Establishing Néel order in the square-lattice antiferromagnetic Heisenberg model: a bootstrap proof
Authors: Nisarg Chadha, Taige Wang, Pavel A. Nosov, Michael G. Scheer, Eslam Khalaf
Abstract
Rigorously establishing spontaneous symmetry breaking (SSB) in the thermodynamic limit is generally challenging. This is exemplified by the two-dimensional (2D) $S = 1/2$ antiferromagnetic Heisenberg model where the existence of Néel order is not rigorously established despite overwhelming evidence from a variety of methods. We show that a bootstrap approach can be used to rigorously establish SSB and use it to prove the existence of Néel order in the 2D $S = 1/2$ XXZ model by deriving a strictly positive lower bound on the Néel order parameter for a set of discrete values of the anisotropy parameter $Δ$ in the range $[0,2]$ including the isotropic (Heisenberg) point $Δ= 1$. Our approach employs a generalization of the Kennedy--Lieb--Shastry inequalities, which use an infrared bound to reduce the existence of long-range order to bounds on certain finite-distance correlators. We compute these bounds for all sufficiently large systems with periodic boundary conditions using the bootstrap. An important ingredient in our proof is a novel method for obtaining exactly certified variational upper bounds on the energy density in the thermodynamic limit from matrix product states. Our approach can be readily generalized to other local lattice models for which infrared susceptibility bounds can be obtained.
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Criticality enabled long-range order in a U(1)-symmetric spin-1 Heisenberg chain with biquadratic interactionsN. Chepiga, F. Essler · 2026 · cited 2 times · Not summarized yet
The spin-1 chain study finds U(1) symmetry breaking at a transition, whereas this paper proves Néel order in a two-dimensional spin-1/2 antiferromagnet.
Abstract
In a recent paper Nahum argued on the basis of a renormalization-group analysis that, contrary to standard lore, ground states of 1D spin chains with short-range interactions can spontaneously break U(1) ``easy-plane''spin rotation symmetry. True long-range order of $(S^x,S^y)$ arises at the phase transition between two quasi-long-range-ordered phases. Here we present detailed numerical results for an anisotropic spin-1 chain model. We find a magnetization exponent $\beta\approx0.29$, consistent with the $\epsilon$-expansion prediction $\beta\approx0.28$, a divergence of the Luttinger parameter on approaching the transition from both sides with an exponent that we relate to Nahum's transitio…
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Existence of Néel order in some spin-1/2 Heisenberg antiferromagnetsT. Kennedy, E. Lieb, B. Shastry · 1988 · Journal of Statistical Physics · cited 136 times · Not summarized yet
Existence of Néel order in selected spin-1/2 Heisenberg antiferromagnets is earlier work on the same order question.
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Defect Bootstrap: Tight Ground State Bounds in Spontaneous Symmetry Breaking PhasesMichael G. Scheer, Nisarg Chadha, Da-Chuan Lu et al. · 2025 · cited 6 times · Not summarized yet
Defect Bootstrap develops a bootstrap framework for tighter ground-state bounds in symmetry-broken phases, while this paper uses bootstrap to prove Néel order.
Abstract
The recent development of bootstrap methods based on semidefinite relaxations of positivity constraints has enabled rigorous two-sided bounds on local observables directly in the thermodynamic limit. However, these bounds inevitably become loose in symmetry broken phases, where local constraints are insufficient to capture long-range order. In this work, we identify the origin of this looseness as order parameter defects which are difficult to remove using local operators. We introduce a $\textit{defect bootstrap}$ framework that resolves this limitation by embedding the system into an auxiliary $\textit{defect model}$ equipped with ancilla degrees of freedom. This construction effectively e…
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High-precision ground state parameters of the two-dimensional spin-1/2 Heisenberg model on the square latticeA. Sandvik · 2026 · Journal of Statistical Mechanics: Theory and Experiment · cited 11 times · Not summarized yet
The quantum Monte Carlo study estimates ground-state energy and Néel order parameters for the same square-lattice spin-1/2 Heisenberg antiferromagnet.
