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arXiv:2402.19311 · Physical review. B./Physical review. B

Anisotropy-induced spin parity effects

Shuntaro Sumita, Akihiro Tanaka, Yusuke S. Kato

Published Sep 6, 2024• 0 citations
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Abstract

Spin parity effects refer to those special situations where a dichotomy in the physical behavior of a system arises, solely depending on whether the relevant spin quantum number is integral or half-odd integral. As is the case with the Haldane conjecture in antiferromagnetic spin chains, their pursuit often derives deep insights and invokes new developments in quantum condensed matter physics. Here, we put forth a simple and general scheme for generating such effects in any spatial dimension through the use of anisotropic interactions, and a setup within reasonable reach of state-of-the-art cold-atom implementations. We demonstrate its utility through a detailed analysis of the magnetization behavior of a specific one-dimensional spin chain model, an anisotropic antiferromagnet in a transverse magnetic field, unraveling along the way the quantum origin of finite-size effects observed in the magnetization curve that had previously been noted but not clearly understood.

DOI
10.1103/physrevb.110.l100403

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Spin parity effects refer to those special situations where a dichotomy in the physical behavior of a system arises, solely depending on whether the relevant spin quantum number is integral or half-odd integral. As is the case with the Haldane conjecture in antiferromagnetic spin chains, their pursuit often derives deep insights and invokes new developments in quantum condensed matter physics. Here, we put forth a simple and general scheme for generating such effects in any spatial dimension through the use of anisotropic interactions, and a setup within reasonable reach of state-of-the-art cold-atom implementations. We demonstrate its utility through a detailed analysis of the magnetization behavior of a specific one-dimensional spin chain model, an anisotropic antiferromagnet in a transverse magnetic field, unraveling along the way the quantum origin of finite-size effects observed in the magnetization curve that had previously been noted but not clearly understood.
While spin parity effects, physics crucially depending on whether the spin quantum number SSS is half-odd integral or integral, have for decades been a source of new developments for the quantum physics of antiferromagnetic spin chains, the investigation into their possible ferromagnetic counterparts has remained largely unchartered, especially in the fully quantum (as opposed to the semiclassical) regime. Here we present such studies for monoaxial chiral ferromagnetic spin chains. We start by examining magnetization curves for finite-sized systems, where a magnetic field is applied perpendicular to the helical axis. For half-odd integer SSS, the curves feature discontinuous jumps identified as a series of level crossings, each accompanied by a shift of the crystal momentum kkk by an amount of $\ensuremath{\pi}$. The corresponding curves for integer valued SSS are continuous and exhibit crossover processes. For the latter case, k=0k=0k=0 throughout. These characteristics are observed numerically when the strength of the Dzyaloshinskii-Moriya interaction (DMI) DDD is comparable to or larger than that of the ferromagnetic exchange interaction JJJ. Solitons are known to be responsible for stepwise changes seen in magnetization curves in the classical limit. These findings therefore prompt us to revise the notion of a soliton, for arbitrary SSS, into a quantum mechanical entity. To unravel this phenomenon at the fully quantum level as is appropriate to spin chains with small SSS, we examine in detail special limiting Hamiltonians amenable to rigorous analysis, consisting of only the DMI and the Zeeman energy. Dubbed the DHDHDH model (for S=12S=\frac{1}{2}S=21​) and the projected DHDHDH (pDH\mathrm{p}DHpDH) model (for general SSS), they have a set of 2S2S2S conserved quantities, each of which is the number of solitons of a specific integer-valued height (as measured in the Sz{S}_{z}Sz​ basis), which ranges from 1 to 2S2S2S. We discuss how to determine the exact crystal momentum of the lowest-energy state belonging to a sector with a given set of the 2S2S2S soliton numbers. Combined with energetic considerations, this information enables us to reproduce the spin parity effect in the magnetization curves. Finally, we show that the ground states of the special models have substantial numerical overlap with those for generic systems with a finite exchange interaction, suggesting the same physics to be valid there as well.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
