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Physical review. B./Physical review. B

Spin fractionalization and zero modes in the spin- 1 2 XXZ chain with boundary fields

Parameshwar R. Pasnoori, Yicheng Tang, Junhyun Lee, J. H. Pixley, Natan Andrei, P. Azaria

Published Jul 17, 2025• 4 citations
Publisher

Abstract

In this work, we argue that the antiferromagnetic spin-$\frac{1}{2}$ XXZ chain in the gapped phase with boundary magnetic fields hosts fractional spin $\frac{1}{4}$ at its edges. Using a combination of Bethe ansatz and the density matrix renormalization group we show that these fractional spins are sharp quantum observables in both the ground and the first excited state as the associated fractional spin operators have zero variance. In the limit of zero edge fields, we argue that these fractional spin operators, once projected onto the low-energy subspace spanned by the ground state and the first excited state, identify with the strong zero-energy mode discovered by Fendley [J. Phys. A: Math. Theor. 49, 30LT01 (2016)].

DOI
10.1103/thlq-h58t

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Physical review. B./Physical review. B•Jun 12, 2023•15 citations
In this work, we argue that the antiferromagnetic spin-12\frac{1}{2}21​ XXZ chain in the gapped phase with boundary magnetic fields hosts fractional spin 14\frac{1}{4}41​ at its edges. Using a combination of Bethe ansatz and the density matrix renormalization group we show that these fractional spins are sharp quantum observables in both the ground and the first excited state as the associated fractional spin operators have zero variance. In the limit of zero edge fields, we argue that these fractional spin operators, once projected onto the low-energy subspace spanned by the ground state and the first excited state, identify with the strong zero-energy mode discovered by Fendley [J. Phys. A: Math. Theor. 49, 30LT01 (2016)].
Spin fractionalization and zero modes in the spin-12\frac{1}{2}21​ XXZ chain with boundary fields
In this work we argue that the antiferromagnetic spin 12\frac{1}{2}21​ XXZ chain in the gapped phase with boundary magnetic fields hosts fractional spin 14\frac{1}{4}41​ at its edges. Using a combination of Bethe ansatz and the density matrix renormalization group we show that these fractional spins are sharp quantum observables in both the ground and the first excited state as the associated fractional spin operators have zero variance. In the limit of zero edge fields, we argue that these fractional spin operators once projected onto the low energy subspace spanned by the ground state and the first excited state, identify with the strong zero energy mode discovered by P. Fendley \cite{Fendley}.
We show that a gapped spin-SSS chain with antiferromagnetic (AFM) order exhibits, in the thermodynamic limit, exponentially localized fractional $\ifmmode\pm\else\textpm\fi{}\frac{S}{2}$ edge modes when the system possesses U(1) symmetry. We show this for integrable and nonintegrable spin chains, both analytically and numerically. Through exact analytical solutions, we show that an AFM spin-12\frac{1}{2}21​ chain with explicitly broken Z2{\mathbb{Z}}_{2}Z2​ symmetry and an integrable AFM spin-1 chain with spontaneously broken Z2{\mathbb{Z}}_{2}Z2​ symmetry have $\ifmmode\pm\else\textpm\fi{}\frac{1}{4}$ and $\ifmmode\pm\else\textpm\fi{}\frac{1}{2}$ fractionalized edge modes, respectively. Furthermore, employing the density matrix renormalization group technique, we extend this analysis to generic $XXZ\text{\ensuremath{-}}S$ chains with $S\ensuremath{\le}3$ and demonstrate that these fractional spins are robust quantum observables, substantiated by the observation of variance of the associated fractional spin operators that is consistent with a vanishing functional form in the thermodynamic limit. Moreover, we find that the edge modes are robust to a disorder that couples to the N\'eel order parameter.
Boundary Quantum Phase Transitions in the Spin 12\frac{1}{2}21​ Heisenberg Chain with Boundary Magnetic Fields
We consider the spin 12\frac{1}{2}21​ Heisenberg chain with boundary magnetic fields and analyze it using a combination of Bethe ansatz and density matrix renormalization group (DMRG) techniques. We show that the system exhibits several different ground states which depend on the orientation of the boundary magnetic fields. When both the boundary fields take equal values greater than a critical field strength, each edge in the ground state accumulates a fractional spin which saturates to spin 14\frac{1}{4}41​, which is similar to systems exhibiting symmetry protected topological phases (SPT). Unlike in SPT systems, the fractional boundary spin in the Heisenberg spin chain is not a genuine quantum number since the variance of the associated operator does not vanish, this is due to the absence of a bulk gap. The system exhibits high energy bound states when the boundary fields take values greater than the critical field. All the excitations in the system can be sorted out into towers whose number depends on the number of bound states exhibited by the system. As the boundary fields are varied, in addition to the ground state phase transition, we find that the system may undergo an eigenstate phase transition (EPT) where the number of towers of the Hilbert space changes. We further inquire how the EPT reflects itself on local ground state properties by computing the magnetization profile ⟨Sjz⟩\langle S^z_j \rangle⟨Sjz​⟩ using DMRG. We identify a clear qualitative change from low edge fields to high edge fields when crossing the critical field. We though are unable to conclude on the basis of our data that EPT corresponds to a genuine phase transition in the ground state.
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We derive exact strong zero mode (ESZM) operators for integrable spin-S chains with open boundary conditions and a boundary field. Their locality properties are generally weaker than in the previously known cases, but they still imply infinite coherence times in the vicinity of the edges. We explain how such integrable chains possess multiple ground states describing a first-order quantum phase transition, and that the odd number of such states for integer S makes the weaker locality properties necessary. We make contact with more traditional approaches by showing how the ESZM for S=1/2 acts on energy eigenstates given by solutions of the Bethe equations.
We consider the spin-1/21/21/2 Heisenberg chain with boundary magnetic fields and analyze it using a combination of Bethe ansatz and density matrix renormalization group (DMRG) techniques. We show that the system exhibits several different ground states, which depend on the orientation of the boundary magnetic fields. When both the boundary fields take equal values greater than a critical field strength, each edge in the ground state accumulates a fractional spin which saturates to spin 1/41/41/4, which is similar to systems exhibiting symmetry protected topological phases (SPT). Unlike in SPT systems, the fractional boundary spin in the Heisenberg spin chain is not a genuine quantum number since the variance of the associated operator does not vanish, this is due to the absence of a bulk gap. The system exhibits high-energy bound states when the boundary fields take values greater than the critical field. All the excitations in the system can be sorted out into towers whose number depends on the number of bound states exhibited by the system. As the boundary fields are varied, in addition to the ground-state phase transition, we find that the system may undergo a boundary eigenstate phase transition (BEPT) where the number of towers of the Hilbert space changes. We further inquire how the EPT reflects itself on local ground-state properties by computing the magnetization profile $\ensuremath{\langle}{S}_{j}^{z}\ensuremath{\rangle}$ using DMRG. We identify a clear qualitative change from low edge fields to high edge fields when crossing the critical field. We though are unable to conclude on the basis of our data that EPT corresponds to a genuine phase transition in the ground state.