# Spin chain

A spin chain is a one-dimensional array of quantum spins, indexed along a line, whose energy is described by a Hamiltonian coupling neighboring (or, in some models, more distant) spins. Spin chains are the simplest setting in which quantum many-body physics, magnetism, entanglement, and integrability can be studied in detail, and they serve as standard models for quasi-one-dimensional magnetic insulators.<sup>[1](https://academic.oup.com/book/43700/chapter-abstract/367113390)</sup>

| Key fact | Detail |
|---|---|
| Hilbert space | A chain of L spin-1/2 sites has dimension 2^L, so exact diagonalization is limited to short chains.<sup>[2](https://people.phys.ethz.ch/~nbeisert/lectures/Int-16HS-Notes.pdf)</sup> |
| Canonical Hamiltonian | Heisenberg chain, H = J Σ_j **S**_j · **S**_{j+1}, introduced by Heisenberg in 1928; J > 0 is antiferromagnetic.<sup>[3](https://advanced.onlinelibrary.wiley.com/doi/10.1002/qute.202400196)</sup> |
| Exact solution | Bethe's 1931 ansatz gives the exact eigenstates of the spin-1/2 antiferromagnetic Heisenberg chain.<sup>[4](https://www.nature.com/articles/s41535-018-0090-7)</sup> |
| Spin dependence | Integer-S antiferromagnetic chains are gapped (the Haldane gap, proportional to J); half-integer-S chains are gapless and critical.<sup>[5](https://comptes-rendus.academie-sciences.fr/chimie/articles/10.1016/j.crci.2019.05.005/)</sup> |
| Entanglement | Gapped 1D ground states obey an area law (bounded entanglement entropy), which justifies matrix-product-state descriptions and the DMRG algorithm.<sup>[6](https://www.nature.com/articles/s41467-020-18055-x)</sup> |
| Numerics | The density matrix renormalization group, proposed by White in 1992, captures ground states of large quantum spin chains with modest numerical effort.<sup>[7](https://doi.org/10.1146%2Fannurev-conmatphys-040721-022705)</sup> |
| Experimental platforms | Quasi-one-dimensional magnets such as Sr₂Cu(PO₄)₂ realize the spin-1/2 Heisenberg chain; trapped-ion simulators implement tunable long-range spin models.<sup>[4](https://www.nature.com/articles/s41535-018-0090-7)</sup><sup> • </sup><sup>[8](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.93.025001)</sup> |

## Models and Hamiltonians

A spin chain of length L is built from L copies of a single-spin [Hilbert space](https://www.edgepedia.org/hilbert-space); for a spin-1/2 site each site carries states |↑⟩ and |↓⟩, and the full chain is the tensor product of the L spaces, with dimension 2^L.<sup>[2](https://people.phys.ethz.ch/~nbeisert/lectures/Int-16HS-Notes.pdf)</sup> The Hamiltonian is typically homogeneous and acts on nearest neighbors, H = Σ_k H_{k,k+1}.<sup>[2](https://people.phys.ethz.ch/~nbeisert/lectures/Int-16HS-Notes.pdf)</sup>

The most studied case is the Heisenberg chain, H = J Σ_j **S**_j · **S**_{j+1}, introduced by Heisenberg in 1928.<sup>[3](https://advanced.onlinelibrary.wiley.com/doi/10.1002/qute.202400196)</sup> In a ferromagnet, equal alignment of neighboring spins has low energy; in an antiferromagnet, opposite alignment does.<sup>[2](https://people.phys.ethz.ch/~nbeisert/lectures/Int-16HS-Notes.pdf)</sup> Anisotropic variants distinguish the spin components. The XXZ chain has Hamiltonian

\[ \mathcal{H}_{XXZ} = J \sum_{i=1}^{n} \left( \sigma_i^x \sigma_{i+1}^x + \sigma_i^y \sigma_{i+1}^y + \Delta \, \sigma_i^z \sigma_{i+1}^z \right), \]

where Δ is the anisotropy parameter; the isotropic Δ = 1 case is the XXX model, and Δ ≠ 1 gives the XXZ model.<sup>[9](https://diposit.ub.edu/bitstreams/409a276a-f7b8-4b03-9b3a-abbd768bfda4/download)</sup><sup> • </sup><sup>[10](https://www.yu.edu/sites/default/files/legacy/uploadedFiles/Faculty/Lea_Ferreira_dos_Santos/Computer_Codes/JoelKollmarSantos.pdf)</sup> The Ising chain, which keeps only the S^z_n S^z_{n+1} term, was considered by Ising in 1925 as an infinite chain of magnetic moments with nearest-neighbor interaction.<sup>[4](https://www.nature.com/articles/s41535-018-0090-7)</sup>

