$$H = J\sum_{i=1}^{4} \mathbf S_i\mathbin{\cdot}\mathbf S_{i+1}, \qquad \mathbf S_5\equiv\mathbf S_1.$$

The four undirected bonds are $(1,2)$, $(2,3)$, $(3,4)$, and $(4,1)$. Set $J=1$ and apply no constant offset.

The complete $16$-state spectrum is

$E/J$	degeneracy	diagnostic content
$-2$	1	singlet ground state
$-1$	3	spin-one multiplet
$0$	7	remaining singlet and triplet states
$+1$	5	fully symmetric spin-two multiplet

The trace moments are

$$\operatorname{Tr}H=0, \qquad \frac{1}{16}\operatorname{Tr}H^2 = \frac{3}{4}J^2.$$

The fully polarized state supplies a one-line bond check: every bond has $\mathbf S_i\cdot\mathbf S_{i+1}=1/4$, so its total energy is $J$.

Required checks

1. Reproduce all four distinct energies and their degeneracies.
2. Check $[H,S^z_{\mathrm{tot}}]=0$ and $[H,\mathbf S_{\mathrm{tot}}^2]=0$.
3. Verify that states grouped by total spin are degenerate to the declared tolerance.
4. Reconstruct the full trace moments from any $S^z_{\mathrm{tot}}$ blocks used in the calculation.
5. Compare the lowest energies obtained in the full basis and in the $S^z_{\mathrm{tot}}=0$ block.

What failures diagnose

A spectrum larger by a factor of four almost always signals the use of Pauli matrices where spin operators were intended. An isolated error in the polarized energy suggests a bond-counting problem. Correct eigenvalues but incorrect multiplet labels point to a faulty total-spin operator or an inconsistent tensor-factor ordering.

The thermodynamic Bethe-ansatz energy density,

$$\lim_{L\to\infty} \frac{E_0(L)}{LJ} = \frac14-\ln2,$$

is a valuable later test, but it is not part of the tiny-ring acceptance condition. The canonical Heisenberg page owns its interpretation.

MB-B003: Hubbard Dimer

The Hubbard Dimer dossier supplies the complete finite-model record, exact observables, and physical interpretation. This section remains the authority for the stable numerical contract.

Contract

Use two sites, two spin species, and

$$\begin{aligned} H ={}& -t\sum_{\sigma} \left( c_{1\sigma}^\dagger c_{2\sigma} + c_{2\sigma}^\dagger c_{1\sigma} \right) \\ &+ U\sum_{i=1}^{2} n_{i\uparrow}n_{i\downarrow}. \end{aligned}$$

Fix the fermionic mode order to

$$(1\uparrow,1\downarrow,2\uparrow,2\downarrow)$$

and benchmark the total-particle sector $N=2$, whose dimension is $\binom42=6$. The $N_\uparrow=N_\downarrow=1$ block has dimension $4$ and contains the $S^z=0$ member of the triplet.

With suitable phases for the singlet and even/odd doublon states, the singlet block is

$$H_{S=0} = \begin{pmatrix} 0 & -2t & 0\\ -2t & U & 0\\ 0 & 0 & U \end{pmatrix}.$$

The full $N=2$ spectrum is

$$\begin{gathered} E_- = \frac{U-\sqrt{U^2+16t^2}}{2}, \\ E_T=0 \quad\text{with degeneracy }3, \\ E_D=U, \qquad E_+ = \frac{U+\sqrt{U^2+16t^2}}{2}. \end{gathered}$$

At $t=1$ and $U=4$,

$$\begin{aligned} E_0/t &=-0.828427124746190, \\ \Delta_{S-T}/t &=0.828427124746190, \\ \langle D\rangle_0 &=0.146446609406726, \end{aligned}$$

where

$$D = \sum_i n_{i\uparrow}n_{i\downarrow}.$$

The last target follows independently from Feynman–Hellmann differentiation,

$$\langle D\rangle_0 = \frac{\partial E_-}{\partial U} = \frac12 \left( 1- \frac{U}{\sqrt{U^2+16t^2}} \right).$$

Required checks

1. Confirm the dimensions $6$ for fixed $N=2$ and $4$ for fixed $N_\uparrow=N_\downarrow=1$.
2. Reproduce the complete analytic spectrum in the full sector.
3. Verify the three triplet states are degenerate at zero energy.
4. Compare direct evaluation of $\langle D\rangle_0$ with $\partial E_-/\partial U$.
5. Repeat after changing the mode ordering and confirm that observables and eigenvalues remain invariant when all fermionic signs are transformed consistently.

