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At least 91 records · Page 5
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DOE OSTI2025

Interacting spin and charge density waves in the kagome metal FeGe

Unveiling the interplay between spin density wave (SDW) and charge density wave (CDW) orders in correlated electron materials is important in obtaining a comprehensive understanding of their electronic, structural, and magnetic properties. Kagome lattice materials are interesting because their flat electronic bands, Dirac points, and Van Hove singularities can enable a variety of exotic electronic and magnetic phenomena. The kagome metal FeGe (the B35 phase), which exhibits a CDW order deep within an A-type antiferromagnetic (AFM) phase, was found to respond dramatically to postgrowth annealing—with the ability to tune the CDW repeatedly from long-range order to negligible order. Additionally, neutron scattering studies suggest that incommensurate magnetic peaks that onset at 𝑇 Canting = 𝑇 SDW ≈ 60 K in the system arise from a SDW order instead of the AFM double-cone structure. Here, in this study, we use inelastic neutron scattering to show that two distinct spin excitations exist below 𝑇 Canting corresponding to two coexisting magnetic orders in the system in both sets of annealed samples with and without CDW. While CDW order or negligible order can dramatically affect the onset temperature of 𝑇 Canting and elastic incommensurate magnetic scattering, its impact on low-energy spin fluctuations is more limited. In both samples, a pair of gapless incommensurate spin excitations arising from the SDW order wave vector coexist with gapped commensurate spin waves from the A-type AFM order across 𝑇 Canting . The low-energy spin excitations for both samples couple dynamically to the lattice through enhanced magnetic scattering intensity on cooling below 𝑇 CDW , regardless of the status of the static long-range CDW order. The incommensurate SDW order in the long-range CDW ordered sample also induces a tiny in-plane lattice distortion of the kagome lattice that is absent in the negligible CDW ordered sample, in a way that is different from the previously known SDW and CDW ordering materials.

charge density waves↗
DOE OSTI2025

Role of Nematic Fluctuations on Superconductivity in FeSe 0.47 ⁢Te 0.53 Revealed by Nuclear Magnetic Resonance under Pressure

The relationship between antiferromagnetic (AFM) spin fluctuations (SF), nematic fluctuations, and superconductivity (SC) has been central to understanding the pairing mechanism in iron-based superconductors (IBSCs). Iron chalcogenides, which hold the simplest crystal structure in IBSCs, provide a good platform to investigate the relationship. Here, we report 77 Se and 125 Te nuclear magnetic resonance studies of FeSe 0.47⁢ Te 0.53 , which is located close to a nematic quantum critical point (QCP), under pressures up to 1.35 GPa. Further, both the superconducting critical temperature and AFMSF were found to be enhanced under pressure, which suggests a correlation between SC and AFMSF in FeSe 0.47 ⁢Te 0.53 . However, the contribution of AFMSF to SC in FeSe 0.47 ⁢Te 0.53 was found to be much less compared to that in FeSe 1−𝑥 ⁢S 𝑥 , suggesting that nematic fluctuations play a dominant role in the SC in FeSe 1−𝑥⁢ Te 𝑥 around the nematic QCP.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Magnons and magnetic fluctuations in atomically thin MnBi 2 Te 4

Electron band topology is combined with intrinsic magnetic orders in MnBi 2 Te 4 , leading to novel quantum phases. Here we investigate collective spin excitations (i.e. magnons) and spin fluctuations in atomically thin MnBi 2 Te 4 flakes using Raman spectroscopy. In a two-septuple layer with non-trivial topology, magnon characteristics evolve as an external magnetic field tunes the ground state through three ordered phases: antiferromagnet, canted antiferromagnet, and ferromagnet. The Raman selection rules are determined by both the crystal symmetry and magnetic order while the magnon energy is determined by different interaction terms. Using non-interacting spin-wave theory, we extract the spin-wave gap at zero magnetic field, an anisotropy energy, and interlayer exchange in bilayers. We also find magnetic fluctuations increase with reduced thickness, which may contribute to a less robust magnetic order in single layers.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Pressure dependent magnetic properties on bulk CrBr 3 single crystals

