References and provenance¶
Use any executable's --references option for its relevant literature and
applicability. Ordinary help and no-argument usage omit the bibliography.
Bibliographic metadata and tool selections live in
data/citations.json; the bibliography
below and the C++ registry are generated from it. The provenance notes remain
hand-written. See maintaining citations.
The paper archive provides selected reference PDFs with edition, source, license, and checksum records. Other papers remain linked; being cited here does not grant permission to redistribute a copy.
The bibliography also contains literature for the model catalogue and proposals. Those references do not imply implemented methods and are not selected by executable help unless the corresponding tool actually supports the cited calculation.
The Heisenberg ground-state solver succeeds Ian McCulloch's
misc/heisenberg-energy.cpp in the Matrix Product Toolkit (copyright
2015-2023). The original is maintained there, rather than duplicated here.
The successor retains its GPL-3.0-or-later notice.
The logarithmic equations, quantum numbers, energy convention, and real-root iteration follow karbach-1998, Eqs. (6)-(9). Table I supplies the N=16 regression reference, including its N/4 energy shift.
The sector-state selection extends the consecutive quantum-number construction in that paper (Eq. (16)). For odd-chain one-spinon states and hole labeling, see groha-2017, Sec. 3. The present solver concerns the unperturbed integrable chain, not the decay calculation.
The zero-field thermodynamic XXX and gapless XXZ dispersions follow
the notes caux-xxx-spinons and
caux-xxz-spinons. The thermodynamic XXZ curve has its own
bethe-xxz-dispersion frontend, not a mode of the finite-chain XXZ executable. We use positive spinon
momentum k in [0,pi], reversing the sign of the notes' convention. At finite
odd N our explicit convention is k=pi/2-2piI_h/N, related to lattice momentum
by P=piM+pi/2-k modulo 2pi. The bulk reference energy is J*(1/4-log(2)).
The massive XXZ dispersion follows caux-mossel-perez-castillo-2008, Eqs. (21)-(23). Our continuum bounds minimize/maximize Eq. (32), keeping unfolded constituent momentum and the pi-shifted union explicitly distinct. The endpoint lower branch is m+epsilon(Q), as described in the prose after Eq. (33); the printed omega-plus branch there is inconsistent with the endpoint construction. Native theta constants and modular transformation retain the small gap. The bulk-energy integral/series is the zero-temperature result in bortz-gohmann-2005, with J*Delta/4 restored; near the isotropic point we Poisson-resum the root density before integrating. zauner-stauber-2018 supplies the symmetry-resolved topological-excitation interpretation for MPS comparisons. We implement neither structure factors nor an MPS solver.
Research using these tools should acknowledge the Matrix Product Toolkit and Uni20 where relevant, and cite the methods used in the calculation.
The free-end spin-1 pure biquadratic solver implements the spectral mapping
of barber-batchelor-1989. Its reference XXZ equations
and normalization follow albertini-2000, Eqs. (2)–(10): our
reference Hamiltonian is half Eq. (3), after a staggered rotation of the xy
exchange, so E_b=2*E_ref-7*(N-1)/4. The opposite XXZ end fields are essential;
the physical spin-1 model has free ends. We implement even-chain TL module
minima and real-root label scans in 1<=I<=N-M, including the global singlet,
not complex-root levels or the odd-chain spinon band. Finite regular Bethe
states are assigned ell=N-2M without counting auxiliary descendants again.
The TL module/multiplicity
separation follows aufgebauer-klumper-2010,
Secs. 2.3 and 3, specialized to generic singlet-projector spin-chain
representations. It does not implement periodic twists, full-spectrum Bethe
enumeration, physical-spin decomposition, or thermodynamics. See the
biquadratic/TL guide for the explicit conventions.
The ferromagnetic option reverses the physical Hamiltonian, not the auxiliary
XXZ equations: H=+sum(S.S)^2=(N-1)+sum e_i. The one-defect module has
E-E0=3+2*cos(pi*j/N), j=1,...,N-1. The exact OBC gap above the entire
ground space, 3-2*cos(pi/N), follows from
koma-nachtergaele-1997, Proposition 2 and Eq. (3.30),
using TL equivalence and rescaling their normalized XXZ Hamiltonian by
2*Delta=3. The sign convention and ground-space degeneracies are also
discussed in zhou-2025-biquadratic.
Real-family selection reverses energy ordering before truncation. Q-system
discoveries are reordered in physical energy without strengthening their
completeness claim. See the ferromagnetic excitation guide.
The targeted bound-pair solver uses an empty-sea two-string regularization of bajnok-2020, Eq. (5.12), with both signs of the finite deviation. Its phase label and deviation sign must satisfy N-J odd for positive deviation, even for negative deviation. The thermodynamic two-defect threshold 5/3 follows from nachtergaele-spitzer-starr-2007, Theorem 2.1, after the same factor-three rescaling. These are targeted finite-chain modes, not an ideal-string substitution, full two-defect spectrum or form-factor calculation.
The free-end XXX equations follow the rational limit of Eqs. (11)-(12) and
footnote 2 in mei-2017.
Our rapidity is z=2*lambda; our spin-1/2 Hamiltonian is the paper's XXX
Hamiltonian divided by four and shifted by N/4. This gives the ferromagnetic
reference (N-1)/4. Taking the positive-root logarithmic branch gives the
integer quantum-number convention documented in the
open-chain guide.
For the highest-weight multiplet and string classification relevant to real-root excitation scans, see caux-xxx-states. The scans enumerate only the supported all-real quantum-number window; they do not implement the string sectors described in those references.
The finite-size periodic XXZ sector solver follows the logarithmic equations, energy and momentum in vlijm-2016, Eqs. (1), (3)-(5). We set J=1, h=0 and add NDelta/4 to the paper's shifted Hamiltonian. Our coordinate z=tanh(lambda)/tan(gamma/2), Delta=cos(gamma), rewrites the equations algebraically into the rational form documented in the XXZ guide; it has the smooth XXX limit z=2lambda_XXX. The excitation scan's one-rapidity infinity bound is derived by taking lambda_j to infinity in Eq. (3): theta_1 tends to pi-gamma and theta_2 to pi-2gamma, giving I_infinity=[N-M+1-(N-2M+2)*gamma/pi]/2. We intersect the strict bound with the conventional XXX window and exclude a precision-dependent rounding band at the infinity threshold. This deliberately restricted family is not a complete classification of XXZ solutions. We do not implement the paper's correlation functions.
For periodic XXZ at Delta>1, the trigonometric equations follow
dugave-2015, Eqs. (1.1)-(1.2), at zero twist and field.
Our spin operators are sigma/2, so our J=1 energy is one quarter of the
paper's J=1 Pauli energy. We rewrite real rapidities as
z=tan(lambda)/tanh(eta/2), Delta=cosh(eta), retaining the scattering phase's
winding with atan2. The consecutive sector-minimum labels are checked against
finite-chain energy and momentum ED, including odd lengths. This does not
extend the existing excitation window or implement the paper's form factors.
The internal negative-Delta ground-root engine uses the periodic equations in kozlowski-2017, Eqs. (0.4), (0.7), and the open reflection equations of mei-2017. Kozlowski explicitly assumes even rings; we do not extend that ground-state identification to odd periodic chains. The implementation note derives the rank-subtracted, scaled equations and records the unresolved odd-ring and public-API work. This internal checkpoint does not extend the executable help's scope.
The internal polynomial XXZ equations start from
caux-xxz-coordinate, Eqs. xxz.be and xxz.e,
at zero twist and J=1, restoring the ferromagnetic energy N*Delta/4.
The divided-Horner recurrence, analytic coefficient derivatives, and collision
regularization are our algebraic implementation. This is a tested equation
building block, not a complete odd-ring ground-state or complex-string solver.
The internal polynomial root recovery uses Ehrlich-Aberth iteration as reviewed in bini-robol-2013, equation (3), with Gauss-Seidel updates. Native-real error-free transforms and compensated complex Horner evaluation follow the numerical strategy discussed in cameron-graillat-2022, Sections 2-4. Our fixed-precision implementation and diagnostic stopping rules are not MPSolve or a reproduction of the latter paper's validated error bounds. It does not perform arbitrary-precision escalation or interval certification.