What failures diagnose

The dimer is especially sensitive to fermionic sign conventions. Incorrect singlet mixing, a split triplet, or a missing factor of two in the square root usually identifies an inconsistent hopping sign or an incomplete spin sum. Agreement only after taking absolute values of matrix entries is not a pass.

MB-B004: Four-Site Hubbard Chain

The Hubbard Chain dossier supplies the periodic thermodynamic model record and explains why this open finite-chain contract must be interpreted separately.

Contract

Use the same Hamiltonian and site-major mode ordering as MB-B003, now on an open chain of $L=4$ sites. The bonds are $(1,2)$, $(2,3)$, and $(3,4)$. Fix

$$t=1, \qquad U=4, \qquad N_\uparrow=N_\downarrow=2.$$

The symmetry-block dimension is

$$\dim\mathcal H_{2,2} = \binom42\binom42 =36.$$

This benchmark is deliberately larger than the dimer but still small enough for complete dense diagonalization. The trusted ground-state data are

$$\begin{aligned} E_0 &=-1.953145308684556, \\ E_1 &=-1.412898695919094, \\ E_1-E_0 &=0.540246612765462, \\ \langle D\rangle_0 &=0.339585650339361. \end{aligned}$$

Here $D=\sum_i n_{i\uparrow}n_{i\downarrow}$. The energy decomposition provides two additional checks:

$$\begin{aligned} E_{\mathrm{int}} &=U\langle D\rangle_0, \\ &=1.358342601357445, \\ E_{\mathrm{kin}} &=E_0-E_{\mathrm{int}}, \\ &=-3.311487910042001. \end{aligned}$$

Over the complete $36$-state block,

$$\begin{aligned} \operatorname{Tr}H &=144, \\ \frac{1}{36}\operatorname{Tr}H^2 &=25.3333333333333. \end{aligned}$$

Required checks

1. Verify the combinatorial dimension before constructing the Hamiltonian.
2. Check Hermiticity and closure of every hopping transition inside the fixed particle-number block.
3. Reproduce $E_0$, $E_1$, the trace, and the second moment.
4. Compute $\langle D\rangle_0$ directly and verify $E_0=E_{\mathrm{kin}}+U\langle D\rangle_0$.
5. Cross-check the result with two independent representations when practical, such as a generic occupation-bit implementation and factorized up/down bit strings.

The quoted $E_1-E_0$ is the first excitation gap within the fixed $(N_\uparrow,N_\downarrow)=(2,2)$ block. It is not, by itself, the thermodynamic charge gap, spin gap, or unrestricted global gap. Those quantities require specified neighboring sectors and a size sequence.

What failures diagnose

• Dimension or closure failures indicate sector enumeration errors.
• A correct trace but wrong low spectrum often indicates hopping signs or bond orientation.
• A correct energy with wrong double occupancy suggests an observable-basis mismatch.
• Agreement between two solvers using the same matrix does not independently test matrix assembly; an independent representation is stronger evidence.