The van der Waals class of materials offer an approach to two-dimensional magnetism as their spin fluctuations can be tuned upon exfoliation of layers. Moreover, it has recently been shown that spin-lattice coupling and long-range magnetic ordering can be modified with pressure in van der Waals materials. In this work, the magnetic properties of quasi two-dimensional CrBr 3 are reported applying hydrostatic pressure. The application of pressure up to 0.844 GPa shows a 1.77% decrease in saturation magnetization with a decrease in the Curie temperature from 33.05 to 30.41 K. Density functional theory calculations with pressure up to 1 GPa show a reduction in volume and interplanar distance as pressure increases. To further understand the magnetic properties with applied pressure, the magnetocrystalline anisotropy energy (MAE) and exchange coupling parameter (J) are calculated. There is small decrease in MAE and the first nearest neighbor interaction (J 1 ) (U = 2.7 eV and J = 0.7 eV) is increasing with respect to increasing pressure. Overall, CrBr 3 displays ferromagnetic interlayer coupling and the calculated exchange coupling and MAE parameters match well with the observations from the experimental work.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Interorbital nematicity and the origin of a single electron Fermi pocket in FeSe

The electronic structure of the enigmatic iron-based superconductor FeSe has puzzled researchers since spectroscopic probes failed to observe the expected electron pocket at the Y point in the 1-Fe Brillouin zone. It has been speculated that this pocket, essential for an understanding of the superconducting state, is either absent or incoherent. In this study, we perform a theoretical study of the preferred nematic order originating from nearest-neighbor Coulomb interactions in an electronic model relevant for FeSe. We find that at low temperatures the dominating nematic components are of interorbital d x z – d x y and d y z – d x y character, with spontaneously broken amplitudes for these two components. This interorbital nematic order naturally leads to distinct hybridization gaps at the X and Y points of the 1-Fe Brillouin zone, and may thereby produce highly anisotropic Fermi surfaces with only a single electron pocket at one of these momentum-space locations. The associated superconducting gap structure obtained with the generated low-energy electronic band structure from spin-fluctuation mediated pairing agrees well with that measured experimentally. Finally, from a comparison of the computed spin susceptibility to available neutron scattering data, we discuss the necessity of additional self-energy effects, and explore the role of orbital-dependent quasiparticle weights as a minimal means to include them.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Quasi-one-dimensional uniform spin- 1 2 Heisenberg antiferromagnet KNaCuP 2 O 7 probed by P 31 and Na 23 NMR

Here we present the structural and magnetic properties of KNaCuP 2 O 7 investigated via x-ray diffraction, magnetization, specific heat, and P 31 and Na 23 NMR measurements and complementary electronic structure calculations. The temperature-dependent magnetic susceptibility and P 31 NMR shift could be modeled very well by the uniform spin- 1 2 Heisenberg antiferromagnetic chain model with a nearest-neighbor interaction J / k B ≃ 58.7 K . The corresponding mapping using first-principles electronic structure calculations leads to J DFT / k B ≃ 59 K with negligibly small interchain couplings, further confirming that the system is indeed a one-dimensional uniform spin- 1 2 Heisenberg antiferromagnet. The diverging trend of NMR spin-lattice relaxation rates ( 1 / T 1 31 and 1 / T 1 23 ) implies the onset of a magnetic long-range ordering at around T N ≃ 1 K . From the value of T N , the average interchain coupling is estimated to be J ' / k B ≃ 0.28 K . Moreover, the NMR spin-lattice relaxation rates show the dominant contributions from uniform ( q = 0 ) and staggered ( q = ± π / a ) spin fluctuations in the high- and low-temperature regimes, respectively, mimicking one-dimensionality of the spin lattice. We have also demonstrated that 1 / T 1 31 in high temperatures varies linearly with 1 / H , reflecting the effect of spin diffusion on the dynamic susceptibility. The temperature-dependent unit cell volume could be described well using the Debye approximation with a Debye temperature of Θ D ≃ 294 K , consistent with the heat capacity data.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Magnetic order and fluctuations in the quasi-two-dimensional planar magnet Sr( Co 1-x Ni x ) 2 As 2