The massive free-end XXZ ground-state module also follows the reflection
equations in mei-2017. Its distinguished boundary root and
finite-size deviation are motivated by grijalva-2019,
Sec. 4.3.2; the module guide derives the implemented
regularization. This is a ground-state solver, not a boundary correlation
function, general string solver, or massive excitation scan. The existing
bethe-xxz-obc frontend selects it for Delta>1 ground states and sector minima.
The free-end XXZ solver uses mei-2017, Eqs. (11)-(12),
with the paper's Hamiltonian divided by four and shifted by N*Delta/4.
The resulting ferromagnetic reference is (N-1)*Delta/4. In the massless
regime, the cosh ratio in Eq. (11) supplies the boundary reflection phase;
in our scaled coordinate it contributes 4*atan(c*z),
c=(1-Delta)/(1+Delta). This term remains nonzero at the XX point, producing
the open-chain standing-wave denominator N+1. Taking one rapidity to infinity
gives I_infinity=N-M+1-(N-2*M+1)*gamma/pi. As in the periodic case, we intersect
this strict bound with the conventional XXX window and exclude a rounding
band at the threshold; this is a restricted positive finite-root family.
See the free-end XXZ guide
for the transformed equations and supported excitation window.
The periodic Hubbard solver uses lieb-wu-2003,
Eqs. (1), (11), and (14)-(18): unshifted Hamiltonian, energy, logarithmic
equations, quantum-number parity, and the consecutive ground-state labels.
We set t=1, use u=U/4, and write the scattering phases as positive 2*atan
terms with the signs shown in the Hubbard guide.
The paper's Eq. (18) supplies the centered labels for even N and odd M,
including our supported doped sectors. At half filling with even M we use
its parity rules and a full charge Brillouin zone with the +pi endpoint,
together with centered spin labels. This extension is checked against small-ring fermionic
exact diagonalization, including momentum; Eq. (18) alone is not a statement
of both parity branches. The reflection reduction, analytic Jacobian, and
damped-Newton continuation are our numerical implementation. We implement
neither the full excited-state string classification nor correlation functions.
The Hubbard tutorial uses osborne-mcculloch-2025, Secs. III D and IV B, to explain how excitation-variance minimization can target an elementary chargon embedded in a multiparticle continuum. This is an MPS comparison reference, not a method implemented by the Bethe energy solvers.
The Hubbard full and partial particle-hole transformations are reviewed in
rylands-2022, Sec. II, Eqs. (4)-(7). We apply them to fixed
particle/spin sectors on even rings and open chains of either length parity.
The unshifted-Hamiltonian energy offsets
are derived from n_down -> 1-n_down; momentum offsets follow from the
complementary occupied momenta under k -> pi-k. Both are independently
tested against fermionic exact diagonalization. See the
sector guide for the explicit conventions and the
distinction between physical observables and auxiliary repulsive roots.
We do not implement that paper's quench dynamics or overlap formulas.
The free-end Hubbard equations follow deguchi-yue-1997, Eqs. (2.5)-(2.8) and (3.1)-(3.2), with all boundary potentials and fields set to zero. The charge boundary factor supplies the length L+1; both direct and reflected spin self-scattering terms are excluded. We use positive roots and consecutive integer labels, as documented in the open Hubbard guide. The one-dimensional open-chain spin ordering of lieb-mattis-1962 underlies selecting the minimum-spin branch in each fixed-Sz sector, without the periodic shell-parity restrictions. The Jacobian, continuation scheme, and symmetry-sector bookkeeping are our implementation, tested against independent fermionic exact diagonalization. We implement neither boundary fields nor the papers' conformal-dimension analysis.
The repulsive periodic Bose gas follows lieb-liniger-1963. The explicit Hamiltonian, logarithmic equations, label parity, energy and momentum were cross-checked against essler-de-klerk-2023, Eqs. (4), (24)-(27). We set hbar^2/(2m)=1 and retain the paper's coupling 2c. Its root-density relation (32)-(33), restricted to a filled symmetric interval, provides an independent test-only bulk benchmark. The complementary weak-coupling equation, scaled residual, Newton implementation, and finite-window scan are our numerical choices; see the model guide. lieb-1963-excitations supplies excitation background, not an implemented thermodynamic dispersion mode. No matrix elements or finite-temperature thermodynamics from the cited papers are implemented.
The periodic SU(3) permutation chain follows
sutherland-1975, with explicit nested equations, labels,
and energy checked against doikou-nepomechie-1998,
Eqs. (2.17)-(2.19) and (2.24)-(2.29). We implement only the all-real,
filled-sea balanced singlet of Sec. 2.3 for L divisible by three. The reference
uses sum(P-1)/2; our H=sum P gives E=2*E_paper+L.
The density-inversion seed, reflection reduction, Jacobian, and damped Newton
solve are our implementation. The SU(3) guide derives the
six-site singlet oracle and the spin-1 ULS energy shift, independently checked
in color/spin space. No excited strings, scattering matrices, boundary fields,
or general-rank SU(n) solver are implied by citing this paper.
The periodic Gaudin–Yang gas follows gaudin-1967 and yang-1967, with the explicit Hamiltonian and nested equations from oelkers-2006, Eqs. (1), (2), (7), (8), (25). We implement the repulsive even-N, odd-minority filled-sea branch, plus exact free gases, using the paper's kinetic units and coupling 2c. Its weak/strong limits (12), (15)-(16) supplement the independent two-body jump condition and dilute-Hubbard discretization tests. Reflection reduction, scaled spin coordinates, continuation, complementary phases, and residual normalization are our numerical choices; see the Gaudin–Yang guide. Other periodic shell branches, attraction, hard walls, excited states, and thermodynamic integral equations are not implemented by this tool.
Bibliography¶
Generated from data/citations.json; edit the registry, not this section.
barber-batchelor-1989¶
Michael N. Barber and Murray T. Batchelor. Spectrum of the biquadratic spin-1 antiferromagnetic chain. Phys. Rev. B 40, 4621-4626 (1989).
DOI.
Relevant tool modes:
bethe-biquadratic-obc: Original free-end spin-1 biquadratic/TL spectral correspondence. Even-chain ground states, TL module minima and restricted real-root excitations of H=-sum (S.S)^2, with representation multiplicities.
albertini-2000¶
Giuseppe Albertini. Is the purely biquadratic spin 1 chain always massive?. arXiv:cond-mat/0012439 (2000).
Relevant tool modes:
bethe-biquadratic-obc: Hamiltonians and energy shift (2)-(5), real-root Bethe equations (6)-(10), and integer-label window 1\<=I\<=N-M. Our spin-half reference is half of (3), after a staggered rotation. The odd-chain spinon band is not implemented.
aufgebauer-klumper-2010¶
Britta Aufgebauer and Andreas Klümper. Quantum spin chains of Temperley-Lieb type: periodic boundary conditions, spectral multiplicities and finite temperature. J. Stat. Mech. 2010, P05018 (2010).
Relevant tool modes:
bethe-biquadratic-obc: TL loop weight and quantum-group XXZ end fields, Sec. 2.3; open spin-chain module multiplicities, Sec. 3, especially (48)-(55). Physical-spin decomposition is available for multiplicity spaces whose dimension fits uint64. Periodic twists and thermodynamics are not implemented.
essler-korepin-1999¶
Fabian H. L. Essler and Vladimir E. Korepin. Form factors in the half-filled Hubbard model. Phys. Rev. B 59, 1734-1738 (1999).
Relevant tool modes:
bethe-hubbard-continuum: Half-filled spinon line Eq. (4), two-spinon continuum edges Eq. (18) and following paragraph; our U is four times the paper's U. No form factors or spectral weights.bethe-hubbard-dispersion: Half-filled spinon and holon dispersions, Eqs. (4)-(5), with our U equal to four times the paper's U; no form factors or spectral weights are implemented.
melzer-1995¶
Ezer Melzer. On the scaling limit of the 1D Hubbard model at half filling. Nucl. Phys. B 443, 553-564 (1995).