MB-B005: Two-Site Bose–Hubbard Model

Contract

Use

$$\begin{aligned} H ={}& -J \left( b_1^\dagger b_2 + b_2^\dagger b_1 \right) \\ &+ \frac{U}{2} \sum_{i=1}^{2} n_i(n_i-1). \end{aligned}$$

in the fixed-$N=2$ basis

$$\bigl\{ |2,0\rangle, |1,1\rangle, |0,2\rangle \bigr\}.$$

The matrix is

$$H = \begin{pmatrix} U & -\sqrt2J & 0\\ -\sqrt2J & 0 & -\sqrt2J\\ 0 & -\sqrt2J & U \end{pmatrix},$$

with spectrum

$$\begin{aligned} E_- &= \frac{U-\sqrt{U^2+16J^2}}{2}, \\ E_D &=U, \\ E_+ &= \frac{U+\sqrt{U^2+16J^2}}{2}. \end{aligned}$$

At $J=1$ and $U=2$,

$$\begin{aligned} E_-/J &=1-\sqrt5, \\ E_D/J &=2, \\ E_+/J &=1+\sqrt5. \end{aligned}$$

or numerically

$$\begin{aligned} E_-/J &=-1.236067977499790, \\ E_D/J &=2, \\ E_+/J &=3.236067977499790. \end{aligned}$$

Define the total on-site pair indicator

$$D_B = \frac12\sum_i n_i(n_i-1).$$

Its ground-state expectation is

$$\begin{aligned} \langle D_B\rangle_0 &= \frac12 \left( 1- \frac{U}{\sqrt{U^2+16J^2}} \right) \\ &=0.276393202250021. \end{aligned}$$

at the benchmark point.

Required checks

1. Confirm that the fixed-number basis has dimension $3$.
2. Generate hopping amplitudes from ladder operators rather than hard-coding them.
3. Reproduce the full spectrum and the reflection parity of all three eigenstates.
4. Verify $\langle D_B\rangle_0=\partial E_0/\partial U$.
5. Check the noninteracting limit $U=0$, where the two bosons occupy the bonding orbital and $E_0=-2J$.

What failures diagnose

Off-diagonal entries of magnitude $J$ instead of $\sqrt2J$ reveal missing bosonic enhancement factors. An interaction diagonal of $2U$ on $|2,0\rangle$ indicates that the factor $1/2$ in $n(n-1)$ was omitted. The Bose–Hubbard Dimer dossier owns the exact finite-model interpretation and additional invariant checks; the general sparse construction lives in Occupation-Number Representation.

MB-B006: Ideal Bose Gas

Contract

Use a uniform, spinless, three-dimensional ideal Bose gas in the thermodynamic limit. Define

$$\lambda_T = \sqrt{ \frac{2\pi\hbar^2}{mk_{\mathrm B}T} }, \qquad z=e^{\beta\mu},$$

and, above the condensation point,

$$n\lambda_T^3 = \operatorname{Li}_{3/2}(z), \qquad 0<z<1.$$

The implementation should reproduce

$$\begin{aligned} \zeta(3/2) &=2.612375348685488\ldots, \\ \zeta(5/2) &=1.341487257250917\ldots. \end{aligned}$$

For the normal-state target

$$n\lambda_T^3=1,$$

the physical root is

$$z=0.698614359135065.$$

For a second target below the transition, set $T/T_c=1/2$. The thermodynamic condensate fraction is

$$\begin{aligned} \frac{N_0}{N} &= 1- \left( \frac{T}{T_c} \right)^{3/2} \\ &=0.646446609406726. \end{aligned}$$

Required checks

1. Reproduce the two zeta values independently of the root solver.
2. Solve the normal-state number equation on the physical interval $0<z<1$ and report its residual.
3. Refine quadrature order or arithmetic precision until the quoted digits stabilize.
4. Below $T_c$, separate the ground mode and pin the excited-state fugacity to its thermodynamic limiting value $z=1$.
5. Verify that the excited fraction scales as $(T/T_c)^{3/2}$ for several temperatures below $T_c$.

What failures diagnose

A root $z>1$ is unphysical for the continuum ideal Bose gas and usually reflects unconstrained root finding. Failure below $T_c$ often occurs when the code keeps trying to satisfy the entire density with excited states. In a finite box the chemical potential remains below the ground-state energy; the exact $z=1$ prescription is a thermodynamic-limit contract, not a finite-volume identity.

