99.9...% of the SU(2) BFSS ground state
Authors: Alexander Michel, Jaroslav Scheinpflug, Xi Yin
Abstract
We solve for the ground-state wavefunction of the SU(2) BFSS matrix model by applying the Rayleigh-Ritz method to an auxiliary operator $H_{\mathrm{aux}}$. We show that this operator agrees with the Hamiltonian on the ground-state subspace and has a spectral gap. For SU(2) specifically, the representation theory of the singlet sector gives an astronomical reduction in the number of degrees of freedom. This allows us to construct an efficient variational basis whose inner products we compute exactly. We demonstrate the method by computing a state that satisfies rigorous bounds on its energy $E\,g_{\mathrm{YM}}^{-2/3}\leq 4.18\times10^{-8}$ and ground-subspace overlap $\|P_{\ker H}ψ\|^2\geq 0.999997$. We compute ground-state moments with high precision, finding values consistent with the bounds from our matrix bootstrap. Additionally, the state accurately reproduces both the exponent and coefficient of the asymptotic radial probability density predicted by the three-graviton amplitude.
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Closest papers
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On the ground state wave function of matrix theoryYing-Hsuan Lin, Yin Xi · 2014 · Journal of High Energy Physics · cited 21 times · Not summarized yet
On the ground state wave function of matrix theory proposes leading asymptotic terms for the BFSS SU(N) ground state, while this paper constructs and tests an SU(2) ground state globally.
Abstract
A bstractWe propose an explicit construction of the leading terms in the asymptotic expansion of the ground state wave function of BFSS SU(N ) matrix quantum mechanics. Our proposal is consistent with the expected factorization property in various limits of the Coulomb branch, and involves a different scaling behavior from previous suggestions. We comment on some possible physical implications.
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Asymptotic Search for Ground States of SU(2) Matrix TheoryM. B. Halpern, C. Schwartz · 1997 · International Journal of Modern Physics A · cited 65 times · Not summarized yet
Asymptotic Search for Ground States of SU(2) Matrix Theory seeks normalizable zero-energy SU(2) solutions but produces candidates that require global stability tests.
Abstract
We introduce a complete set of gauge-invariant variables and a generalized Born–Oppenheimer formulation to search for normalizable zero-energy asymptotic solutions of the Schrodinger equation of SU(2) matrix theory. The asymptotic method gives only ground state candidates, which must be further tested for global stability. Our results include a set of such ground state candidates, including one state which is a singlet under spin (9).
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Asymptotic Ground State for 10 Dimensional Reduced Supersymmetric SU(2) Yang Mills TheoryG. M. Graf, J. Hoppe · 1998 · arXiv: High Energy Physics - Theory · preprint, not peer reviewed · cited 14 times · Not summarized yet
Asymptotic Ground State for 10 Dimensional Reduced Supersymmetric SU(2) Yang Mills Theory derives the invariant SU(2) ground-state wavefunction’s asymptotic decay and form.
Abstract
We calculate the power law decay, and asymptotic form of a (unique) SO(9) and SU(2) invariant wave function satisfying, to leading and sub-leading order, $Q_{\hat{\beta}} \psi = 0$ for all 16 supercharges of the matrix model corresponding to supermembranes in 11 space-time dimensions.
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Towards the Ground State of the SupermembraneY. Michishita, M. Trzetrzelewski · 2012 · Nuclear Physics · cited 10 times · Not summarized yet
Towards the Ground State of the Supermembrane studies the SU(2) ground-state wavefunction near the origin through its Taylor expansion.
Abstract
The explicit, near the origin, form of the ground state of the SU(2) supermembrane matrix model is studied. We evaluate the 2nd order terms of the Taylor expansion of the wave-function, which together with the 0th and the 1st order terms can be used to determine other terms by recurrence equations coming from �
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Asymptotic Factorisation of the Ground-State for SU(N)-invariant Supersymmetric Matrix-ModelsD. Hasler, J. Hoppe · 2002 · arXiv: High Energy Physics - Theory · preprint, not peer reviewed · cited 18 times · Not summarized yet
Asymptotic Factorisation of the Ground-State for SU(N)-invariant Supersymmetric Matrix-Models proves factorization of the ground-state wavefunction at large eigenvalue separation.
Abstract
We give a simple - straightforward and rigorous - derivation that when the eigenvalues of one of the d = 9 (5,3,2) matrices in the SU(N) invariant supersymmetric matrix model become large (and well separated from each other) the ground-state wavefunction (resp. asymptotic zero-energy solution of the corresponding differential equation) factorizes, for all N > 1, into a product of supersymmetric harmonic oscillator wavefunctions (involving the ‘off-diagonal’ degrees of freedom) and a wavefunction that is annihilated by the free supercharge formed out of all ‘diagonal’ (Cartan sub-algebra) degrees of freedom. During the past few years, zero-energy states in supersymmetric matrix-models have be…
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Construction of the zero-energy state of SU(2)-matrix theory: Near the originJ. Hoppe, D. Lundholm, M. Trzetrzelewski · 2008 · Nuclear Physics · cited 19 times · Not summarized yet
Construction of the zero-energy state of SU(2)-matrix theory constructs a Spin(9) × SU(2) singlet near the origin, whereas this paper gives a global variational state.
Abstract
We explicitly construct a (unique) Spin(9) x SU(2) singlet state, phi, involving only the fermionic degrees of freedom of the supersymmetric matrix-model corresponding to reduced 10-dimensional sup ...