Relevant tool modes:
bethe-hubbard-dispersion: Exact massive-branch K-Bessel series, resummed here into nonoscillatory positive integrals to retain the weak-coupling charge gap; hopping and interaction conventions are converted as documented.
essler-2010¶
Fabian H. L. Essler. Threshold singularities in the one-dimensional Hubbard model. Phys. Rev. B 81, 205120 (2010).
Relevant tool modes:
bethe-hubbard-dispersion: Doped zero-field density and dressed-energy equations (103)-(106), with dressed momenta from (28)-(29); our mu_unshifted equals mu+2u in (105), u=U/4. Elementary lines only, not threshold exponents or continuum minimization.
luo-pu-guan-2024¶
Jia-Jia Luo, Han Pu, and Xi-Wen Guan. Exact results of the one-dimensional repulsive Hubbard model. Rep. Prog. Phys. 87, 117601 (2024).
Relevant tool modes:
bethe-hubbard-dispersion: Real charge particle/hole and spinon excitation interpretation, Sec. II; doped charge-particle is a real-root addition, not the gapped half-filled antiholon or a k-Lambda string.
karbach-1998¶
Michael Karbach, Kun Hu, and Gerhard Müller. Introduction to the Bethe ansatz II. Computers in Physics 12, 565 (1998).
Relevant tool modes:
bethe-xxx-pbc: Periodic XXX equations, energy normalization, and sector quantum numbers; Eqs. (6)-(9), (16), Table I.bethe-xxx-structure-factor: Periodic XXX states and real-root two-spinon triplet window; our z=2*lambda_ABA.bethe-xxz-pbc: XXX limit at Delta=1 and the conventional real-root quantum-number window.bethe-tj-pbc: No-hole reduction to the periodic XXX sector solver: H_tJ=2*H_XXX-L/2. Fermionic translation adds the filled-reference phase (-1)^(L-1).
groha-2017¶
Stefan Groha and Fabian H. L. Essler. Spinon decay in the spin-1/2 Heisenberg chain with weak next nearest neighbour exchange. J. Phys. A 50, 334002 (2017).
Relevant tool modes:
bethe-xxx-pbc: Odd-chain one-spinon states and hole labels, Sec. 3; only the unperturbed integrable chain is used.
caux-xxx-spinons¶
Jean-Sébastien Caux. The Bethe Ansatz: XXX spinons. Online notes.
Relevant tool modes:
bethe-xxx-pbc: Thermodynamic XXX spinon dispersion and the bulk reference for --spinons.
caux-xxz-spinons¶
Jean-Sébastien Caux. The Bethe Ansatz: XXZ spinons. Online notes.
Relevant tool modes:
bethe-xxz-dispersion: Zero-field gapless spinon dispersion, -1\<Delta\<=1; our positive spinon momentum reverses the notes' sign convention.
mei-2017¶
Zhongtao Mei and C. J. Bolech. Derivation of matrix product states for the Heisenberg spin chain with open boundary conditions. Phys. Rev. E 95, 032127 (2017).
Relevant tool modes:
bethe-xxx-obc: Free-end XXX equations: rational limit of Eqs. (11)-(12) and footnote 2; our spin-1/2 normalization differs.bethe-xxz-obc: Free-end XXZ equations and boundary reflection phase, Eqs. (11)-(12); our Hamiltonian is divided by four and shifted. Negative-Delta ground states use an algebraically rank-subtracted, rescaled form of these equations.bethe-experimental-xxx-boundary-dynamics: Positive-real-root free-end equations; z=2*lambda_ABA.bethe-experimental-xxx-boundary-magnetization: Positive-real-root free-end XXX equations in z=2*lambda_ABA.
caux-xxx-states¶
Jean-Sébastien Caux. The Bethe Ansatz: real rapidities, SU(2) descendants, and two-string states. Online notes.
Real rapidities, SU(2) descendants, Two-string states.
Relevant tool modes:
bethe-xxx-pbc: Highest-weight and string-classification background; strings and infinite-root descendants are not implemented.bethe-xxx-obc: SU(2) and string-classification background, not the boundary equations; scans include only the supported finite-real family.bethe-xxz-pbc: Background for the XXX window used to restrict the XXZ scan; not a complete XXZ state classification.
vlijm-2016¶
R. Vlijm, I. S. Eliëns, and J.-S. Caux. Correlations of zero-entropy critical states in the XXZ model: integrability and Luttinger theory far from the ground state. SciPost Phys. 1, 008 (2016).
Relevant tool modes:
bethe-xxz-pbc: Massless periodic XXZ equations, energy, and momentum, Eqs. (1), (3)-(5); our energy includes the N*Delta/4 shift.
bajnok-2020¶
Zoltán Bajnok, Etienne Granet, Jesper Lykke Jacobsen, and Rafael I. Nepomechie. On Generalized Q-systems. JHEP 03 (2020) 177 (2020).
Relevant tool modes:
bethe-biquadratic-obc: Open quantum-group-invariant XXZ Q-system, Sec. 5: Wronskian (5.17), Bethe equations (5.12), and admissibility conditions. Used for Q-system module searches (validated through N=8; larger sizes experimental), the regularized AF two-string singlet, and empty-sea two-/three-string ferro branches on odd/even long chains. Original equations also underlie odd/even selected real roots and our restricted high-label scattering windows. Two-strings retain signed real deviations; three-strings retain complex deviations. Neither numerical completeness nor long-chain energy ordering is rigorously certified.
popkov-2021¶
Vladislav Popkov, Xin Zhang, and Andreas Klümper. Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains. Phys. Rev. B 104, L081410 (2021).
kozlowski-2017¶
Karol K. Kozlowski. On condensation properties of Bethe roots associated with the XXZ chain. arXiv:1508.05741v2 (2017).
Relevant tool modes:
bethe-xxz-pbc: Even-ring negative-anisotropy sector ground states: Eqs. (0.4), (0.7), with Pauli exchange divided by four. We use rank-subtracted equations scaled near Delta=-1; the reference explicitly assumes even length and does not justify odd-ring ground-state selection.
lieb-wu-2003¶
Elliott H. Lieb and F. Y. Wu. The one-dimensional Hubbard model: A reminiscence. Physica A 321, 1-27 (2003).
Relevant tool modes:
bethe-hubbard-pbc: Lieb-Wu ground-state equations and quantum-number parity, Eqs. (1), (11), (14)-(18); t=1 and unshifted U*n_up*n_down.
rylands-2022¶
Colin Rylands, Bruno Bertini, and Pasquale Calabrese. Integrable quenches in the Hubbard model. J. Stat. Mech. 2022, 103103 (2022).
Relevant tool modes:
bethe-hubbard-pbc: Full particle-hole and partial particle-hole (Shiba) transformations, Sec. II, Eqs. (4)-(7); used for sector mappings, not quench dynamics.bethe-hubbard-obc: Bipartite particle-hole and Shiba transformations, Sec. II; used for physical-to-auxiliary sector mappings, not quench dynamics.
deguchi-yue-1997¶
Tetsuo Deguchi and Ruihong Yue. Exact solutions of 1-D Hubbard model with open boundary conditions and the conformal dimensions under boundary magnetic fields. arXiv:cond-mat/9704138 (1997).
Relevant tool modes:
bethe-hubbard-obc: Free-end Hubbard energy and nested reflection equations, Eqs. (2.5)-(2.8), (3.1)-(3.2), with all boundary potentials zero; t=1 and unshifted interaction.
lieb-mattis-1962¶
Elliott Lieb and Daniel Mattis. Theory of Ferromagnetism and the Ordering of Electronic Energy Levels. Phys. Rev. 125, 164-172 (1962).
DOI.
Relevant tool modes:
bethe-hubbard-obc: Spin ordering for the nearest-neighbour open chain: the sector minimum has S=|Sz|. No periodic-shell spin-branch selection is required.
lieb-liniger-1963¶
Elliott H. Lieb and Werner Liniger. Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State. Phys. Rev. 130, 1605 (1963).
DOI.
Relevant tool modes:
bethe-bose-fermi-pbc: Pure-boson ground-state reduction on a ring.bethe-lieb-liniger-dispersion: Repulsive zero-temperature root-density equation and bulk ground-state energy, in units hbar=2m=1.bethe-lieb-liniger-pbc: Repulsive Bose gas on a ring, H=-sum d_j^2+2c sum delta, and its ground state; c>0.
lieb-1963-excitations¶
Elliott H. Lieb. Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum. Phys. Rev. 130, 1616 (1963).