MB-B007: Ideal Fermi Gas

Contract

Use a uniform three-dimensional gas with spin degeneracy $g=2$ and units

$$\frac{\hbar^2}{2m}=1.$$

Choose

$$n=\frac{1}{3\pi^2},$$

so that $k_F=1$ and $\epsilon_F=1$. With

$$f(\epsilon) = \frac{1}{e^{(\epsilon-\mu)/(k_{\mathrm B}T)}+1},$$

the number and energy densities can be evaluated directly as

$$\begin{aligned} n &= \frac{1}{\pi^2} \int_0^\infty k^2 f(k^2)\,dk, \\ u &= \frac{1}{\pi^2} \int_0^\infty k^4 f(k^2)\,dk. \end{aligned}$$

At zero temperature the exact targets are

$$\begin{aligned} \mu &=\epsilon_F=1, \\ \frac{U}{N} &=\frac35\epsilon_F=0.6, \\ \frac{P}{n\epsilon_F} &=\frac25. \end{aligned}$$

For $\theta=T/T_F\to0$, the Sommerfeld coefficients provide a refinement benchmark:

$$\begin{aligned} \frac{\mu}{\epsilon_F} &= 1- \frac{\pi^2}{12}\theta^2 +O(\theta^4), \\ \frac{U}{N\epsilon_F} &= \frac35 \left[ 1+ \frac{5\pi^2}{12}\theta^2 +O(\theta^4) \right], \\ \frac{C_V}{Nk_{\mathrm B}} &= \frac{\pi^2}{2}\theta +O(\theta^3). \end{aligned}$$

Required checks

1. Reproduce the $T=0$ density, energy per particle, and pressure ratio.
2. At finite $T$, solve the number equation for $\mu$ rather than holding $\mu=\epsilon_F$.
3. Use a decreasing sequence such as $\theta=0.20,0.10,0.05,0.025$ and monitor the scaled corrections.
4. Verify the limits

$$\begin{aligned} \frac{1-\mu/\epsilon_F}{\theta^2} &\longrightarrow \frac{\pi^2}{12}, \\ \frac{ U/(N\epsilon_F)-3/5 }{\theta^2} &\longrightarrow \frac{\pi^2}{4}, \\ \frac{C_V}{Nk_{\mathrm B}\theta} &\longrightarrow \frac{\pi^2}{2}. \end{aligned}$$

5. Record quadrature truncation and root residual separately from the asymptotic $O(\theta^4)$ or $O(\theta^3)$ error.

The low-temperature formulas are convergence laws, not exact finite-$\theta$ targets. Demanding twelve-digit agreement at $\theta=0.1$ would confuse truncation of the Sommerfeld series with numerical error.

What failures diagnose

An incorrect zero-temperature density commonly signals a missing spin degeneracy or phase-space factor. A chemical potential that stays fixed at finite temperature violates the fixed-density contract. Unstable heat capacity estimates may come from differentiating noisy energy data; analytic derivatives, higher precision, or a fitted low-temperature expansion can separate differentiation error from quadrature error.

MB-B008: Mean-Field BCS Gap Equation

Contract

Use a constant density of states within a symmetric energy cutoff $|\xi|\leq\omega_c$. Let

$$\lambda=g\nu(0)=0.25, \qquad \omega_c=1.$$

The nonzero mean-field gap satisfies

$$\frac{1}{\lambda} = \int_0^{\omega_c} \frac{d\xi}{\sqrt{\xi^2+\Delta^2}} \tanh\!\left( \frac{\sqrt{\xi^2+\Delta^2}}{2k_{\mathrm B}T} \right).$$

At $T=0$, the integral is elementary and gives the exact finite-cutoff target

$$\frac{\Delta_0}{\omega_c} = \frac{1}{\sinh(1/\lambda)} = 0.0366435703258656.$$

The weak-coupling approximation is

$$\frac{\Delta_0}{\omega_c} \approx 2e^{-1/\lambda} = 0.0366312777774684,$$

whose relative error at this parameter point is approximately $3.36\times10^{-4}$. A solver should reproduce the exact finite-cutoff result, not be judged against the asymptotic expression at more precision than the approximation warrants.