We use neutron scattering to investigate spin excitations in Sr ( Co 1 - x Ni x ) 2 As 2 , which has a c -axis incommensurate helical structure of the two-dimensional (2D) in-plane ferromagnetic (FM) ordered layers for 0.013 ≤ x ≤ 0.25 . By comparing the wave vector and energy dependent spin excitations in helical ordered Sr ( Co 0.9 Ni 0.1 ) 2 As 2 and paramagnetic SrCo 2 As 2 , we find that Ni doping, while increasing lattice disorder in Sr ( Co 1 - x Ni x ) 2 As 2 , enhances quasi-2D FM spin fluctuations. However, our band structure calculations within the combined density functional theory and dynamic mean field theory ( DFT + DMFT ) failed to generate a correct incommensurate wave vector for the observed helical order from nested Fermi surfaces. Furthermore, since transport measurements reveal increased in-plane and c -axis electrical resistivity with increasing Ni doping and associated lattice disorder, we conclude that the helical magnetic order in Sr ( Co 1 - x Ni x ) 2 As 2 may arise from a quantum order-by-disorder mechanism through the itinerant electron mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Molecular beam epitaxy synthesis and electrical transport properties of the correlated kagome metal Ni⁢ 3 In

Ni 3 ⁢In is a paramagnetic intermetallic consisting of 𝐴⁢𝐵-stacked Ni-kagome networks. Correlated electron behaviors deviating from the Fermi-liquid form have recently been observed in Ni 3 ⁢In bulk single crystals, attributed to stabilization of a partially flat electronic band near the Fermi level. Synthesis of this system in thin-film form offers unique opportunities for tuning of materials that could aid in identifying the microscopic origin of the non-Fermi-liquid response and exploring the suspected quantum criticality therein. Here, we report the realization of (001)-oriented epitaxial thin films of Ni 3 ⁢In on single-crystal SrTiO 3 (111) substrates by molecular beam epitaxy. Via control of growth conditions, we fabricate high-quality films with quantum fluctuations strongly influencing the physical properties of the system. Analysis of the electrical transport response reveals that intrinsic spin fluctuations in Ni 3 ⁢In may account for the observed non-Fermi-liquid behavior. Such structures may facilitate driving Ni 3 ⁢In across a potential quantum critical phase transition and uncover the role of unusual flat bands in triggering correlated phenomena.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Inhomogeneous magnetic ordered state and evolution of magnetic fluctuations in Sr⁢(Co 1–x ⁢Ni x ) 2 ⁢P 2 revealed by 31 P NMR

SrCo 2 ⁢P 2 with a tetragonal structure is known to be a Stoner-enhanced Pauli paramagnetic metal being nearly ferromagnetic. Recently J. Schmidt et al. [Phys. Rev. B 108, 174415 (2023)] reported that a ferromagnetic ordered state is actually induced by a small Ni substitution for Co of x = 0.02 in Sr⁢(Co 1–x ⁢Ni x ) 2 ⁢P 2 where an antiferromagnetic ordered phase also appears by further Ni substitution with x = 0.06–0.35. Here, in this work, using nuclear magnetic resonance (NMR) measurements on 31 P nuclei, we have investigated how the magnetic properties change by the Ni substitution in Sr⁢(Co 1–x ⁢Ni x ) 2 ⁢P 2 from a microscopic point of view, especially focusing on the evolution of magnetic fluctuations with the Ni substitution and the characterization of the magnetically ordered states. The temperature dependencies of the 31 P spin-lattice relaxation rate divided by temperature (1/T 1 ⁢T) and Knight shift (K) for SrCo 2 ⁢P 2 are reasonably explained by a model where a double-peak structure for the density of states near the Fermi energy is assumed. Based on a Korringa ratio analysis using the T 1 and K data, ferromagnetic spin fluctuations are found to dominate in the ferromagnetic Sr⁢(Co 1–x ⁢Ni x ) 2 ⁢P 2 as well as the antiferromagnets where no clear antiferromagnetic fluctuations are observed. We also found the distribution of the ordered Co moments in the magnetically ordered states from the analysis of the 31 P-NMR spectra exhibiting a characteristic rectangular-like shape.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Superdiffusion from Nonabelian Symmetries in Nearly Integrable Systems