DOI.
Relevant tool modes:
bethe-lieb-liniger-dispersion: Thermodynamic type-I particle and type-II hole excitation branches; energies are fixed-N gaps relative to the ground state. No spectral weights or finite-temperature TBA.bethe-lieb-liniger-pbc: Excited-state background; we solve finite-volume states in explicit label windows, not a thermodynamic dispersion calculation.
gaudin-1971¶
M. Gaudin. Boundary Energy of a Bose Gas in One Dimension. Phys. Rev. A 4, 386 (1971).
DOI.
Relevant tool modes:
bethe-lieb-liniger-obc: Repulsive Bose gas with Dirichlet walls, positive real roots and reflected scattering. We implement finite-volume states and bounded label windows, not general boundary potentials.
gaudin-1967¶
M. Gaudin. Un système à une dimension de fermions en interaction. Phys. Lett. A 24, 55-56 (1967).
DOI.
Relevant tool modes:
bethe-gaudin-yang-pbc: Original spin-1/2 continuum fermion solution; we implement repulsive periodic ground states in selected sectors, not attraction.
yang-1967¶
C. N. Yang. Some Exact Results for the Many-Body Problem in One Dimension with Repulsive Delta-Function Interaction. Phys. Rev. Lett. 19, 1312 (1967).
DOI.
Relevant tool modes:
bethe-gaudin-yang-pbc: Original multicomponent delta-gas solution; our implementation has two spin components only.
yang-1968¶
C. N. Yang. S Matrix for the One-Dimensional N-Body Problem with Repulsive or Attractive delta-Function Interaction. Phys. Rev. 168, 1920 (1968).
DOI.
sutherland-1975¶
Bill Sutherland. Model for a multicomponent quantum system. Phys. Rev. B 12, 3795 (1975).
DOI.
Relevant tool modes:
bethe-su3-pbc: Original multicomponent permutation-chain solution; the implementation selects only the fundamental SU(3) periodic balanced ground state.
essler-korepin-1992¶
Fabian H. L. Essler and Vladimir E. Korepin. Higher conservation laws and algebraic Bethe Ansätze for the supersymmetric t-J model. Phys. Rev. B 46, 9147 (1992).
Relevant tool modes:
bethe-tj-pbc: Projected t-J Hamiltonian (1.3)-(1.5), Sutherland BFF equations (3.73), and energy (3.75). We remove the shift 2*N_e-L: E=2*N_h-sum 1/(lambda^2+1/4). Doped mixed-spin coverage is restricted to odd N_up and N_down; J=2t=2.
babujian-1982¶
H. M. Babujian. Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spins S. Phys. Lett. A 90, 479-482 (1982).
DOI.
Relevant tool modes:
bethe-tb-pbc: Original integrable higher-spin chain family; only the periodic even-length spin-1 singlet ground state is implemented.
vlijm-caux-2014¶
Rogier Vlijm and Jean-Sébastien Caux. Computation of dynamical correlation functions of the spin-1 Babujan-Takhtajan chain. J. Stat. Mech. 2014, P05009 (2014).
Relevant tool modes:
bethe-tb-dispersion: Hamiltonian (1.2), energy (1.4), and spinon construction in Sec. 2 fix the normalization and allowed spin sectors. Our H=J sum[S.S-(S.S)^2] has J_paper=4J and spinon energy 2*pi*J*sin(p). Two-/four-spinon envelopes are thermodynamic kinematics, not finite-size ideal-string levels or dynamical weights.bethe-tb-pbc: Spin-1 Hamiltonian and complex Bethe equations (1.2)-(1.4), filled two-string sea, and finite-deviation equations (3.8)-(3.11). Our bilinear coefficient is 1, i.e. J=4 in this paper. Equation (3.18) supplies only an initial guess; finite-size deviations are solved, not dropped. No dynamical correlations or excitations are implemented.
dukelsky-2004¶
J. Dukelsky, S. Pittel, and G. Sierra. Colloquium: Exactly solvable Richardson-Gaudin models for many-body quantum systems. Rev. Mod. Phys. 76, 643-662 (2004).
Relevant tool modes:
bethe-richardson: Reduced BCS pairing model and blocked-level sectors. We support attractive uniform pairing with distinct doubly degenerate single-particle levels, not arbitrary degeneracies or the full Gaudin family.
faribault-2011¶
Alexandre Faribault, Omar El Araby, Christoph Sträter, and Vladimir Gritsev. Gaudin models solver based on the correspondence between Bethe ansatz and ordinary differential equations. Phys. Rev. B 83, 235124 (2011).
Relevant tool modes:
bethe-richardson: Eigenvalue variables (7), quadratic equations in Sec. II, and Richardson equations (32). We use e_i=2*epsilon_i and y_i=g*Lambda(e_i), continue the filled lowest levels from g=0, and include diagonal pair scattering. Energy is sum e_i*y_i-g*M*(L-M+1), plus blocked single-particle energies. No pair-root reconstruction or form factors are implemented.bethe-central-spin: Quadratic eigenvalue-variable method and central-spin realization, Secs. II and IV.3. We compactify the inverse-field continuation to reach B=0; the seed minimizes the central-spin Hamiltonian, not the Richardson energy.
faribault-schuricht-2013¶
Alexandre Faribault and Dirk Schuricht. Spin decoherence due to a randomly fluctuating spin bath. Phys. Rev. B 88, 085323 (2013).
Relevant tool modes:
bethe-central-spin: Spin-half central-spin Hamiltonian and all-down-reference Bethe equations, Eqs. (2)-(6), with no bath field: H=B*S0^z+sum A_j*S0.Sj. Only fixed-magnetization ground energies for distinct nonzero couplings are implemented; no dynamics or form factors.
sutherland-1968¶
Bill Sutherland. Further Results for the Many-Body Problem in One Dimension. Phys. Rev. Lett. 20, 98-100 (1968).
DOI.
Relevant tool modes:
bethe-sun-fermions-pbc: Original multicomponent delta-gas solution. We implement equal-mass repulsive fermions on a periodic ring, not arbitrary mixtures or statistics.
lee-2011¶
J. Y. Lee, X. W. Guan, and M. T. Batchelor. Yang-Yang method for the thermodynamics of one-dimensional multi-component interacting fermions. J. Phys. A: Math. Theor. 44, 165002 (2011).
Relevant tool modes:
bethe-sun-fermions-pbc: Nested finite-size Bethe equations (2)-(3), with c'=c/2, and logarithmic signs from (13)-(19). Our Hamiltonian is -sum d_j^2+2c sum delta with c>=0 and no Zeeman term. Only the centered all-odd occupied-population ground branch is implemented; no strings, TBA or attraction.
imambekov-demler-2006¶
Adilet Imambekov and Eugene Demler. Exactly solvable case of a one-dimensional Bose-Fermi mixture. Phys. Rev. A 73, 021602(R) (2006).
Relevant tool modes:
bethe-bose-fermi-pbc: Original equal-coupling integrable mixture and ground-state study.
imambekov-demler-2006-applications¶
Adilet Imambekov and Eugene Demler. Applications of exact solution for strongly interacting one-dimensional Bose-Fermi mixture: Low-temperature correlation functions, density profiles, and collective modes. Ann. Phys. 321, 2390 (2006).
Relevant tool modes:
bethe-bose-fermi-pbc: Equal-mass, equal-repulsion Bose-Fermi Hamiltonian (3) and periodic nested equations (28)-(34). Odd-fermion mixed ground shells; no auxiliary-auxiliary scattering. No correlation functions or trapped-gas approximation.
wang-1999¶
Yupeng Wang. Exact solution of a spin-ladder model. Phys. Rev. B 60, 9236 (1999).