In Heisenberg models with exchange anisotropy, transverse spin components are not conserved and can decay not only by transport, but also by dephasing.Here, we utilize ultracold atoms to simulate the dynamics of 1D Heisenberg spin chains and observe fast, local spin decay controlled by the anisotropy.However, even for isotropic interactions, we observe dephasing due to a new effect: an effective magnetic field created by superexchange.If spatially uniform, it leads only to uniform spin precession and is, therefore, typically ignored.However, we show through experimental studies and extensive numerical simulations how this superexchange-generated field is relevant and leads to additional dephasing mechanisms over the exchange anisotropy: There is dephasing due to (i) inhomogeneity of the effective field from variations of lattice depth between chains; (ii) a twofold reduction of the field at the edges of finite chains; and (iii) fluctuations of the effective field due to the presence of mobile holes in the system.The latter is a new coupling mechanism between holes and magnons.All these dephasing mechanisms have not been observed before with ultracold atoms and illustrate basic properties of the underlying Hubbard model.
Low dimensionality enhances quantum fluctuations, triggering novel states of quantum matter to emerge. In real materials, low dimensionality usually arises from spatially strongly anisotropic couplings. Here, we demonstrate a different mechanism: in two-dimensional systems with coupled alternating ferromagnetic (FM) and antiferromagnetic (AFM) spin-1/21/21/2 chains, an applied magnetic field may drive a dimensional reduction. Under magnetic field, the FM chains polarize and stiffen, suppressing the propagation of transverse AFM fluctuations from one chain to another, and effectively induce one-dimensional behavior at low energies. For a model describing botallackite, Cu2_22​(OH)3_33​Br, quantum Monte Carlo dynamics show that beyond a critical magnetic field, the low-energy spectrum reduces to that of a one-dimensional AFM Heisenberg spin-1/21/21/2 chain with field-dependent incommensurate two-spinon fluctuations, providing clear signatures for inelastic neutron scattering.
Finite-size quantum spin systems can be magnetized by the application of a symmetry-breaking field, but in general their symmetry is expected to be restored once the field is turned off adiabatically. Recently (F. Caleca et al., arXiv:2412.15493) we have shown that systems of half-integer spins with an odd number of sites and a parity-preserving Hamiltonian can retain a finite magnetization, hence exhibiting spontaneous symmetry breaking (SSB) at finite size. Here we generalize this phenomenon to spin chains whose low-energy physics (in zero field) realizes a Luttinger-liquid phase. We observe that odd-sized chains can exhibit a phenomenon of finite-size quasi-SSB, in which a net sub-extensive magnetization, M∼N1−1/(4K)M \sim N^{1-1/(4K)}M∼N1−1/(4K) is retained, where NNN is the number of sites and KKK the Luttinger exponent. Interestingly, the states prepared by turning off the symmetry-breaking field quasi-adiabatically display scalable spin squeezing -- namely stronger the bigger the system -- regardless of the parity of NNN. The scaling of the squeezing parameter is dictated again by the Luttinger exponent, ξR2∼N−1+1/(2K)ξ_R^2 \sim N^{-1+1/(2K)}ξR2​∼N−1+1/(2K). This result shows that scalable quantum correlations with metrological significance, associated typically with high-dimensional systems, can be found as well in gapless one-dimensional ones; and they are a direct consequence of the critical nature of Luttinger liquids.
We study the effect of symmetry-breaking perturbations in the one-dimensional SU(4)\mathrm{SU}(4)SU(4) spin-orbital model. We allow the exchange in spin (J1){(J}_{1})(J1​) and orbital (J2){(J}_{2})(J2​) channel to be different and thus reduce the symmetry to $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2).$ A magnetic field hhh along the Sz{S}^{z}Sz direction is also applied. Using the formalism developed by Azaria et al. [Phys. Rev. Lett. 83,83,83, 624 (1999)] we extend their analysis of the isotropic J1=J2,{J}_{1}{=J}_{2},J1​=J2​, h=0h=0h=0 case and obtain the low-energy effective theory near the SU(4)\mathrm{SU}(4)SU(4) point in the generic case ${J}_{1}\ensuremath{\ne}{J}_{2},$ $h\ensuremath{\ne}0.$ In zero magnetic field, we retrieve the same qualitative low-energy physics as in the isotropic case. In particular, the isotropic massless behavior found on the line J1=J2<K/4{J}_{1}{=J}_{2}<K/4J1​=J2​<K/4 extends in a large anisotropic region. We discover, however, that the anisotropy plays its trick in allowing nontrivial scaling behaviors of the physical quantities. For example, the mass gap MMM has two different scaling behaviors depending on the anisotropy. In addition, we show that in some regions, the anisotropy is responsible for anomalous finite-size effects and may change qualitatively the shape of the computed critical line in a finite system. When a magnetic field is present the effect of the anisotropy is striking. In addition to the usual commensurate-incommensurate phase transition that occurs in the spin sector of the theory, we find that the field may induce a second transition of the KT type in the remaining degrees of freedom to which it does not couple directly. In this sector, we find that the effective theory is that of an SO(4) Gross-Neveu model with an $h\ensuremath{-}\mathrm{dependent}$ coupling that may change its sign as hhh varies.