The Heisenberg spin chain is a canonical integrable model. As such, it features stable ballistically propagating quasiparticles, but spin transport is subballistic at any nonzero temperature: An initially localized spin fluctuation spreads in time t to a width t 2/3 . This exponent as well as the functional form of the dynamical spin correlation function suggest that spin transport is in the Kardar–Parisi–Zhang (KPZ) universality class. However, the full counting statistics of magnetization is manifestly incompatible with KPZ scaling. A simple two-mode hydrodynamic description, derivable from microscopic principles, captures both the KPZ scaling of the correlation function and the coarse features of the full counting statistics, but remains to be numerically validated. These results generalize to any integrable spin chain invariant under a continuous nonabelian symmetry and are surprisingly robust against moderately strong integrability-breaking perturbations that respect the nonabelian symmetry.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

EuAuSb: An odd-parity helical variation of altermagnetism

EuAuSb is a triangular-lattice Dirac semimetal in which a topological Hall effect has been observed to develop in association with a magnetically ordered phase. Our single-crystal neutron diffraction measurements have identified an incommensurate helical order in which individual ferromagnetic Eu 2+ layers rotate in-plane by ∼ 120° from one layer to the next. An in-plane magnetic field distorts the incommensurate order, eventually leading to a first order transition to a state that is approximately commensurate and that is continuously polarized as the bulk magnetization approaches saturation. From an analysis of the magnetic diffraction intensities versus field, we find evidence for a dip in the ordered in-plane moment at the same field where the topological Hall effect is a maximum, and we propose that this is due to field-induced quantum spin fluctuations. Our electronic structure calculations yield exchange constants compatible with the helical order and show that the bands near the Fermi level lose their spin degeneracy via a mechanism similar to that in the collinear altermagnets. In conclusion, we find that, unlike the even symmetry seen in the altermagnets, the spin splitting in EuAuSb has odd-wave symmetry similar to that recently found in a number of coplanar magnetic materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Electronic Structure Correspondence of Singlet-Triplet Scale Separation in Strained Sr2RuO4

At a temperature of roughly 1 K, Sr 2 RuO 4 undergoes a transition from a normal Fermi liquid to a superconducting phase. Even while the former is relatively simple and well understood, the superconducting state has not even been understood after 25 years of study. More recently, it has been found that critical temperatures can be enhanced by the application of uniaxial strain, up to a critical strain, after which it falls off. In this work, we take an “instability” approach and seek divergences in susceptibilities. This provides an unbiased way to distinguish tendencies to competing ground states. We show that in the unstrained compound, the singlet and triplet instabilities of the normal Fermi liquid phase are closely spaced. Under uniaxial strain, electrons residing on all orbitals contributing to the Fermiology become more coherent, while the electrons of the Ru-dxy character become heavier, and the electrons of the Ru d x z , y z characters become lighter. In the process, Im χ(q,ω) increases rapidly around q = ( 0.3 , 0.3 , 0 ) 2 π / a and q = ( 0.5 , 0.25 , 0 ) 2 π / a , while it gets suppressed at all other commensurate vectors, in particular at q = 0, which is essential for spin-triplet superconductivity. We observe that the magnetic anisotropy under strain drops smoothly, which is concomitant with the increment in singlet instability. Thus, the triplet superconducting instability remains the lagging instability of the system, and the singlet instability enhances under strain, leading to a large energy-scale separation between these competing instabilities. However, since this happens even without spin-orbit coupling, we believe it is primarily the enhancement in the spin fluctuation glue around quasi-anti-ferromagnetic vectors that drives the Cooper pairing instead of the magnetic anisotropy. At large strain, an instability to a spin density wave overtakes the superconducting one. The analysis relies on a high-fidelity, ab initio description of the one-particle properties and two-particle susceptibilities, based on the quasiparticle self-consistent GW approximation augmented by dynamical mean field theory. This approach is described and its high fidelity confirmed by comparing to observed one- and two-particle properties.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Anisotropic 2D van der Waals Magnets Hosting 1D Spin Chains