Relevant tool modes:
bethe-ladder-pbc: Permutation-form ladder Hamiltonian (2), rung basis (3), conserved color populations and chemical-potential form (4), and three nested rational equations (5). We use spin-1/2 operators, leg coefficient 1, four-spin coefficient 4, and J_r=2*J in (2)/(4), restoring E=E_perm-L/4+J_r*(L/4-N_s)-h*(N_+-N_-). Longitudinal field enters through the conserved populations; periodic ground states and singlet-count or fixed-magnetization sector minima only.
hakobyan-2004¶
Tigran Hakobyan. The ordering of energy levels for SU(n) symmetric antiferromagnetic chains. Nucl. Phys. B 699, 575-594 (2004).
Relevant tool modes:
bethe-ladder-pbc: SU(n) multiplets, Young diagrams and weight spaces. The periodic energy-ordering theorem has a row-parity restriction; we do not assume each population's own highest weight is lowest. Compatible dominant multiplets and displaced packed real seas are compared, not the full excited-state spectrum.
bogoliubov-1997¶
N. M. Bogoliubov, A. G. Izergin, and N. A. Kitanine. Correlation functions for a strongly correlated boson system. arXiv:solv-int/9710002 (1997).
Relevant tool modes:
bethe-q-boson-pbc: Deformed Fock algebra (1.2)-(1.8), free/phase limits, and exact phase-model momenta (3.4). Our Hamiltonian is twice (1.1), with the +2N shift included.
zhang-klumper-popkov-2024¶
Xin Zhang, Andreas Klümper, and Vladislav Popkov. Pedestrian's way to Baxter's Bethe ansatz for the periodic XYZ chain. Phys. Rev. B 109, 115411 (2024).
Relevant tool modes:
bethe-xyz-dispersion: Theta-ratio coupling convention, Eq. (2), divided by four for S=sigma/2; same Hamiltonian as bethe-xyz-pbc, not that frontend's finite-size root solver.bethe-xyz-pbc: Rectangular XYZ coupling convention (2), regular Bethe equations (46)-(48), and energy normalized to S=sigma/2. Even periodic symmetric ground branch only; no singular-pair or excited-spectrum completeness claim.
baxter-1973¶
R. J. Baxter. Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. III. Eigenvectors of the transfer matrix and Hamiltonian. Ann. Phys. 76, 48-71 (1973).
DOI.
Relevant tool modes:
bethe-xyz-pbc: Eight-vertex/XYZ Bethe-ansatz foundation; this frontend implements only the even-chain regular ground branch.
haldane-1988¶
F. D. M. Haldane. Exact Jastrow-Gutzwiller resonating-valence-bond ground state of the spin-1/2 antiferromagnetic Heisenberg chain with 1/r^2 exchange. Phys. Rev. Lett. 60, 635 (1988).
DOI.
Relevant tool modes:
bethe-haldane-shastry-pbc: Original periodic spin-1/2 inverse-square chain; we use H=(pi/N)^2 sum_{i\<j} S_i.S_j/sin^2(pi*(i-j)/N), J=1.
shastry-1988¶
B. Sriram Shastry. Exact solution of an S=1/2 Heisenberg antiferromagnetic chain with long-ranged interactions. Phys. Rev. Lett. 60, 639 (1988).
DOI.
Relevant tool modes:
bethe-haldane-shastry-pbc: Independent exact solution of the long-range antiferromagnet; finite even and odd rings, not nearest-neighbor XXX.
jiang-lamers-miao-2026¶
Yunfeng Jiang, Jules Lamers, and Yuan Miao. Norms, overlaps and Yangian descendants for the Haldane-Shastry spin chain. arXiv:2606.20168v2 (2026).
Relevant tool modes:
bethe-haldane-shastry-pbc: Motif spectral rules and Yangian multiplicities, Secs. 2.2-2.3. Convert their H=sum(1-P)/(4*sin^2) by H_ours=E_ferro-2*(pi/N)^2*H_theirs. No wavefunctions, norms or overlaps are implemented.
sutherland-1971¶
Bill Sutherland. Exact Results for a Quantum Many-Body Problem in One Dimension. Phys. Rev. A 4, 2019 (1971).
DOI.
Relevant tool modes:
bethe-sutherland-pbc: Original trigonometric inverse-square gas. We select periodic scalar bosons with collision behavior |x_i-x_j|^lambda, lambda>=0; the coupling coefficient alone does not fix this domain.
gurappa-panigrahi-1999¶
N. Gurappa and Prasanta K. Panigrahi. Equivalence of the Sutherland Model to Free Particles on a Circle. arXiv:hep-th/9908127v2 (1999).
Relevant tool modes:
bethe-sutherland-pbc: Hamiltonian (18), ground energy (19), and the partition spectrum after (23), with beta=lambda and hbar=2m=1. Ascending integer labels include common boosts. Exact energies and momenta only; finite label windows do not claim global spectral completeness.
andrei-1980¶
N. Andrei. Diagonalization of the Kondo Hamiltonian. Phys. Rev. Lett. 45, 379 (1980).
DOI.
Relevant tool modes:
bethe-kondo-response: Exact solution of the single-channel Kondo problem.
hegedus-2026¶
Arpad Hegedus. NLIE formulations for the generalized Gibbs ensemble in the sine-Gordon model. Nucl. Phys. B 1025, 117385 (2026).
Relevant tool modes:
bethe-sine-gordon-vacuum: Integer-coupling D-type TBA used for independent p=2 vacuum validation, equations (2.6)-(2.8).
rutkevich-2020¶
S. B. Rutkevich. On the ground-state energy of the finite sine-Gordon ring. J. Stat. Mech. 2020, 103101 (2020).
Relevant tool modes:
bethe-sine-gordon-bethe-yang: The existing vacuum Fourier kernel fixes chi'=2*pi*G in the same physical p and soliton-mass convention. Vacuum finite-volume energies are separate from this asymptotic excitation calculation.bethe-sine-gordon-excited: Separately converged untwisted vacuum NLIE supplies the vacuum-relative gap, using the same physical soliton mass and bulk subtraction.bethe-sine-gordon-vacuum: Bulk-subtracted finite-ring vacuum scaling function and shifted-contour NLIE; soliton-mass convention.
destri-de-vega-1992¶
C. Destri and H. J. de Vega. New thermodynamic Bethe ansatz equations without strings. Phys. Rev. Lett. 69, 2313-2317 (1992).
DOI.
Relevant tool modes:
bethe-sine-gordon-vacuum: Nonlinear integral equation approach to finite-volume sine-Gordon energies.
gwa-spohn-1992¶
Leh-Hun Gwa and Herbert Spohn. Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. Phys. Rev. A 46, 844 (1992).
DOI.
Relevant tool modes:
bethe-asep-pbc: Original periodic asymmetric-exclusion relaxation-gap analysis.bethe-tasep-pbc: Original periodic asymmetric-exclusion relaxation-gap analysis. This frontend restricts hopping to the totally asymmetric case.
yang-yang-1966¶
C. N. Yang and C. P. Yang. One-Dimensional Chain of Anisotropic Spin-Spin Interactions. I. Proof of Bethe's Hypothesis for Ground State in a Finite System. Phys. Rev. 150, 321 (1966).
DOI.
shastry-sutherland-1990¶
B. Sriram Shastry and Bill Sutherland. Twisted boundary conditions and effective mass in Heisenberg-Ising and Hubbard rings. Phys. Rev. Lett. 65, 243 (1990).
DOI.
de-vega-gonzalez-ruiz-1994¶
H. J. de Vega and A. González-Ruiz. Boundary K-matrices for the XYZ, XXZ and XXX spin chains. J. Phys. A: Math. Gen. 27, 6129-6138 (1994).
essler-de-klerk-2023¶
F. H. L. Essler and A. J. J. M. de Klerk. Statistics of matrix elements of local operators in integrable models. arXiv:2307.12410v1 (2023).
Relevant tool modes:
bethe-lieb-liniger-pbc: Explicit normalization and finite-ring equations, Eqs. (4), (24)-(27); root-density equation (32)-(33) used for validation, not a thermodynamics API. No matrix elements are implemented.
yang-yang-1969¶
C. N. Yang and C. P. Yang. Thermodynamics of a One-Dimensional System of Bosons with Repulsive Delta-Function Interaction. J. Math. Phys. 10, 1115 (1969).
DOI.