The transition temperature follows from the linearized equation

$$\frac{1}{\lambda} = \int_0^{\omega_c} \frac{d\xi}{\xi} \tanh\!\left( \frac{\xi}{2k_{\mathrm B}T_c} \right).$$

For the benchmark parameters,

$$\frac{k_{\mathrm B}T_c}{\omega_c} = 0.0207674786897134,$$

and therefore

$$\frac{2\Delta_0}{k_{\mathrm B}T_c} = 3.52893780447368.$$

The slight difference from the asymptotic weak-coupling value $2\pi/e^\gamma\approx3.52775$ is primarily the finite-cutoff correction retained in $\Delta_0$.

Required checks

1. Evaluate the $T=0$ integral and reproduce the exact nonzero root.
2. Report both the gap error and the residual of the integral equation.
3. Bracket the physical root on $\Delta>0$; do not let the trivial normal solution masquerade as convergence below $T_c$.
4. Solve the linearized equation for $T_c$ and compare the finite-cutoff ratio with the weak-coupling limit.
5. Check that the superconducting stationary point has lower mean-field free energy than the normal state for $T<T_c$.
6. If the calculation is performed at fixed density rather than fixed chemical potential, solve and report the number equation as an additional contract.

What failures diagnose

False convergence to $\Delta=0$ usually reflects an unbracketed nonlinear solve or use of the undivided stationarity equation without phase selection. A correct root with a large integral residual suggests cancellation or quadrature error. Agreement with $2e^{-1/\lambda}$ but not $1/\sinh(1/\lambda)$ can simply mean the implementation silently replaced the finite cutoff by the weak-coupling approximation.

MB-B009: Finite-Size Scaling of the Ising Gap

Contract

Return to the open transverse-field Ising chain of MB-B001 at its critical point,

$$h=J>0.$$

The gap between the two lowest states of the complete finite Hilbert space is

$$\Delta_L = 4J \sin\!\left( \frac{\pi}{4L+2} \right).$$

Useful exact values are

$L$	$\Delta_L/J$
2	1.236067977499790
3	0.890083735825258
4	0.694592710667721
5	0.569259353093141
6	0.482146721021292
7	0.418113853070614
8	0.369073437853208

The boundary-shifted scaling variable exposes the known correction structure. With $\ell=L+1/2$,

$$\begin{aligned} \frac{\ell\Delta_L}{\pi J} &= \frac{\sin x}{x}, \\ x &= \frac{\pi}{4\ell}, \end{aligned}$$

so that

$$\frac{\ell\Delta_L}{\pi J} = 1- \frac{\pi^2}{96\ell^2} +O(\ell^{-4}).$$

This single contract tests both spectral extraction and the scaling analysis. The leading critical law is

$$\Delta_L\sim\frac{\pi J}{L},$$

corresponding to dynamical exponent $z=1$, while the exact formula reveals how open-boundary corrections approach that limit.

Required checks

1. Reproduce the exact gaps for every size used in the fit.
2. State whether the gap is global or restricted to a symmetry sector. This contract uses the global gap.
3. Verify that $\Delta_L$ decreases monotonically over the benchmark sequence.
4. Fit at least two lower-size cutoffs $L_{\min}$ and report the drift of the inferred exponent and amplitude.
5. Compare an unshifted fit in $L$ with the boundary-aware variable $\ell=L+1/2$.
6. Test the normalized residual

$$R_L = \frac{\Delta_L}{ 4J\sin[\pi/(4L+2)] }-1$$

before interpreting any extrapolation.

What failures diagnose

An exact-formula residual at small $L$ is a spectrum, boundary, or sector error rather than a scaling uncertainty. A good log–log line with drifting exponent can result from omitted boundary corrections. Using a parity-restricted gap without saying so changes the physical excitation and invalidates comparison with this contract.

Passing the fit only confirms that the analysis can recover a known $z=1$ sequence over the tested window. It does not validate an ansatz for a different universality class.