Abstract The exploration of 1D magnetism, frequently portrayed as spin chains, constitutes an actively pursued research field that illuminates fundamental principles in many‐body problems and applications in magnonics and spintronics. The inherent reduction in dimensionality often leads to robust spin fluctuations, impacting magnetic ordering and resulting in novel magnetic phenomena. Here, structural, magnetic, and optical properties of highly anisotropic 2D van der Waals antiferromagnets that uniquely host spin chains are explored. First‐principle calculations reveal that the weakest interaction is interchain, leading to essentially 1D magnetic behavior in each layer. With the additional degree of freedom arising from its anisotropic structure, the structure is engineered by alloying, varying the 1D spin chain lengths using electron beam irradiation, or twisting for localized patterning, and spin textures are calculated, predicting robust stability of the antiferromagnetic ordering. Comparing with other spin chain magnets, these materials are anticipated to bring fresh perspectives on harvesting low‐dimensional magnetism.

1D magnetism↗
DOE OSTI2022

Intertwined spin, charge, and pair correlations in the two-dimensional Hubbard model in the thermodynamic limit

Significance The high-temperature superconducting cuprates are governed by intertwined striped magnetic and charge orders, in addition to superconductivity. Remarkably similar behavior has also been seen in numerical calculations for the Hubbard model describing the copper–oxygen layers in these materials. Finite-cluster methods typically find that spin- and charge-stripe order dominates, while embedded quantum-cluster methods, which access the thermodynamic limit, often conclude that superconductivity does. Here, we report the observation of fluctuating spin and charge stripes in an embedded cluster calculation for the Hubbard model. This discovery demonstrates that striped states survive in the thermodynamic limit and allows us to study their influence on the model’s superconducting properties, where we find evidence for pair-density-wave correlations intertwined with the stripe correlations.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

Pulling Order Back from the Brink of Disorder: Observation of a Nodal-Line Spin Liquid and Fluctuation Stabilized Order in K 2 ⁢IrCl 6

Competing interactions in frustrated magnets can give rise to highly degenerate ground states from which correlated liquidlike states of matter often emerge. The scaling of this degeneracy influences the ultimate ground state, with extensive degeneracies potentially yielding quantum spin liquids, while subextensive or smaller degeneracies yield static orders. A long-standing problem is to understand how ordered states precipitate from this degenerate manifold and what echoes of the degeneracy survive ordering. Here, we use neutron scattering to experimentally demonstrate a new “nodal-line” spin liquid, where spins collectively fluctuate within a subextensive manifold spanning one-dimensional lines in reciprocal space. Realized in the spin-orbit-coupled, face-centered-cubic iridate K 2 ⁢IrCl 6 , we show that the subextensive degeneracy is robust, but remains susceptible to fluctuations or longer-range interactions which cooperate to select a magnetic order at low temperatures. Proximity to the nodal-line spin liquid influences the ordered state, enhancing the effects of quantum fluctuations that in turn act to stabilize the sublattice magnetization through the self-consistent opening of a large spin-wave gap. Our results demonstrate how quantum fluctuations can act counterintuitively in frustrated materials: Even in a case where fluctuations are ineffective at selecting an ordered state from a degenerate manifold, at the brink of the nodal spin liquid, they can act to protect the ordered state and dictate its low-energy physics.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Monte Carlo study of the pseudogap and superconductivity emerging from quantum magnetic fluctuations