Relevant tool modes:
bethe-lieb-liniger-thermal: Finite-temperature thermodynamics of repulsive continuum bosons, at fixed chemical potential or by inversion at fixed density; hbar=2m=k_B=1.
voros-penc-2021¶
Dániel Vörös and Karlo Penc. The dynamical structure factor of the SU(3) Heisenberg chain: The variational Monte Carlo approach. arXiv:2107.09588 (2021).
Relevant tool modes:
bethe-su3-dispersion: Eq. (64) fixes the two elementary dispersions; Eq. (14) gives the factor-two exchange conversion. Use Eq. (64)'s momentum ranges, not the interchanged ranges in the prose below Eq. (65). Two- and four-soliton kinematic envelopes are derived from these bands; no spectral intensities are calculated.
doikou-nepomechie-1998¶
Anastasia Doikou and Rafael I. Nepomechie. Bulk and Boundary S Matrices for the SU(N) Chain. Nucl. Phys. B 521, 547-572 (1998).
Relevant tool modes:
bethe-su3-dispersion: Sec. 2.3 fixes the elementary hole representations, triality constraint and additive thermodynamic energies. H=J sum P has twice the excitation energies of their Hamiltonian at J=1. No finite-size excited-state enumeration or scattering matrices are implemented here.bethe-su3-pbc: Nested equations and energy, Eqs. (2.17)-(2.19), logarithmic labels (2.24)-(2.29), and the filled-sea singlet in Sec. 2.3. Our H=sum P gives E=2*E_paper+L. No strings, S matrices or boundary fields are implemented.
oelkers-2006¶
N. Oelkers, M. T. Batchelor, M. Bortz, and X.-W. Guan. Bethe Ansatz study of one-dimensional Bose and Fermi gases with periodic and hard wall boundary conditions. J. Phys. A 39, 1073-1098 (2006).
Relevant tool modes:
bethe-gaudin-yang-pbc: Hamiltonian and periodic fermion equations (1), (2), (7), (8), (25); even N, odd minority population as in Sec. 5. Weak and strong limits (12), (15)-(16) provide checks. Hard walls, attraction and excitations are not implemented.
grijalva-2019¶
Sebastian Grijalva, Jacopo De Nardis, and Veronique Terras. Open XXZ chain and boundary modes at zero temperature. SciPost Phys. 7, 023 (2019).
Relevant tool modes:
bethe-xxz-obc: Massive open XXZ boundary root and finite-size deviation, Sec. 4.3.2, specialized to zero boundary fields; used for Delta>1 ground states, not boundary correlations or excitation scans.bethe-experimental-xxx-boundary-dynamics: Boundary scalar-product and local endpoint form-factor construction in Sec. 3, specialized to free-end XXX and spin operators; no Pauli or periodic Fourier normalization.bethe-experimental-xxx-boundary-magnetization: Section 3 boundary scalar products and local operator reconstruction. Our rational, spin-normalized specialization retains the directed product and raising-gauge signs for same-sector endpoint Sz amplitudes.
dugave-2015¶
M. Dugave, F. Göhmann, K. K. Kozlowski, and J. Suzuki. On form-factor expansions for the XXZ chain in the massive regime. J. Stat. Mech. (2015) P05037 (2015).
Relevant tool modes:
bethe-xxz-pbc: Massive periodic XXZ Hamiltonian and trigonometric Bethe equations, Eqs. (1.1)-(1.2), with zero twist and field; Pauli exchange is divided by four. Used for Delta>1 sector ground states, not form factors, strings or excitations.
caux-xxz-coordinate¶
Jean-Sébastien Caux. The Bethe Ansatz: coordinate wavefunctions and periodic XXZ equations. Online notes.
General M, Two-magnon scattering.
korepin-2009¶
Vladimir E. Korepin. Norm of Bethe Wave Function as a Determinant. arXiv:0911.1881 (historical account of the 1982 norm formula) (2009).
arXiv, HTML, Original 1982 paper.
bini-robol-2013¶
Dario A. Bini and Leonardo Robol. Solving secular and polynomial equations: a multiprecision algorithm. Author manuscript, May 10, 2013 (2013).
cameron-graillat-2022¶
Thomas R. Cameron and Stef Graillat. On a compensated Ehrlich-Aberth method for the accurate computation of all polynomial roots. Electronic Transactions on Numerical Analysis 55, 401-423 (2022).
koma-nachtergaele-1997¶
Tohru Koma and Bruno Nachtergaele. The spectral gap of the ferromagnetic XXZ chain. Lett. Math. Phys. 40, 1-16 (1997).
Relevant tool modes:
bethe-biquadratic-obc: Ferromagnetic OBC gap and one-defect band: Proposition 2 and Eq. (3.30), transferred through TL equivalence at Delta=3/2 and multiplied by 2*Delta=3. Gap above the entire ground space is 3-2*cos(pi/N), not a zero-mode splitting.
zhou-2025-biquadratic¶
Huan-Qiang Zhou, Qian-Qian Shi, Ian P. McCulloch, and Murray T. Batchelor. Goldstone modes and the golden spiral in the ferromagnetic spin-1 biquadratic model. J. Phys. A: Math. Theor. 58, 39LT01 (2025).
Relevant tool modes:
bethe-biquadratic-obc: Ferromagnetic convention H=+sum (S.S)^2, Eq. (3), and Fibonacci ground-space multiplicities under free ends. Ground-space zero modes are distinct from the positive-energy TL defect band; entanglement and ground-state wavefunctions are not implemented.
nachtergaele-spitzer-starr-2007¶
Bruno Nachtergaele, Wolfgang Spitzer, and Shannon Starr. Droplet Excitations for the Spin-1/2 XXZ Chain with Kink Boundary Conditions. Ann. Henri Poincare 8, 165-201 (2007).
Relevant tool modes:
bethe-biquadratic-obc: Droplet interpretation and thermodynamic module-edge limit, Theorem 2.1; rescaling by 2*Delta=3 gives thresholds 5/3 for two defects and 2 for three. Finite-chain pair/triple modes solve the original Bethe equations with finite deviations, not a bulk/ideal-string substitution. Physical-spin decomposition is available within uint64 dimension limits; form factors are not implemented.
reichert-2019¶
Benjamin Reichert, Grigori E. Astrakharchik, Aleksandra Petkovic, and Zoran Ristivojevic. Exact Results for the Boundary Energy of One-Dimensional Bosons. Phys. Rev. Lett. 123, 250602 (2019).
Relevant tool modes:
bethe-lieb-liniger-obc: Equations (7)-(8) fix the hard-wall ground-state convention and exclusion of self-image scattering; the finite-volume solver extends the logarithmic labels to selected excitations. No boundary-energy integral-equation or thermodynamics API is implemented.
franchini-2011¶
Fabio Franchini. Notes on Bethe Ansatz Techniques. SISSA lecture notes, May 15, 2011 (2011).
Relevant tool modes:
bethe-lieb-liniger-thermal: Section 2.11: thermal filling and entropy of Lieb-Liniger Bethe states. No attractive strings, trapped gases or dynamical correlations.bethe-lieb-liniger-dispersion: Dressed-energy equation and Fermi boundary condition (2.87)-(2.88), related particle-hole energies (2.95); our phase is +2 atan(k/c).
caux-lieb-liniger¶
Jean-Sébastien Caux. The Bethe Ansatz: The ground state and the Lieb equation. Online lecture notes, integrability.org.
Relevant tool modes:
bethe-lieb-liniger-dispersion: Pedagogical ground-state integral equation and normalization at fixed density.
pozsgay-2014-q-boson¶
B. Pozsgay. Quantum quenches and Generalized Gibbs Ensemble in a Bethe Ansatz solvable lattice model of interacting bosons. J. Stat. Mech. (2014) P10045 (2014).
Relevant tool modes:
bethe-q-boson-pbc: Periodic q-boson Hamiltonian (2.1), Bethe equations (2.13), and shifted energy 4 sum sin^2(k/2). Fixed-N ground states and canonical real-root excitation scans; no quenches or correlation functions.
guan-batchelor-2011¶
X.-W. Guan and M. T. Batchelor. Polylogs, thermodynamics and scaling functions of one-dimensional quantum many-body systems. J. Phys. A: Math. Theor. 44, 102001 (2011).