Coverage Matrix

The suite is most useful when failures are interpreted jointly.

Implementation feature	Primary contract	Independent reinforcement
spin-bit indexing and tensor order	MB-B001	MB-B002
bond enumeration and boundaries	MB-B001	MB-B004, MB-B009
spin normalization and multiplets	MB-B002	symmetry commutators
fermionic anticommutation signs	MB-B003	MB-B004
fixed-particle sector enumeration	MB-B003	MB-B004, MB-B005
bosonic ladder factors	MB-B005	Feynman–Hellmann check
constrained thermodynamic roots	MB-B006	MB-B007, MB-B008
low-temperature asymptotics	MB-B007	zero-temperature limits
nonlinear self-consistency	MB-B008	free-energy ordering
size-window and correction analysis	MB-B009	exact pointwise residuals

A failed row should be repaired at its lowest relevant layer. For example, do not tune a finite-size fit while MB-B001 still reports the wrong open-chain spectrum.

Recommended Execution Ladder

Run and promote evidence in the following order:

1. Structural tests: dimensions, basis uniqueness, sector closure, Hermiticity, and symmetry commutators.
2. Exact finite clusters: complete spectra, degeneracies, traces, and simple observables.
3. Independent identities: Feynman–Hellmann derivatives, energy decompositions, and known limiting cases.
4. Numerical refinement: solver tolerance, quadrature order, arithmetic precision, and fit window.
5. Independent implementations: a second basis representation, solver, or integration route.
6. Scaling claims: only after every finite-size datum passes its pointwise contract.

This ladder distinguishes correctness of the encoded problem from convergence of the numerical method and from interpretation of the physical limit.

Benchmark Record

Every run should preserve a compact record such as the following:

Field	Example
benchmark_id	MB-B004
canonical_target	four-site open Hubbard chain
implementation_version	commit or immutable archive identifier
environment	language, package versions, hardware notes
representation	site-major fermion modes, fixed $(N_\uparrow,N_\downarrow)$
parameters	$L=4$, $t=1$, $U=4$, open boundaries
method	complete dense diagonalization
tolerance	declared eigenvalue, residual, and observable thresholds
observed	values with more digits than the pass threshold requires
refinement	solver or precision sequence
status	passed, warning, failed, or skipped
horizon	no thermodynamic or charge-gap claim

Store raw values at higher precision than displayed plots, but do not imply more physical accuracy than the reference and method support. A screenshot of a passing table is weaker evidence than machine-readable results plus the environment and test logic that produced them.

The planned Reproducible Notebooks index maps each future artifact to the applicable MB-Bxxx contracts and withholds evidential status until clean execution and validation are recorded.

Regression and Promotion Rules

A benchmark result can become a regression target after:

1. its Hamiltonian and conventions have been reviewed against the canonical page;
2. its reference values have an analytic derivation or an independent construction;
3. all applicable structural checks pass;
4. its tolerance has a documented numerical rationale;
5. the environment and implementation version are recorded;
6. the evidence horizon is stated.

Never update a stored reference merely because a new code version disagrees with it. Diagnose the discrepancy first. A legitimate convention change should receive a new contract revision or explicit migration note, while the stable benchmark ID continues to identify an unambiguous problem.

Common Mistakes

Treating a ground energy as sufficient

One scalar can agree accidentally. Include dimensions, moments, degeneracies, residuals, and at least one observable whenever the contract supplies them.

Comparing different sectors

“The gap” is incomplete language. Record particle number, magnetization or parity, boundary condition, and whether the gap is global or sector restricted.

Hiding normalization conventions

Pauli matrices and spin-$1/2$ operators differ by a factor of two. A density of states may be per spin or spin summed. These choices belong in the contract, not in undocumented code defaults.

Choosing tolerance after inspection

A post hoc tolerance converts validation into description. Set it from arithmetic, conditioning, solver behavior, stochastic uncertainty, and intended use before opening the result.