Abstract
Motivated by the need for high-precision benchmark results for basic quantum many-body models, several ground state properties of the square lattice S=1/2 Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures T sufficiently low to realize the T → 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L × L lattices with L∈[6,96] are tabulated versus L and extrapolations to infinite system …
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Bootstrapping Classical Systems with Quenched DisorderMinjae Cho, Yaprak Önder, E. Khalaf et al. · 2026 · cited 2 times · Not summarized yet
The disorder study uses bootstrap positivity constraints to bound observables, applying the method to classical disordered systems rather than antiferromagnetic order.
Abstract
We introduce a bootstrap method for deriving bounds on disorder-averaged observables in classical statistical systems with quenched disorder. The three main ingredients of the bootstrap are the positivity of ensembles of probability measures, disorder-averaged equations of motion, and explicit expectation values of random coupling variables. Together, these ingredients lead to a linear programming problem over the space of disorder averages. By further employing the positivity of the two-replica Gram matrix, we also formulate a semidefinite programming problem that produces bounds on two-replica correlators. We demonstrate the method using two examples: the two-dimensional random site-dilute…
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Bootstrapping the Ising Model on the LatticeMinjae Cho, Barak Gabai, Ying-Hsuan Lin et al. · 2022 · cited 31 times · Not summarized yet
The lattice Ising study uses bootstrap positivity constraints to rigorously bound correlators, but studies classical Ising spins rather than quantum Néel order.
Abstract
We study the statistical Ising model of spins on the infinite lattice using a bootstrap method that combines spin-flip identities with positivity conditions, including reflection positivity and Griffiths inequalities, to derive rigorous two-sided bounds on spin correlators through semi-definite programming. For the 2D Ising model on the square lattice, the bootstrap bounds based on correlators supported in a 13-site diamond-shaped region determine the nearest-spin correlator to within a small window, which for a wide range of coupling and magnetic field is narrower than the precision attainable with Monte Carlo methods. We also report preliminary results of the bootstrap bounds for the 3D Is…
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Long-Range Order, “Tower” of States, and Symmetry Breaking in Lattice Quantum SystemsH. Tasaki · 2018 · Journal of statistical physics · cited 96 times · Not summarized yet
This work rigorously relates long-range order to spontaneous symmetry breaking in finite quantum lattice systems, providing conceptual groundwork for the paper’s order proof.
Abstract
In a quantum many-body system where the Hamiltonian and the order operator do not commute, it often happens that the unique ground state of a finite system exhibits long-range order (LRO) but does not show spontaneous symmetry breaking (SSB). Typical examples include antiferromagnetic quantum spin systems with Néel order, and lattice boson systems which exhibits Bose–Einstein condensation. By extending and improving previous results by Horsch and von der Linden and by Koma and Tasaki, we here develop a fully rigorous and almost complete theory about the relation between LRO and SSB in the ground state of a finite system with continuous symmetry. We show that a ground state with LRO but witho…
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Continuous Symmetry Breaking in a Two-dimensional Rydberg ArrayCheng Chen, Guillaume Bornet, Marcus Bintz et al. · 2022 · Quantum Gases · cited 225 times · AI summary available
The Rydberg-array study examines continuous symmetry breaking in a two-dimensional system, a neighbouring setting to the paper’s proof of antiferromagnetic Néel order.
Abstract
Spontaneous symmetry breaking underlies much of our classification of phases of matter and their associated transitions. It provides an example of the power of many-body interactions, enabling a collection of individual degrees of freedom to align its behavior across large spatial and temporal scales. Crucially, the nature of the underlying symmetry being broken determines many of the qualitative properties of the phase; this is illustrated by the case of discrete versus continuous symmetry breaking. Indeed, in contrast to the discrete case, the breaking of a continuous symmetry is governed by Goldstone's theorem, which predicts the existence of gapless modes that mediate power-law correlati…
Prior work it builds on
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Bootstrapping Flatband Superconductors: Rigorous Lower Bounds on Superfluid Stiffness.Qiang Gao, Zhao-Yu Han, E. Khalaf · 2025 · Physical Review Letters · cited 11 times · Not summarized yet
The flatband-superconductor study applies a quantum many-body bootstrap to bound superfluid stiffness rather than antiferromagnetic order.