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On asymptotic hamiltonian for SU(N) matrix theoryA. Konechny · 1998 · Journal of High Energy Physics · cited 24 times · Not summarized yet
On asymptotic hamiltonian for SU(N) matrix theory derives an effective Hamiltonian at large separation and discusses ground-state candidates.
Abstract
We compute the leading contribution to the effective Hamiltonian of SU(N) matrix theory in the limit of large separation. We work with a gauge fixed Hamiltonian and use generalized Born-Oppenheimer approximation, extending the recent work of Halpern and Schwartz for SU(2) [8]. The answer turns out to be a free Hamiltonian for the coordinates along the flat directions of the potential. Applications to finding ground state candidates and calculation of the correction (surface) term to Witten index are discussed.
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Uniqueness, Nondegeneracy, and Asymptotic Expansions of Ground States for the Dirac-Coulomb SystemPan Chen, Qi Guo, Xiao-Yu Zeng · 2026 · Not summarized yet
Uniqueness, Nondegeneracy, and Asymptotic Expansions of Ground States for the Dirac-Coulomb System studies ground states of a different quantum system.
Abstract
We study ground states of the following Dirac-Coulomb system $$\mathscr{D}_cu_c- \left(\abs{x}^{-1}\ast \abs{u_c}^2\right)u_c=\omega_cu_c,$$ under the constraint $\norm[L^2]{u_c}^2=1.$For sufficiently large \(c\), we prove uniqueness of ground states modulo translations, phase transformations and spinorial rotations, and determine the kernel of the linearized operator at ground state. We also obtain asymptotic expansions of ground state in the nonrelativistic limit regime.
Prior work it builds on
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Witten index and threshold bound states of D-branesPiljin Yi · 1997 · Nuclear Physics · cited 166 times · Not summarized yet
Witten index and threshold bound states of D-branes counts threshold bound states in SU(2) super Yang-Mills quantum mechanics rather than constructing their wavefunction.
Abstract
We consider the Witten index I = Tr(−1)F of SU(2) super Yang-Mills quantum mechanics (SYMQ) with N = 16, 8, 4 supersymmetries. The theory governs the interactions between a pair of D-branes under various circumstances, and our goal is to count the number of the threshold bound states directly from the low-energy effective theory. String theory and M-theory have predicted that I = 1 for N = 16, which in fact forms an underlying hypothesis of the M(atrix)-theory formulation. Also the consistency of conifold transitions in type II theories is known to require I = 0 for N = 8 and 4. Here, the bulk contribution to I is computed explicitly, and for N = 16, 8, 4, found to be 54, 14, 14 respectively…
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Bound states at threshold in supersymmetric quantum mechanicsMassimo Porrati, A. Rozenberg · 1997 · Nuclear Physics · cited 91 times · Not summarized yet
Bound states at threshold in supersymmetric quantum mechanics develops an existence method and applies it to D0-brane quantum mechanics.
Abstract
Abstract We propose a general method that allows the detection of the existence of normalizable ground states in supersymmetric quantum mechanical systems with a non-Fredholm spectrum. We apply our method to show the existence of bound states at threshold in two important cases: (i) the quantum mechanical system describing H-monopoles; (ii) the quantum mechanics of D0 branes.
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Bootstrap bounds on D0-brane quantum mechanicsHenry W. Lin · 2023 · Journal of High Energy Physics · cited 38 times · Not summarized yet
Bootstrap bounds on D0-brane quantum mechanics derives bounds on BFSS correlation functions, which this paper compares with its computed ground-state moments.
Abstract
We derive simple bootstrap bounds on correlation functions of the BFSS matrix theory/D0-brane quantum mechanics. The result strengthens and extends Polchinski’s virial theorem bound to finite energies and gives the first non-trivial bound on ⟨Tr X2⟩. Despite their simplicity, the bounds hint at some features of the dual black hole geometry. Our best lower bounds are already a factor of ∼ 2 from existing Monte Carlo results.
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Hamiltonian truncation study of supersymmetric quantum mechanics: S-matrix and metastable statesBruno Balthazar, V. Rodríguez, Xi Yin · 2016 · Journal of High Energy Physics · cited 7 times · Not summarized yet
Hamiltonian truncation study of supersymmetric quantum mechanics uses Rayleigh–Ritz truncation to study scattering and resonances in other supersymmetric systems.
Abstract
We implement the Rayleigh-Ritz method in supersymmetric quantum mechanics with flat directions, and extract the S-matrix and metastable resonances. The effectiveness of the method is demonstrated in two strongly coupled systems: an N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 toy supermembrane model, and an N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{u…
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Normalized vacuum states in N=4 supersymmetric Yang–Mills quantum mechanics with any gauge groupV. Kac, A. Smilga · 1999 · Nuclear Physics · cited 72 times · Not summarized yet
Normalized vacuum states in N=4 supersymmetric Yang–Mills quantum mechanics studies normalized vacua in a different supersymmetry and gauge-theory setting.
Abstract
Abstract We study the question of existence and the number of normalized vacuum states in N =4 super-Yang–Mills quantum mechanics for any gauge group. The mass deformation method is the simplest and clearest one. It allowed us to calculate the number of normalized vacuum states for all gauge groups. For all unitary groups, #vac=1, but for the symplectic groups [starting from Sp(6) ], for the orthogonal groups [starting from SO(8)] and for all the exceptional groups, it is greater than one. We also discuss at length the functional integral method. We calculate the “deficit term” for some non-unitary groups and predict the value of the integral giving the “principal contribution”. The issues l…
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Updated on 10 November 2026.