A rigid commensurate host and a defect that switches: vacancies in the frustrated alternating spin-(1, 1/2) Heisenberg chain

The gist The alternating spin-(1, 1/2) Heisenberg chain leaves its ferrimagnetic phase by a direct collapse from S = L/4 to S = 0, and that the singlet phase it enters carries a modulation locked at the commensurate wavevector 3π/4, within 1.1 × 10−4 in the linear extrapolation at the three couplings measured at four sizes, J2 = 0.28, 0.30 and 0.32, and is gapless, its singlet–triplet gap closing as 1/L and its entanglement entropy growing with a central charge c≃ 1.8 > The gist R L Silva,1,∗ R C Silva,1 and A R Pereira2 The alternating spin-(1, 1/2) Heisenberg chain with a frustrating next-nearest-neighbor coupling J2 loses its Lieb–Mattis moment through a discontinuous transition to a singlet whose nature was left open.

Model and Ground State Collapse

The system under study is defined by the Hamiltonian H = J1 X i−j=1 Si · Sj + J2 X i−j=2 Si · Sj on an open chain of L sites, with S = 1 on odd sites (sublattice A) and S = 1/2 on even sites (sublattice B), where Ncell = L/2. The ground state transition is characterized by a direct collapse from the Lieb–Mattis value S = L/4 to S = 0, occurring between J2 = 0.22 and J2 = 0.24 at L = 80. The energy E(S) curve does not show an interior minimum, indicating that a partially polarized phase is not predicted by this study.

Properties of the Singlet Phase

The singlet phase is described as critical, exhibiting several key features. Its singlet–triplet gap closes as 1/L over L = 80 to 200, and its entanglement entropy grows logarithmically with a central charge c≃ 1.8. The modulation wavevector, extrapolated from L = 80 to 200, is found to be commensurate at q = 3π/4 and independent of J2 to five times ten-five. The singlet phase is gapless, as evidenced by the singlet–triplet gap closing as 1/L.

The Role of the Defect

A single impurity in this one-dimensional interacting system acts as a singular perturbation, potentially opening resonant channels and changing the fate of the chain. For a single S = 1 site removed, the transmission through it depends on whether J2 is below or above a critical coupling J∗2. Below this coupling, the doped chain is a triplet, but this triplet is described as a magnon of the host confined to one segment, not a moment bound to the vacancy. The defect transition at J∗2 = 0.3102 at L = 80 marks where the ground state of the doped chain switches from a triplet to a singlet.

Transmission and Observables

The transmission of correlations through the vacancy, TIC(L), is defined as the ratio of Fourier amplitudes of symmetric-pair correlator C(v − r, v + r) over clean. On the moment side (J2 = 0.28 and 0.30), the transmission saturates at a finite value, T(∞) = 0.102(15) and 0.036(9). On the singlet side (J2 = 0.32), the transmission keeps falling with a slope near 0.6–0.7 that sizes up to L = 200 cannot resolve into a plateau or a decay to zero.

Host Rigidity and Defect Switching

The central result is that A defect transition in a rigid commensurate host is the central result of this work. The host registers J∗2 in nothing that survives to the thermodynamic limit, as evidenced by the sign change in the finite-size correction to its pitch, which moves with the pair of sizes and vanishes from resolution on the largest. The control defect (S = 1/2 vacancy) crosses elsewhere near J2 = 0.262, meaning the transition at J∗2 belongs to the removed S = 1 site and not to doping in general.

Conclusion on Physics

The study concludes that the phase above the first-order boundary is a period-eight commensurate singlet, rigid against J2, and gapless. The vacancy's effect is to switch the ground state from a triplet to a singlet at J∗2 = 0.3102 at L = 80, while the host shows no feature at that coupling.

Experimental Signatures

Any probe coupling to the local spectral weight across a vacancy, such as NMR relaxation or end-state spectroscopy, should see the low-lying triplet of the doped chain appear and disappear as frustration is tuned. The invariant correlator ⟨Si · Sj ⟩ is presented as the only safe route whenever the defect carries a moment.