Abstract The origin of the pseudogap behavior, found in many high- T c superconductors, remains one of the greatest puzzles in condensed matter physics. One possible mechanism is fermionic incoherence, which near a quantum critical point allows pair formation but suppresses superconductivity. Employing quantum Monte Carlo simulations of a model of itinerant fermions coupled to ferromagnetic spin fluctuations, represented by a quantum rotor, we report numerical evidence of pseudogap behavior, emerging from pairing fluctuations in a quantum-critical non-Fermi liquid. Specifically, we observe enhanced pairing fluctuations and a partial gap opening in the fermionic spectrum. However, the system remains non-superconducting until reaching a much lower temperature. In the pseudogap regime the system displays a “gap-filling" rather than “gap-closing" behavior, similar to the one observed in cuprate superconductors. Our results present direct evidence of the pseudogap state, driven by superconducting fluctuations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Static and dynamic magnetic properties of the spiin-$\frac{5}{2}$triangle lattice antiferromagnet Na 3 Fe(PO 4 ) 2 studied by 31 P NMR

We report 31 P nuclear magnetic resonance (NMR) measurements have been carried out to investigate the magnetic properties and spin dynamics of Fe 3+ (S = 5/2) spins in the two-dimensional triangular lattice (TL) compound Na 3 Fe(PO 4 ) 2 . The temperature (T) dependence of nuclear spin-lattice relaxation rates (1/T 1 ) shows a clear peak around Néel temperature, T N = 10.9 K, corresponding to an antiferromagnetic (AFM) transition. From the temperature dependence of NMR shift (K) above T N , an exchange coupling between Fe 3+ spins was estimated to be J/k B ≃ 1.9 K using the spin-5/2 Heisenberg isotropic-TL model. The temperature dependence of 1/T 1 T divided by the magnetic susceptibility (χ), 1/T 1 T χ , above T N proves the AFM nature of spin fluctuations below ~50 K in the paramagnetic state. In the magnetically ordered state below T N , the characteristic rectangular shape of the NMR spectra is observed, indicative of a commensurate AFM state in its ground state. The strong temperature dependence of 1/T 1 in the AFM state is well explained by the two-magnon (Raman) process of the spin waves in a 3D antiferromagnet with a spin-anisotropy energy gap of 5.7 K. The temperature dependence of sublattice magnetization is also well reproduced by the spin waves. Those results indicate that the magnetically ordered state of Na 3 Fe(PO 4 ) 2 is a conventional 3D AFM state, and no obvious spin frustration effects were detected in its ground state

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Antiferro- and metamagnetism in the S = 7 / 2 hollandite analog Eu Ga 2 Sb 2

Recent work analyzing the impact of nonsymmorphic symmetries on electronic states has given rise to the discovery of multiple types of topological matter. Here we report the single-crystal synthesis and magnetic properties of Eu Ga 2 Sb 2 , a Eu-based antiferromagnet structurally consisting of pseudo-1D chains of Eu ions related by a nonsymmorphic glide plane. Here we find the onset of antiferromagnetic order at T N = 8K. Above T N the magnetic susceptibility is isotropic. Curie-Weiss analysis suggests competing ferromagnetic and antiferromagnetic interactions, with p eff = 8.1 μ B as expected for 4f 7 J = S = 7/2 Eu 2+ ions. Below T N and at low applied magnetic fields, an anisotropy develops linearly, reaching χ ⊥ /χ ∥ = 6 at T = 2K. There is concomitant metamagnetic behavior along χ ∥ , with a magnetic field of μ 0 H ≈ 0.5 T sufficient to suppress the anisotropy. Independent of crystal orientation, there is a continuous evolution to a field-polarized paramagnetic state with M = 7μ B /Eu 2+ at μ 0 H = 2 T as T → 0 K. Specific-heat measurements show a recovered magnetic entropy of ΔS mag ≈ 16.4 Jmol –1 K –1 from T ~ 0 K to T = T N , close to the expected value of R ln(8) for an S = 7/2 ion, indicating negligible low-dimensional spin fluctuations above T N . We find no evidence of unusual behaviors arising either from the dimensionality or the presence of the nonsymmorphic symmetries.

1-dimensional spin chains↗