## Exact solutions

In 1931 [Hans Bethe](https://www.edgepedia.org/hans-bethe) introduced his ansatz method to find the exact quantum ground state of the antiferromagnetic [Heisenberg model](https://www.edgepedia.org/heisenberg-model) in one dimension.<sup>[4](https://www.nature.com/articles/s41535-018-0090-7)</sup> In the [Bethe ansatz](https://www.edgepedia.org/bethe-ansatz), eigenstates are built from Bethe roots u_k satisfying a set of algebraic equations; in the limit of long chains the equations turn into integral equations that approximate the antiferromagnetic ground state.<sup>[2](https://people.phys.ethz.ch/~nbeisert/lectures/Int-16HS-Notes.pdf)</sup> Using Bethe's solution, Hulthén calculated the exact ground-state energy of the antiferromagnetic spin-1/2 chain in 1938.<sup>[11](https://kups.ub.uni-koeln.de/1105/1/Dissertation_Windt.pdf)</sup> The Heisenberg chain and its anisotropic generalizations remain the central examples of integrable quantum systems, in which quantities such as the spectrum can be calculated exactly from the microscopic Hamiltonian.<sup>[12](https://scholarship.miami.edu/esploro/outputs/doctoral/Integrable-quantum-spin-chains/991031447817602976)</sup>

Thermal order is restricted in one dimension: according to the Mermin-Wagner theorem, put forward in 1966, isotropic Heisenberg spin systems in one or two dimensions are incapable of developing ferromagnetic or antiferromagnetic order at any temperature above zero.<sup>[4](https://www.nature.com/articles/s41535-018-0090-7)</sup>

## Phases: the Haldane distinction

The physics of the antiferromagnetic Heisenberg chain depends crucially on the spin magnitude S. For integer S the excitation spectrum is gapped and the system can host topological edge states; for half-integer S the chain is critical with gapless excitations.<sup>[3](https://advanced.onlinelibrary.wiley.com/doi/10.1002/qute.202400196)</sup> This distinction was admitted by Haldane in 1983.<sup>[4](https://www.nature.com/articles/s41535-018-0090-7)</sup> The gap in the integer-spin case, the Haldane gap, is proportional to the exchange coupling J, and the integer-spin chain has short-range magnetic order with a nonmagnetic singlet ground state.<sup>[5](https://comptes-rendus.academie-sciences.fr/chimie/articles/10.1016/j.crci.2019.05.005/)</sup>

The uniform spin-1/2 chain is gapless and has fractionalized excitations, domain walls carrying spin S = 1/2.<sup>[4](https://www.nature.com/articles/s41535-018-0090-7)</sup> These fractional excitations are known as spinons, each carrying spin 1/2, in contrast with the usual magnon (spin-wave) excitations, which carry spin 1.<sup>[3](https://advanced.onlinelibrary.wiley.com/doi/10.1002/qute.202400196)</sup> In the XXZ chain, the region Δ ∈ (−1, 1] is critical (the XY phase) with conformal scaling of the entropy and central charge c = 1; quantum phase transitions occur at Δ = 1 (a Kosterlitz-Thouless transition) and at Δ = −1.<sup>[9](https://diposit.ub.edu/bitstreams/409a276a-f7b8-4b03-9b3a-abbd768bfda4/download)</sup>

## Entanglement and computation

The Hilbert space dimension 2^L makes exact diagonalization impractical beyond short chains, and spin chains became the testing ground for numerical many-body methods.<sup>[2](https://people.phys.ethz.ch/~nbeisert/lectures/Int-16HS-Notes.pdf)</sup> The density matrix renormalization group (DMRG), proposed by Steven White in 1992, revealed an ability to capture the ground state of large quantum spin chains with only modest numerical effort.<sup>[7](https://doi.org/10.1146%2Fannurev-conmatphys-040721-022705)</sup> Shortly afterward, Östlund and Rommer identified the fixed point of the infinite DMRG algorithm with matrix product states (MPS), and Dukelsky and colleagues connected DMRG to a variational search over these states.<sup>[7](https://doi.org/10.1146%2Fannurev-conmatphys-040721-022705)</sup>

The efficiency of MPS rests on entanglement structure. An MPS with bond dimension D has entanglement entropy less than log D across any cut of the system, while generic quantum states require exponentially large bond dimensions, D = exp[O(n)].<sup>[6](https://www.nature.com/articles/s41467-020-18055-x)</sup> For gapped one-dimensional ground states the area law holds, bounding the entanglement entropy by a constant, and this rigorously justifies the efficient MPS description of ground states and the DMRG algorithm.<sup>[6](https://www.nature.com/articles/s41467-020-18055-x)</sup> At quantum critical points the entropy instead scales logarithmically with the block size L, with a coefficient given by the central charge of the conformal theory describing the transition.<sup>[13](https://arxiv.org/abs/quant-ph/0211074)</sup> For critical chains simulated with periodic boundary conditions, MPS simulations split into a finite-size-scaling regime and a finite-entanglement-scaling regime; in the thermodynamic limit only MPS in the finite-size-scaling regime maintain a finite overlap with the exact ground state.<sup>[14](https://arxiv.org/abs/1204.3934)</sup>