Relevant tool modes:
bethe-lieb-liniger-thermal: Yang-Yang pseudoenergy, root density and pressure equations (2)-(5). We solve the full integral equations, not the finite-c polylog expansion.
golinelli-mallick-2006¶
O. Golinelli and K. Mallick. The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics. J. Phys. A: Math. Gen. 39, 12679-12705 (2006).
Relevant tool modes:
bethe-asep-pbc: Periodic ASEP Bethe equations (66) and Markov eigenvalues (69), with arbitrary nonnegative hopping rates. Leading relaxation branch followed by continuation; no full-spectrum claim.
golinelli-mallick-2005¶
O. Golinelli and K. Mallick. Spectral gap of the totally asymmetric exclusion process at arbitrary filling. J. Phys. A: Math. Gen. 38, 1419-1425 (2005).
Relevant tool modes:
bethe-asep-pbc: Finite-size TASEP leading-relaxation branch used as the continuation seed.bethe-tasep-pbc: Periodic TASEP finite-size Bethe equations (2)-(8) and leading relaxation branch (12)-(14) at arbitrary filling. Complex Markov eigenvalues, not energies; no partial asymmetry or full-spectrum claim.
barcza-2020¶
G. Barcza, K. Bauerbach, F. Eickhoff, F. B. Anders, F. Gebhard, and O. Legeza. Symmetric single-impurity Kondo model on a tight-binding chain: A comparison of analytical and numerical ground-state approaches. Phys. Rev. B 101, 075132 (2020).
Relevant tool modes:
bethe-kondo-response: Universal zero-temperature magnetization, equations (124)-(126). Full Zeeman splitting b=2B and T_B=2T1; impurity energy change obtained by integrating the response.
gainutdinov-hao-nepomechie-sommese-2015¶
A. M. Gainutdinov, W. Hao, R. I. Nepomechie, and A. J. Sommese. Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity. J. Phys. A: Math. Theor. 48, 494003 (2015).
Relevant tool modes:
bethe-xxz-qg-obc: Quantum-group-invariant open XXZ Hamiltonian and Bethe equations (1.1)-(1.3). Our spin-half Hamiltonian is one quarter of the Pauli normalization with spatially reflected boundary fields. Positive finite-real roots at 0\<Delta\<1 have no completeness claim; at Delta=0, a separate free-fermion construction gives complete fixed-Sz spectra and Jordan-block sizes, consistent with Appendices C and D.
gainutdinov-nepomechie-2016¶
A. M. Gainutdinov and R. I. Nepomechie. Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity. Nucl. Phys. B 909, 796-839 (2016).
Relevant tool modes:
bethe-xxz-qg-obc: Root-of-unity complete strings and generalized eigenvectors explain why regular eigenvalues alone do not determine Jordan structure. Delta=0 Hamiltonian block sizes are implemented separately; explicit generalized eigenvectors and other root-of-unity extensions remain follow-ups.
essler-korepin-1994-scattering¶
F. H. L. Essler and V. E. Korepin. Scattering matrix and excitation spectrum of the Hubbard model. Phys. Rev. Lett. 72, 908-911 (1994).
Relevant tool modes:
bethe-hubbard-continuum: Elementary SO(4) spinon and charge doublets and additive spinon-holon, spinon-antiholon and neutral holon-antiholon scattering energies and momenta. Charge-family extrema are resolved numerically, not certified global bounds; our U is four times the paper's U.
bajnok-el-deeb-pearce-2015¶
Zoltan Bajnok, Omar el Deeb, and Paul A. Pearce. Finite-Volume Spectra of the Lee-Yang Model. JHEP 04 (2015), 073 (2015).
Relevant tool modes:
bethe-lee-yang-excited: Source-free periodic vacuum TBA and bulk-subtracted energy, Eqs. (133), (212), (218), used as the excitation-gap reference. The CFT central charge is not inferred from the excited level.bethe-lee-yang-vacuum: Periodic source-free massive Lee-Yang ground-state TBA and bulk-subtracted finite-volume energy: equations (133), (212), (218). Equations (151)-(152) distinguish c_eff=2/5 from c=-22/5 with h_min=-1/5. No excited-state, boundary or defect equations are implemented.
caux-mossel-perez-castillo-2008¶
Jean-Sébastien Caux, Jorn Mossel, and Isaac Pérez Castillo. The two-spinon transverse structure factor of the gapped Heisenberg antiferromagnetic chain. J. Stat. Mech. 2008, P08006 (2008).
Relevant tool modes:
bethe-xxx-structure-factor: Transverse integrated and first-frequency sum rules, Eqs. (62)-(63), evaluated at Delta=1; thermodynamic fractions are not finite-size identities.bethe-xxz-structure-factor-thermo: Zero-field massive transverse two-spinon DSF, published Eqs. (44), (51), (56). Both spectral sheets and their unequal intensities; physical k=pi-Q. S-+(k,omega), integrals use domega/(2*pi), no sum-rule rescaling. Longitudinal and higher-spinon responses are not included.bethe-xxz-dispersion: Massive spinon dispersion, Eqs. (21)-(23), and two-spinon kinematics, Eq. (32). Both unfolded and two-site-folded continuum edges are implemented, not structure factors or spectral weights.
bortz-gohmann-2005¶
Michael Bortz and Frank Göhmann. Exact thermodynamic limit of short-range correlation functions of the antiferromagnetic XXZ-chain at finite temperatures. Eur. Phys. J. B 46, 399-408 (2005).
Relevant tool modes:
bethe-xxz-dispersion: Zero-temperature bulk energy integral/series, with J*Delta/4 restored. Near Delta=1 the massive root density is Poisson-resummed for numerical evaluation; no finite-temperature correlations are implemented.
zauner-stauber-2018¶
Valentin Zauner-Stauber, Laurens Vanderstraeten, Jutho Haegeman, Ian P. McCulloch, and Frank Verstraete. Topological nature of spinons and holons: Elementary excitations from matrix product states with conserved symmetries. Phys. Rev. B 97, 235155 (2018).
Relevant tool modes:
bethe-xxz-dispersion: Interpretation of a single spinon as a topological, symmetry-resolved MPS excitation; momentum folding and asymptotic-vacuum conventions, not an MPS implementation.
johnson-krinsky-mccoy-1973¶
James D. Johnson, Samuel Krinsky, and Barry M. McCoy. Vertical-Arrow Correlation Length in the Eight-Vertex Model and the Low-Lying Excitations of the X-Y-Z Hamiltonian. Phys. Rev. A 8, 2526-2547 (1973).
Relevant tool modes:
bethe-xyz-dispersion: Thermodynamic XYZ spinons, Eq. (7.8), and bound branches, Eqs. (7.11)-(7.12); exchange signs, axis labels and Pauli normalization are mapped to our theta-ratio Hamiltonian. Bound existence and parity/momentum shifts are explicit; no structure factors or complete finite-ring enumeration.
dasmahapatra-kedem-mccoy-melzer-1994¶
Srinandan Dasmahapatra, Rinat Kedem, Barry M. McCoy, and Ezer Melzer. Virasoro Characters from Bethe Equations for the Critical Ferromagnetic Three-State Potts Model. J. Stat. Phys. 74, 239-274 (1994).
Relevant tool modes:
bethe-potts-pbc: Critical ferromagnetic Hamiltonian, physical versus ghost excitations and c=4/5 conformal spectrum. Our clock Hamiltonian is sqrt(3)/2 times the paper's normalization. Direct Potts complex-root enumeration is not used by this frontend.
nichols-2006¶
A. Nichols. The Temperley-Lieb algebra and its generalizations in the Potts and XXZ models. J. Stat. Mech. 2006, P01003 (2006).
Relevant tool modes:
bethe-potts-pbc: Sections 6.1-6.2 distinguish periodic Potts from the unprojected twisted-XXZ spectrum and identify the pi/3 and pi twist sectors. Extra XXZ states must not be counted as Potts levels.
fukai-kleinemuhl-pozsgay-vernier-2024¶
Kohei Fukai, Raphael Kleinemuhl, Balazs Pozsgay, and Eric Vernier. On correlation functions in models related to the Temperley-Lieb algebra. SciPost Phys. 16, 003 (2024).