Confusing approximation error with numerical error

The Sommerfeld series and weak-coupling BCS formula are asymptotic approximations. A highly accurate solver should disagree with their truncated forms by the expected higher-order correction.

Sharing one implementation path

Two eigensolvers applied to the same incorrectly assembled matrix are not independent checks of the Hamiltonian. Independence should reach the layer under suspicion.

Overclaiming from a passing suite

The nine contracts do not validate frustrated sign structures, continuum renormalization, two-dimensional thermodynamic extrapolation, real-time stability, or spectral analytic continuation. Add benchmarks targeted to those claims.

Exercises

Exercise 1: The factor-of-four failure

A four-site Heisenberg implementation returns the levels $-8J$, $-4J$, $0$, and $4J$ with the correct degeneracies. Identify the likely error and give a direct one-state test.

Solution

The spectrum is four times the target, so the code probably used Pauli matrices $\boldsymbol\sigma$ in

$$J\sum_i \boldsymbol\sigma_i\mathbin{\cdot}\boldsymbol\sigma_{i+1}$$

instead of spin operators $\mathbf S_i=\boldsymbol\sigma_i/2$. Test the fully polarized state. Each intended bond contributes $J/4$, and the four-site ring must therefore have energy $J$. A result $4J$ confirms the normalization error.

Exercise 2: Double occupancy without eigenvectors

Use the exact Hubbard-dimer ground energy to derive its total double occupancy. Evaluate the result at $U=4t$.

Solution

Feynman–Hellmann gives

$$\left\langle \sum_i n_{i\uparrow}n_{i\downarrow} \right\rangle_0 = \frac{\partial E_-}{\partial U}.$$

Differentiating

$$E_- = \frac{U-\sqrt{U^2+16t^2}}{2}$$

yields

$$\langle D\rangle_0 = \frac12 \left( 1- \frac{U}{\sqrt{U^2+16t^2}} \right).$$

At $U=4t$ this is

$$\begin{aligned} \langle D\rangle_0 &= \frac12 \left( 1-\frac{1}{\sqrt2} \right) \\ &=0.146446609406726\ldots. \end{aligned}$$

Exercise 3: Why the sector gap is not the charge gap

Explain why $E_1-E_0$ in the four-site $(N_\uparrow,N_\downarrow)=(2,2)$ Hubbard block is not a charge gap. Write a finite-system expression that probes particle addition and removal instead.

Solution

Both states in the quoted difference have the same particle numbers, so the excitation does not test the energy cost of changing the total charge. A common finite-system charge-gap estimator at total particle number $N$ is

$$\begin{aligned} \Delta_c(N) ={}& E_0(N+1) + E_0(N-1) \\ &-2E_0(N). \end{aligned}$$

with spin sectors chosen and reported consistently. Other conventions use pair addition and removal to preserve spin balance. In either case, neighboring particle-number sectors are essential.

Exercise 4: Recover the bosonic enhancement

Compute $\langle1,1|b_2^\dagger b_1|2,0\rangle$ and explain why MB-B005 detects a hard-core or spin-style hopping implementation.

Solution

Apply the annihilation operator first:

$$b_1|2,0\rangle = \sqrt2|1,0\rangle.$$

Then

$$b_2^\dagger|1,0\rangle = |1,1\rangle.$$

Therefore

$$\langle1,1| b_2^\dagger b_1 |2,0\rangle = \sqrt2.$$

A hopping routine that moves particles with unit amplitude misses the occupation-dependent ladder factor and returns $-J$ rather than $-\sqrt2J$.

Exercise 5: Boundary-aware Ising scaling

Starting from the exact critical gap, show that the shifted length $\ell=L+1/2$ removes the leading $1/L$ boundary correction from the normalized amplitude.