Abstract
The superfluid stiffness fundamentally constrains the transition temperature of superconductors, especially in the strongly coupled regime. However, accurately determining this inherently quantum many-body property in microscopic models remains a significant challenge. In this Letter, we show how the quantum many-body bootstrap framework, specifically the reduced density matrix (RDM) bootstrap, can be leveraged to obtain rigorous lower bounds on the superfluid stiffness in frustration free interacting models with superconducting ground state. We numerically apply the method to a special class of frustration-free models, which are known as quantum geometric nesting models, for flatband superc…
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Thermal bootstrap of matrix quantum mechanicsMinjae Cho, Barak Gabai, Joshua Sandor et al. · 2024 · Journal of High Energy Physics · cited 27 times · Not summarized yet
The matrix-quantum-mechanics study uses bootstrap constraints to bound observables, but applies them to finite-temperature matrix models.
Abstract
We implement a bootstrap method that combines stationary state conditions, thermal inequalities, and semidefinite relaxations of matrix logarithm in the ungauged one-matrix quantum mechanics, at finite rank N as well as in the large N limit, and determine finite temperature observables that interpolate between available analytic results in the low and high temperature limits respectively. We also obtain bootstrap bounds on thermal phase transition as well as preliminary results in the ungauged two-matrix quantum mechanics.
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The spin-half XXZ antiferromagnet on the square lattice revisited: A high-order coupled cluster treatmentRaymond F. Bishop, P.-H.-Y. Li, R. Zinke et al. · 2016 · Journal of Magnetism and Magnetic Materials · cited 16 times · Not summarized yet
The coupled-cluster study computes ground-state properties of the same square-lattice spin-half XXZ antiferromagnet across anisotropy values.
Abstract
We use the coupled cluster method (CCM) to study the ground-state properties and lowest-lying triplet excited state of the spin-half {\it XXZ} antiferromagnet on the square lattice. The CCM is applied to it to high orders of approximation by using an efficient computer code that has been written by us and which has been implemented to run on massively parallelized computer platforms. We are able therefore to present precise data for the basic quantities of this model over a wide range of values for the anisotropy parameter Δ in the range −1≤Δ 1) regimes, where Δ→∞ represents the Ising limit. We present results for the ground-state energy, the sublattice magnetization, the zero-field transver…
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Long-range order in the XXZ modelH. Nishimori, K. Kubo, Y. Ozeki et al. · 1989 · Journal of Statistical Physics · cited 12 times · Not summarized yet
This earlier work addresses long-range order in the XXZ model, the model family in which the paper proves Néel order.
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Bootstrapping Matrix Quantum Mechanics.Xi-Zhi Han, S. Hartnoll, Jorrit Kruthoff · 2020 · Physical Review Letters · cited 115 times · Not summarized yet
The matrix-quantum-mechanics study uses bootstrap positivity constraints to obtain ground-state results for matrix models rather than lattice antiferromagnets.
Abstract
Large N matrix quantum mechanics is central to holographic duality but not solvable in the most interesting cases. We show that the spectrum and simple expectation values in these theories can be obtained numerically via a "bootstrap" methodology. In this approach, operator expectation values are related by symmetries-such as time translation and SU(N) gauge invariance-and then bounded with certain positivity constraints. We first demonstrate how this method efficiently solves the conventional quantum anharmonic oscillator. We then reproduce the known solution of large N single matrix quantum mechanics. Finally, we present new results on the ground state of large N two matrix quantum mechani…
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Updated on 9 November 2026.