References

[1] E. Lieb and D. Mattis, J. Math. Phys. 3, 749 (1962)

[2] S. K Pati, S Ramasesha, and D Sen, Phys Rev B 55, 8894 (1997)

[3] S Brehmer, H-J Mikeska, and S Yamamoto, J Phys Condens Matter 9, 3921 (1997)

[4] A K Kolezhuk, H-J Mikeska, and S Yamamoto, Phys Rev B 55 R3336 (1997)

[5] N B Ivanov, J Richter, and U Schollwöck, Phys Rev B 58 14456 (1998)

[6] K Hida and K Takano, Phys Rev B 78 064407 (2008)

[7] C L Kane and M P A Fisher, Phys Rev Lett. 68 1220 (1992)

[8] C L Kane and M P A Fisher, Phys Rev B 46 15233 (1992)

[9] C L Kane and M P A Fisher, Phys Rev B 46 7268 (1992)

[10] A Furusaki and N Nagaosa, Phys Rev Lett. 72 892 (1994)

[11] S Eggert and I Affleck, Phys Rev B 46 10866 (1992)

[12] O Kahn, in Theoretical Approaches, Structure and Bonding, Vol. 68 (Springer Berlin Heidelberg, 1987) pp. 89–167

[13] M Hagiwara, K Minami, Y Narumi, K Tatani, and K Kindo, J Phys Soc Jpn. 67 2209 (1998)

[14] W Huang, M Stark, P Greule, K H Au-Yeung, D Sostina et al., Nat Commun. 16 5208 (2025)

[15] S Mishra, G Catarina et al., Nature 598 287 (2021)

[16] S R White, Phys Rev Lett. 69 2863 (1992)

[17] U Schollwöck, Ann Phys (N.Y.) 326 96 (2011)

[18] I P McCulloch and M Gulácsi, Europhys Lett.

1. Improve phase transition detection in frustrated mixed-spin systems by using an SU(2)-symmetric density-matrix renormalization group (DMRG) where the total spin S is a quantum-number label. This allows for the identification of a direct collapse from S = L/4 to S = 0, with no partially polarized window as the mechanism for moment destruction.

2. Develop a tool to precisely map the nature of defect transitions by analyzing the sign of its finite-size correction in correlation functions like the transmission ratio, stating that the cut-chain law T ∝ L −1 is excluded on both sides. This enables distinguishing between a defect transition in a rigid commensurate host and host features.

3. Create an AI system capable of determining the exact nature of the singlet phase by measuring its critical exponents; specifically, it can calculate the central charge c ≃ 1.8 and the correlation exponent η = 1, allowing for characterization beyond a simple single SU(2) chain prediction.

4. Implement an AI system that identifies the precise location of defect transitions by analyzing the scaling of finite-size effects; this system can detect when the coupling at which that correction changes sign is itself a finite-size quantity, moving from 0.3076(43) measured on L = 80, 120 to 0.3263(18) on L = 120, 160 and back below 0.30.

5. Build a predictive model for defect behavior by correlating the position of the vacancy with the ground state sectors; this system can predict whether the triplet is bound to the vacancy based on whether the segment to its left of the vacancy holds an even number of unit cells.

The alternating spin- (1, 12) Heisenberg chain with a frustrating next-nearest-neighbor coupling J 2 loses its Lieb--Mattis moment through a discontinuous transition to a singlet whose nature was left open. Using an SU(2)-symmetric density-matrix renormalization group, in which the total spin is a quantum-number label, we identify that phase and follow a single vacancy through it. The total spin collapses directly from S=L/4 to S=0, with no partially polarized window. The singlet phase is critical: its singlet--triplet gap closes as 1/L, its entanglement entropy grows logarithmically with a central charge c 1.8, and its modulation wavevector, extrapolated from L=80 to 200, is commensurate at 3π/4 and independent of J 2 to 5 times10-5. Removing the central S=1 site switches the ground state of the doped chain from a triplet to a singlet at J 2=0.3068(3), where the host shows no feature; the triplet is not bound to the vacancy but spread over one of the two segments it leaves. The transmission of correlations through the vacancy saturates at a finite value below J 2, T(infinity)=0.102(15) and 0.036(9) at J 2=0.28 and 0.30, and the cut-chain law T proportional to L-1 is excluded on both sides. We also show that the longitudinal correlator is not an observable in a triplet sector.