## Experimental realizations and quantum simulation

Quasi-one-dimensional magnetic insulators, in which the interaction of the magnetic moments along one direction is much larger than in the perpendicular directions, realize spin-chain Hamiltonians with high accuracy.<sup>[1](https://academic.oup.com/book/43700/chapter-abstract/367113390)</sup> Among uniform Heisenberg spin-1/2 chain materials, Sr₂Cu(PO₄)₂ and Ba₂Cu(PO₄)₂ show broad susceptibility maxima at T_max = 92 K (J = 143 K) and T_max = 82 K (J = 132 K) respectively; the Sr compound is regarded as a particularly good realization, with long-range order appearing only at T_N = 0.085 K.<sup>[4](https://www.nature.com/articles/s41535-018-0090-7)</sup> Gapped alternating spin-1/2 chains such as BaCu₂V₂O₈ (J₁ = 460 K) and Haldane-gap spin-1 chains such as PbNi₂V₂O₈ provide realizations of the other phases.<sup>[4](https://www.nature.com/articles/s41535-018-0090-7)</sup>

Beyond solid-state materials, programmable quantum simulators implement spin-chain Hamiltonians directly. In trapped-ion systems, effective spins are represented by internal energy levels of laser-cooled ions and measured with near-perfect efficiency by state-dependent fluorescence; optical dipole forces modulate the Coulomb interaction to produce long-range, tunable spin-spin interactions that can be reconfigured by shaping the laser spectrum and pattern.<sup>[8](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.93.025001)</sup> Such platforms have been used to study non-ergodic phenomena in long-range interacting spin models, including many-body localization memory, prethermalization, and discrete time-crystal oscillations.<sup>[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC5665787/)</sup>

## References

1. [The Anisotropic Heisenberg Quantum Spin Chain, in *Models of Quantum Matter*, Oxford Academic (2019)](https://academic.oup.com/book/43700/chapter-abstract/367113390)
2. [N. Beisert, Introduction to Integrability, Lecture Notes, ETH Zurich, Chapter 4: Integrable Spin Chains (2016)](https://people.phys.ethz.ch/~nbeisert/lectures/Int-16HS-Notes.pdf)
3. [P. Laurell, Witnessing Entanglement and Quantum Correlations in Condensed Matter: A Review, *Advanced Quantum Technologies* (2025)](https://advanced.onlinelibrary.wiley.com/doi/10.1002/qute.202400196)
4. [Milestones of low-D quantum magnetism, *npj Quantum Materials* (2018)](https://www.nature.com/articles/s41535-018-0090-7)
5. [T. Jolicoeur and O. Golinelli, Physics of integer-spin antiferromagnetic chains: Haldane gaps and edge states, *Comptes Rendus Chimie* 22, 445–451 (2019)](https://comptes-rendus.academie-sciences.fr/chimie/articles/10.1016/j.crci.2019.05.005/)
6. [T. Kuwahara and K. Saito, Area law of noncritical ground states in 1D long-range interacting systems, *Nature Communications* 11, 4478 (2020)](https://www.nature.com/articles/s41467-020-18055-x)
7. [M. C. Bañuls, Tensor Network Algorithms: A Route Map, *Annual Review of Condensed Matter Physics* 14, 173–191 (2023)](https://doi.org/10.1146%2Fannurev-conmatphys-040721-022705)
8. [Programmable quantum simulations of spin systems with trapped ions, *Reviews of Modern Physics* 93, 025001 (2021)](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.93.025001)
9. [The Heisenberg XXZ model, University of Barcelona thesis chapter (2018)](https://diposit.ub.edu/bitstreams/409a276a-f7b8-4b03-9b3a-abbd768bfda4/download)
10. [J. Kollmar and L. F. dos Santos, on the XXZ and XXX Heisenberg models, Yeshiva University](https://www.yu.edu/sites/default/files/legacy/uploadedFiles/Faculty/Lea_Ferreira_dos_Santos/Computer_Codes/JoelKollmarSantos.pdf)
11. [S. Windt, dissertation chapter on antiferromagnetic Heisenberg chains, University of Cologne](https://kups.ub.uni-koeln.de/1105/1/Dissertation_Windt.pdf)
12. [Integrable quantum spin chains, University of Miami dissertation](https://scholarship.miami.edu/esploro/outputs/doctoral/Integrable-quantum-spin-chains/991031447817602976)
13. [G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in quantum critical phenomena, arXiv:quant-ph/0211074; *Phys. Rev. Lett.* 90, 227902 (2003)](https://arxiv.org/abs/quant-ph/0211074)
14. [B. Pirvu, G. Vidal, F. Verstraete, and L. Tagliacozzo, Matrix product states for critical spin chains: finite size scaling versus finite entanglement scaling, *Phys. Rev. B* 86, 075117 (2012)](https://arxiv.org/abs/1204.3934)
15. [Non-thermalization in trapped atomic ion spin chains, *Philosophical Transactions of the Royal Society A* (PMC5665787)](https://pmc.ncbi.nlm.nih.gov/articles/PMC5665787/)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum phenomena and measurement › Entanglement and nonlocal correlations › Entanglement in many-body and macroscopic systems*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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