Relevant tool modes:
bethe-potts-pbc: Sections 2.2-2.3 give the twisted-XXZ and Potts Temperley-Lieb representations. Published XXZ root/Potts level comparisons in Tables 3 and 11 check the charged family at four clock sites. Only the vacuum and a selected real-root one-hole family are implemented, not a complete TL-module decomposition or correlation functions.
dorey-tateo-1996¶
Patrick Dorey and Roberto Tateo. Excited states by analytic continuation of TBA equations. Nucl. Phys. B 482, 639-659 (1996).
Relevant tool modes:
bethe-lee-yang-excited: Periodic spin-zero one-particle source terms, quantization and energy, Eqs. (2.3)-(2.7). Only the regular infrared branch at 5\<=mL\<=30 is implemented, not continuation through the source collision near mL=2.53 or higher/moving states. Levels are bulk-subtracted; gaps subtract a separately converged vacuum.
frahm-stahlsmeier-1998¶
Holger Frahm and Martin Stahlsmeier. Spinon statistics in integrable spin-S Heisenberg chains. Phys. Lett. A 250, 293-299 (1998).
Relevant tool modes:
bethe-tb-dispersion: Spin-1/2 elementary spinons and non-Abelian state counting in higher-spin integrable chains. We report spin labels and energy bounds, not a finite-size counting formula or fusion multiplicities; TB's SU(2)_2 content is not XXX's SU(2)_1.
feverati-ravanini-takacs-1999¶
G. Feverati, F. Ravanini, and G. Takacs. Nonlinear Integral Equation and Finite Volume Spectrum of Sine-Gordon Theory. Nucl. Phys. B 540, 543-586 (1999).
Relevant tool modes:
bethe-sine-gordon-excited: Eqs. (3.13), (3.15), (3.18) and Sec. 5.2.2: exact continuum two-hole NLIE, bulk-subtracted energy and UV weights. Selected symmetric same-charge pairs at p>=1, delta=0, I=1/2 or 3/2, winding charge +/-2, momentum zero; no complex/special roots or complete-spectrum claim.
feher-takacs-2011¶
G. Feher and G. Takacs. Sine-Gordon form factors in finite volume. Nucl. Phys. B 852, 441-467 (2011).
Relevant tool modes:
bethe-sine-gordon-dispersion: Physical soliton mass and exact breather masses, Eq. (2.2). Stable-particle dispersions and fixed-content threshold kinematics only; no form factors or spectral weights.
feher-palmai-takacs-2012¶
G. Z. Feher, T. Palmai, and G. Takacs. Sine-Gordon multi-soliton form factors in finite volume. Phys. Rev. D 85, 085005 (2012).
Relevant tool modes:
bethe-sine-gordon-bethe-yang: Eq. (2.1) fixes S_ss=-exp(i*chi); Sec. 3 describes scattering quantization. We implement opposite-rapidity same-charge pairs at p>=1, with half-odd Bethe numbers and no wrapping corrections. Numerical convergence is not exact finite-volume accuracy.
itou-matsumoto-tanizaki-2024¶
Etsuko Itou, Akira Matsumoto, and Yuya Tanizaki. DMRG study of the theta-dependent mass spectrum in the 2-flavor Schwinger model. arXiv:2407.11391 (2024).
Relevant tool modes:
bethe-sine-gordon-dispersion: Leading light-sector sine-Gordon description of two equal small-mass Schwinger flavours at theta=0: p=1/3, triplet and singlet with mass ratio sqrt(3). Not an exact solution of the full massive or finite-spacing Schwinger model.
osborne-mcculloch-2025¶
Jesse J. Osborne and Ian P. McCulloch. Efficient and systematic calculation of arbitrary observables for the matrix product state excitation ansatz. Phys. Rev. Research 7, 023018 (2025).
kitanine-maillet-terras-1999¶
N. Kitanine, J. M. Maillet, and V. Terras. Form factors of the XXZ Heisenberg spin-1/2 finite chain. Nucl. Phys. B 554, 647-678 (1999).
Relevant tool modes:
bethe-xxx-structure-factor: Algebraic Bethe ansatz norm and local spin form-factor determinant foundation, Secs. 3 and 5.
caux-hagemans-maillet-2005¶
Jean-Sebastien Caux, Rob Hagemans, and Jean-Michel Maillet. Computation of dynamical correlation functions of Heisenberg chains: the gapless anisotropic regime. J. Stat. Mech. 2005, P09003 (2005).
Relevant tool modes:
bethe-xxx-structure-factor: Rational XXX limit of normalized transverse form factors and Gaudin matrices, Eqs. (11)-(13). Only zero-field real-root two-spinon states are implemented; no strings or full DSF.
karbach-muller-bougourzi-1996¶
Michael Karbach, Gerhard Müller, and A. Hamid Bougourzi. Two-spinon dynamic structure factor of the one-dimensional S=1/2 Heisenberg antiferromagnet. arXiv:cond-mat/9606068v2 (1996).
Relevant tool modes:
bethe-xxx-structure-factor-thermo: Threshold behavior and two-spinon intensity/first-moment sum rules; no broadening or renormalization to full intensity.bethe-xxx-moments-thermo: Two-spinon intensity and first-moment sum rules; full-response denominators are references, not fitted normalizations.
abada-bougourzi-si-lakhal-1997¶
A. Abada, A. H. Bougourzi, and B. Si-Lakhal. Exact four-spinon dynamical correlation function of the Heisenberg model. Nucl. Phys. B 497, 733-753 (1997).
Relevant tool modes:
bethe-xxx-structure-factor-thermo: General n-spinon contour representation and its isotropic limit, used by the four- and six-spinon density evaluators. Physical normalization is checked independently, not adjusted by the user.bethe-xxx-moments-thermo: General n-spinon contour representation and its isotropic limit, used by the four- and six-spinon density evaluators. Physical normalization is checked independently, not adjusted by the user.bethe-xxz-structure-factor-thermo: Independent reference uses the original vertex q-products in Eqs. (2.6)-(2.7) restricted to two spinons, with the two-vacuum completeness measure. No four-spinon contour or prefactor is used by this tool.
caux-hagemans-2006¶
Jean-Sébastien Caux and Rob Hagemans. The 4-spinon dynamical structure factor of the Heisenberg chain. J. Stat. Mech. 2006, P12013 (2006).
Relevant tool modes:
bethe-xxx-structure-factor-thermo: Thermodynamic sector decomposition, exact two-spinon Szz (Eqs. (9)-(12)), and four-spinon continuum kinematics. Physical higher-sector density normalization is validated independently.bethe-xxx-moments-thermo: Thermodynamic spinon decomposition and the two-spinon transition rate, Eqs. (9)-(12). Higher-sector normalization and moment measures are independently validated.
dugave-gohmann-kozlowski-suzuki-2016¶
Maxime Dugave, Frank Göhmann, Karol K. Kozlowski, and Junji Suzuki. Thermal form factor approach to the ground-state correlation functions of the XXZ chain in the antiferromagnetic massive regime. J. Phys. A: Math. Theor. 49, 394001 (2016).
babenko-gohmann-kozlowski-suzuki-2021¶
Constantin Babenko, Frank Göhmann, Karol K. Kozlowski, and Junji Suzuki. A thermal form factor series for the longitudinal two-point function of the Heisenberg-Ising chain in the antiferromagnetic massive regime. J. Math. Phys. 62, 041901 (2021).
babenko-gohmann-kozlowski-sirker-suzuki-2021¶
Constantin Babenko, Frank Göhmann, Karol K. Kozlowski, Jesko Sirker, and Junji Suzuki. Exact real-time longitudinal correlation functions of the massive XXZ chain. Phys. Rev. Lett. 126, 210602 (2021).
wang-boundary-1997¶
Yupeng Wang. Exact solution of the open Heisenberg chain with two impurities. Phys. Rev. B 56, 14045 (1997).
Relevant tool modes:
bethe-experimental-xxx-boundary-dynamics: Integrable boundary-impurity coupling. Its derivative at the homogeneous point gives the ground-state boundary bond and full local first moment without a spectral truncation.