We review mathematical results about the spectrum of spin- $s$ chains with a nearest neighbor interaction given by the projection onto the singlet state of two spin- $s$ degrees of freedom. Up to normalization, these interaction terms generate a Temperley-Lieb algebra at loop parameter $d=2s+1$ . We also present several new results. First, we prove a ferromagnetic ordering of energy levels across symmetry sectors of the model, thus generalizing a previous result of Nachtergaele, Spitzer and Starr (2004) for the spin- $\tfrac12$ $XXZ$ chain. This enables us to determine a uniform lower bound on the finite volume spectral gap for these spin chains. By combining a variational estimate with Knabe's finite size criterion, we show that the exact spectral gap of the GNS Hamiltonian of the fully polarized state is $1-2/d$ . Furthermore, this value serves as a lower bound for the gap of the maximally mixed ground state. Finally, we show that the ground state projectors on open chains satisfy the Jones-Wenzl recursion, thereby recovering a known result that the degeneracies are the quantum integers $[L+1]_q$ for $q+q^{-1}=d$ . We discuss how the high degeneracy of the ground states allows the model to break a continuous symmetry while remaining gapped, and renders that gap unstable.We review mathematical results about the spectrum of spin-$s$ chains with a nearest neighbor interaction given by the projection onto the singlet state of two spin-$s$ degrees of freedom. Up to normalization, these interaction terms generate a Temperley-Lieb algebra at loop parameter $d=2s+1$. We also present several new results. First, we prove a ferromagnetic ordering of energy levels across symmetry sectors of the model, thus generalizing a previous result of Nachtergaele, Spitzer and Starr (2004) for the spin-$\tfrac12$ $XXZ$ chain. This enables us to determine a uniform lower bound on the finite volume spectral gap for these spin chains. By combining a variational estimate with Knabe's finite size criterion, we show that the exact spectral gap of the GNS Hamiltonian of the fully polarized state is $1-2/d$. Furthermore, this value serves as a lower bound for the gap of the maximally mixed ground state. Finally, we show that the ground state projectors on open chains satisfy the Jones-Wenzl recursion, thereby recovering a known result that the degeneracies are the quantum integers $[L+1]_q$ for $q+q^{-1}=d$. We discuss how the high degeneracy of the ground states allows the model to break a continuous symmetry while remaining gapped, and renders that gap unstable.

arXiv:2609.38074Sep 29, 20260 citations

Proof of the Peierls instability in one dimension

Tom Kennedy, Élliott H. Lieb

Fr\"ohlich and Peierls showed that a one-dimensional system with a half-filled band can lower its ground-state energy by a dimerization from period 1 to period 2. It was an open question whether or not this dimerization was exact, i.e., whether additional symmetry breaking would further lower the energy. We prove that the dimerization is exact for a periodic chain of infinitely massive, harmonically bound atoms with nearest-neighbor electron hopping matrix elements that vary linearly with the nearest-neighbor distance.Fr\"ohlich and Peierls showed that a one-dimensional system with a half-filled band can lower its ground-state energy by a dimerization from period 1 to period 2. It was an open question whether or not this dimerization was exact, i.e., whether additional symmetry breaking would further lower the energy. We prove that the dimerization is exact for a periodic chain of infinitely massive, harmonically bound atoms with nearest-neighbor electron hopping matrix elements that vary linearly with the nearest-neighbor distance.

Physical Review LettersSep 21, 198777 citations

The One-Dimensional Quantum Spin Chain

Naoto Nagaosa

Springer eBooksJan 1, 19990 citations

Zero-Energy Problems for Supersymmetric Hamiltonians on a Chain Are $\mathrm{QMA}_1$ -CompleteZero-Energy Problems for Supersymmetric Hamiltonians on a Chain Are $\mathrm{QMA}_1$-Complete

Yibin Wang

We determine the computational complexity of deciding whether a supersymmetric quantum system on a one-dimensional chain has a zero-energy ground state. We prove $\mathrm{QMA}_1$ -completeness for two input models, formulated in terms of Hermitian and nilpotent supercharges, respectively. In the Hermitian model, both the supercharge terms and the Hamiltonian have bounded range on the chain; in the nilpotent model, this geometric restriction applies only to the Hamiltonian. The hardness reduction from quantum satisfiability on a chain preserves exact zero energy and gives an inverse-polynomial ground-energy lower bound for negative instances. A common verification method establishes membership in $\mathrm{QMA}_1$ while allowing cancellations between supercharge terms.We determine the computational complexity of deciding whether a supersymmetric quantum system on a one-dimensional chain has a zero-energy ground state. We prove $\mathrm{QMA}_1$-completeness for two input models, formulated in terms of Hermitian and nilpotent supercharges, respectively. In the Hermitian model, both the supercharge terms and the Hamiltonian have bounded range on the chain; in the nilpotent model, this geometric restriction applies only to the Hamiltonian. The hardness reduction from quantum satisfiability on a chain preserves exact zero energy and gives an inverse-polynomial ground-energy lower bound for negative instances. A common verification method establishes membership in $\mathrm{QMA}_1$ while allowing cancellations between supercharge terms.

quant-pharXiv:2608.29333Sep 1, 20260 citations

Lower bound for the ground energy of spin systems

L.N. Epele, Huner Fanchiotti A, Carlos A. García Canal +1 more

Physica A Statistical Mechanics and its ApplicationsMay 1, 19915 citations

We review mathematical results about the spectrum of spin-

s

chains with a nearest neighbor interaction given by the projection onto the singlet state of two spin-

s

degrees of freedom. Up to normalization, these interaction terms generate a Temperley-Lieb algebra at loop parameter

d=2s+1

. We also present several new results. First, we prove a ferromagnetic ordering of energy levels across symmetry sectors of the model, thus generalizing a previous result of Nachtergaele, Spitzer and Starr (2004) for the spin-

\tfrac12

XXZ

chain. This enables us to determine a uniform lower bound on the finite volume spectral gap for these spin chains. By combining a variational estimate with Knabe's finite size criterion, we show that the exact spectral gap of the GNS Hamiltonian of the fully polarized state is

1-2/d

. Furthermore, this value serves as a lower bound for the gap of the maximally mixed ground state. Finally, we show that the ground state projectors on open chains satisfy the Jones-Wenzl recursion, thereby recovering a known result that the degeneracies are the quantum integers

[L+1]_q

for

q+q^{-1}=d

. We discuss how the high degeneracy of the ground states allows the model to break a continuous symmetry while remaining gapped, and renders that gap unstable.

Zero-Energy Problems for Supersymmetric Hamiltonians on a Chain Are

\mathrm{QMA}_1

-Complete

We determine the computational complexity of deciding whether a supersymmetric quantum system on a one-dimensional chain has a zero-energy ground state. We prove

\mathrm{QMA}_1

-completeness for two input models, formulated in terms of Hermitian and nilpotent supercharges, respectively. In the Hermitian model, both the supercharge terms and the Hamiltonian have bounded range on the chain; in the nilpotent model, this geometric restriction applies only to the Hamiltonian. The hardness reduction from quantum satisfiability on a chain preserves exact zero energy and gives an inverse-polynomial ground-energy lower bound for negative instances. A common verification method establishes membership in

\mathrm{QMA}_1

while allowing cancellations between supercharge terms.