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arXiv:2406.11955 · Physical review. B./Physical review. B

Edge spin fractionalization in open one-dimensional spin- S quantum antiferromagnets

Pradip Kattel, Yicheng Tang, J. H. Pixley, Natan Andrei

Published Jun 2, 2025• 3 citations
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Abstract

We show that a gapped spin-$S$ 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-$\frac{1}{2}$ chain with explicitly broken ${\mathbb{Z}}_{2}$ symmetry and an integrable AFM spin-1 chain with spontaneously broken ${\mathbb{Z}}_{2}$ 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.

DOI
10.1103/physrevb.111.l220402

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We show that a gapped spin-SSS chain with antiferromagnetic (AFM) order exhibits, in the thermodynamic limit, exponentially localized fractional \ifmmode±\else\textpm\fiS2\ifmmode\pm\else\textpm\fi{}\frac{S}{2}\ifmmode±\else\textpm\fi2S​ 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±\else\textpm\fi14\ifmmode\pm\else\textpm\fi{}\frac{1}{4}\ifmmode±\else\textpm\fi41​ and \ifmmode±\else\textpm\fi12\ifmmode\pm\else\textpm\fi{}\frac{1}{2}\ifmmode±\else\textpm\fi21​ fractionalized edge modes, respectively. Furthermore, employing the density matrix renormalization group technique, we extend this analysis to generic XXZ\ensuremath-SXXZ\text{\ensuremath{-}}SXXZ\ensuremath-S chains with S\ensuremath≤3S\ensuremath{\le}3S\ensuremath≤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.
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}.
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)].
The ground state properties of an Ising chain with nearest (J1{J}_{1}J1​) and next-nearest neighbor (J2{J}_{2}J2​) interactions in a transverse field are investigated using the density matrix renormalization group and cluster mean-field theory methods. Its quantum phase diagram has two ordered regions (separated by a quantum disordered region), one of which has simple ferromagnetic or N\'eel order depending upon the sign of J1{J}_{1}J1​, and the other is the double-staggered antiferromagnetic phase. The presence of the edge modes in these ordered phases is inferred by calculating the end-to-end spin-spin correlations on open chains. It is found that there occur four edge modes in the double-staggered phase, while the ferromagnetic and N\'eel phases support two edge modes, except very near J1=0{J}_{1}=0J1​=0, where it seems they have four edge modes like the exact case at J1=0{J}_{1}=0J1​=0 itself.
We study symmetric gapped ground states of one-dimensional spin SSS chains with a continuous exponential symmetry U(1)⋊Z2U(1)\rtimes\mathbb Z_2U(1)⋊Z2​ with integer base qqq, where the U(1) charge has a modulated unit of charge qL−jq^{L-j}qL−j decaying exponentially with site jjj. The simplest model is a quantum breakdown model that can be viewed as a qqq-deformation of the XXZ Haldane spin chain. We show that the topology of their symmetric ground states crucially depends on SSS and qqq. For integer SSS, odd qqq allows a nontrivial Z2\mathbb Z_2Z2​ Haldane-type symmetry-protected topological (SPT) phase with protected edge modes, while even qqq allows no nontrivial SPT phase. For half-odd integer SSS, odd qqq prohibits any unique symmetric gapped ground state due to a Lieb-Schultz-Mattis obstruction, whereas even qqq enforces a novel symmetric gapped phase which we name as the ``half-SPT" phase. Unconventionally, the half-SPT phase only has a symmetry protected edge mode on one edge, and has half of the string orders of an SPT phase. Our results uncover a new type of symmetry-protected gapped phases beyond existing classification schemes.
We study a kagomelike spin-12\frac{1}{2}21​ Heisenberg ladder with competing ferromagnetic (FM) and antiferromagnetic (AFM) exchange interactions. Using the density-matrix renormalization group based calculations, we obtain the ground-state phase diagram as a function of the ratio between the FM and AFM exchange interactions. Five different phases exist. Three of them are spin-polarized phases: an FM phase and two kinds of ferrimagnetic (FR) phases (referred to as FR1 and FR2 phases). The spontaneous magnetization per site is m=12,13m=\frac{1}{2}, \frac{1}{3}m=21​,31​, and 16\frac{1}{6}61​ in the FM, FR1, and FR2 phases, respectively. This can be understood from the fact that an effective spin-1 Heisenberg chain formed by the upper and lower leg spins has a three-step fractional quantization of the magnetization per site as m=1,12m=1, \frac{1}{2}m=1,21​, and 0. In particular, an anomalous ``intermediate'' state m=12m=\frac{1}{2}m=21​ of the effective spin-1 chain with the reduced Hilbert space of a spin from three to two dimensional is highly unexpected in the context of conventional spin-1 physics. Thus, surprisingly, the effective spin-1 chain behaves like a spin-12\frac{1}{2}21​ chain with SU(2) symmetry. The remaining two phases are spin-singlet phases with translational symmetry breaking in the presence of valence bond formations. One of them is octamer-singlet phase with a spontaneous octamerization long-range order of the system, and the other is period-4 phase characterized by the magnetic superstructure with a period of four structural unit cells. In these spin-singlet phases, we find the coexistence of valence bond structure and gapless chain. Although this may be emerged through the order-by-disorder mechanism, there can be few examples of such a coexistence.
We study the phase diagram of a one-dimensional spin quantum breakdown model, which has an exponential U(1) symmetry with charge unit decaying as 2^{-j} with site position j. By exact diagonalization and density matrix renormalization group, we show that the model with spin S≥2 exhibits an exponential U(1) spontaneous symmetry-breaking (SSB) phase dubbed a quantum breakdown condensate. It exhibits a bulk gap violating the Goldstone theorem and an edge mode only on the left edge if in open boundary condition. In a length L lattice, the condensate has O(2^{L}) number of SSB ground states originating from the O(2^{L}) number of exponential U(1) charge sectors, leading to a finite entropy density ln2. This enforces a first-order SSB phase transition into this phase, as observed numerically and verified in the large S limit on an exactly solvable Rokhsar-Kivelson line. The condensate has an SSB order parameter being the local in-plane spin, which points in angles related by the chaotic Bernoulli (dyadic) map and thus is effectively random. Moreover, we show the condensate exhibits nondecaying local autocorrelations and does not have an off-diagonal long-range order. The quantum breakdown condensate thus behaves as a disorder-free quantum glass and is beyond the existing classifications of phases of matter.