Solution

Write

$$\Delta_L = 4J\sin\!\left( \frac{\pi}{4\ell} \right), \qquad \ell=L+\frac12.$$

With $x=\pi/(4\ell)$,

$$\frac{\ell\Delta_L}{\pi J} = \frac{\sin x}{x}.$$

Expanding gives

$$\begin{aligned} \frac{\sin x}{x} &= 1- \frac{x^2}{6} +O(x^4) \\ &= 1- \frac{\pi^2}{96\ell^2} +O(\ell^{-4}). \end{aligned}$$

There is no term proportional to $1/\ell$. If one normalizes with $L$ instead, expanding $L/(L+1/2)$ reintroduces a leading $1/L$ correction.

Exercise 6: Design a fair low-temperature test

Why should the ideal-Fermi-gas value at one temperature not be compared directly with the truncated Sommerfeld formula at machine precision? Propose a better acceptance test.

Solution

The displayed Sommerfeld formulas omit terms of order $\theta^4$ in $\mu$ and $U/N$, and order $\theta^3$ in $C_V$. Even exact quadrature should therefore differ from the truncated formula at finite $\theta$.

A better test uses a decreasing temperature sequence, solves the number equation at each point, and examines scaled corrections such as

$$a_\mu(\theta) = \frac{1-\mu(\theta)/\epsilon_F}{\theta^2}.$$

The sequence should approach $\pi^2/12$ with drift consistent with $O(\theta^2)$. Quadrature and root residuals should be much smaller than that expected asymptotic drift and reported separately.

References

1. E. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain,” Annals of Physics 16, 407–466 (1961).
2. P. Pfeuty, “The one-dimensional Ising model with a transverse field,” Annals of Physics 57, 79–90 (1970).
3. H. Bethe, “Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette,” Zeitschrift für Physik 71, 205–226 (1931).
4. J. Hubbard, “Electron correlations in narrow energy bands,” Proceedings of the Royal Society A 276, 238–257 (1963).
5. 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).
6. 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).
7. J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of superconductivity,” Physical Review 108, 1175–1204 (1957).
8. E. Dagotto, “Correlated electrons in high-temperature superconductors,” Reviews of Modern Physics 66, 763–840 (1994).
9. A. W. Sandvik, “Computational studies of quantum spin systems,” AIP Conference Proceedings 1297, 135–338 (2010).
10. J. P. F. LeBlanc et al., “Solutions of the two-dimensional Hubbard model: benchmarks and results from a wide range of numerical algorithms,” Physical Review X 5, 041041 (2015).
11. K. Huang, Statistical Mechanics, 2nd ed., Wiley (1987).
12. R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier (2011).
13. J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press (2007).
14. L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM (1997).

Further Study

• Model Encyclopedia for the model conventions, important limits, observables, and solution-status claims that these contracts test.
• Spin-$1/2$ Chain Model Dossier for the shared spin normalization, bond graph, trace-moment audit, and the relation between MB-B001 and MB-B002.
• Transverse-Field Ising Model dossier for the exact baseline record, parity and boundary audit, and the interpretation of MB-B001 and MB-B009.
• Heisenberg Chain dossier for the exact dimer, $SU(2)$ multiplet audit, Bethe fingerprints, and the interpretation of MB-B002.
• Ideal Bose Gas Model Dossier for the exact baseline record, finite-volume caveats, and the interpretation of MB-B006.
• Ideal Fermi Gas Model Dossier for the exact baseline record, ensemble audit, and the interpretation of MB-B007.
• Computational Many-Body Overview for method selection and evidence design.
• Reproducible Notebooks for the status-aware artifact catalog and promotion gates.
• Exact Diagonalization Preview for basis construction, sparse action, residuals, and finite-system observables.
• Symmetry Sectors in Many-Body Numerics for block closure and full-spectrum reconstruction.
• Finite-Size Scaling in Numerics for fit windows, corrections, crossings, and thermodynamic claims.
• Lanczos Method Preview for testing targeted eigenpairs against the exact-cluster contracts.
• Quantum Monte Carlo Preview for stochastic errors, autocorrelation, and finite-size protocols.
• Dynamical Correlation Functions Numerically for frequency moments and cross-method spectral benchmarks.
• Numerical Benchmarks for general convergence, conservation, and regression guidance.