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SciPost Physics

Quantum spin spiral ground state of the ferrimagnetic sawtooth chain

Roman Rausch, Matthias Peschke, Cassian Plorin, Jürgen Schnack, Christoph Karrasch

Published Mar 28, 2023• 9 citations
Publisher

Abstract

The ferrimagnetic phase of the sawtooth chain with mixed ferromagnetic nearest-neighbour interactions J J and antiferromagnetic next-nearest-neighbour interactions J' J ′ (within the isotropic Heisenberg model) was previously characterized as a phase with commensurate order. In this paper, we demonstrate that the system in fact exhibits an incommensurate quantum spin spiral. Even though the ground state is translationally invariant in terms of the local spin expectations ‹\vec{S}_i› < S ⃗ i > , the spiral can be detected via the connected spin-spin correlations ‹{\vec{S}_i\cdot\vec{S}_j}› < S ⃗ i ⋅ S ⃗ j > - ‹{\vec{S}_i}›\cdot‹{\vec{S}_j}› < S ⃗ i > ⋅ < S ⃗ j > between the apical spins. It has a long wavelength that grows with J' J ′ and that soon exceeds finite-system sizes typically employed in numerical simulations. A faithful treatment thus requires the use of state-of-the-art simulations for large, periodic systems. In this work, we are able to accurately treat up to L=400 L = 400 sites (200 unit cells) with periodic boundary conditions using the density-matrix renormaliztion group (DMRG). Exploiting the SU(2) symmetry allows us to directly compute the lowest-energy state for a given total spin. Our results are corroborated by variational uniform matrix product state (VUMPS) calculations, which work directly in the thermodynamic limit at the cost of a lower accuracy.

DOI
10.21468/scipostphys.14.3.052

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The ferrimagnetic phase of the sawtooth chain with mixed ferromagnetic nearest-neighbour interactions J J and antiferromagnetic next-nearest-neighbour interactions J' J ′ (within the isotropic Heisenberg model) was previously characterized as a phase with commensurate order. In this paper, we demonstrate that the system in fact exhibits an incommensurate quantum spin spiral. Even though the ground state is translationally invariant in terms of the local spin expectations ‹\vec{S}_i› < S ⃗ i > , the spiral can be detected via the connected spin-spin correlations ‹{\vec{S}_i\cdot\vec{S}_j}› < S ⃗ i ⋅ S ⃗ j > - ‹{\vec{S}_i}›\cdot‹{\vec{S}_j}› < S ⃗ i > ⋅ < S ⃗ j > between the apical spins. It has a long wavelength that grows with J' J ′ and that soon exceeds finite-system sizes typically employed in numerical simulations. A faithful treatment thus requires the use of state-of-the-art simulations for large, periodic systems. In this work, we are able to accurately treat up to L=400 L = 400 sites (200 unit cells) with periodic boundary conditions using the density-matrix renormaliztion group (DMRG). Exploiting the SU(2) symmetry allows us to directly compute the lowest-energy state for a given total spin. Our results are corroborated by variational uniform matrix product state (VUMPS) calculations, which work directly in the thermodynamic limit at the cost of a lower accuracy.
The ferrimagnetic phase of the sawtooth chain with mixed ferromagnetic nearest-neighbour interactions JJJ and antiferromagnetic next-nearest-neighbour interactions J′J'J′ (within the isotropic Heisenberg model) was previously characterized as a phase with commensurate order. In this paper, we demonstrate that the system in fact exhibits an incommensurate quantum spin spiral. Even though the ground state is translationally invariant in terms of the local spin expectations $\avg{\vec{S}_i}$, the spiral can be detected via the connected spin-spin correlations $\avg{\vec{S}_i\cdot\vec{S}_j}-\avg{\vec{S}_i}\cdot\avg{\vec{S}_j}$ between the apical spins. It has a long wavelength that grows with J′J'J′ and that soon exceeds finite-system sizes typically employed in numerical simulations. A faithful treatment thus requires the use of state-of-the-art simulations for large, periodic systems. In this work, we are able to accurately treat up to L=400L=400L=400 sites (200 unit cells) with periodic boundary conditions using the density-matrix renormaliztion group (DMRG). Exploiting the SU(2) symmetry allows us to directly compute the lowest-energy state for a given total spin. Our results are corroborated by variational uniform matrix product state (VUMPS) calculations, which work directly in the thermodynamic limit at the cost of a lower accuracy.
We study the low-energy properties of a sawtooth chain with spin-1's at the bases of the triangles and spin-12′s\frac{1}{2}\mathrm{'}\mathrm{s}21​′sat the vertices of the triangles. The spins have Heisenberg antiferromagnetic interactions between nearest neighbors, with a coupling J2{J}_{2}J2​ between a spin-1 and a spin-12,\frac{1}{2},21​, and a coupling J1=1{J}_{1}=1J1​=1 between two spin-1's. Analysis of the exact diagonalization data for periodic chains containing up to N=12N=12N=12 unit cells shows that the ground state is a singlet for exchange couplings up to approximately J2=3.8,{J}_{2}=3.8,J2​=3.8, whereas for larger J2{J}_{2}J2​ the system exhibits a ferrimagnetic ground state characterized by a net ferromagnetic moment per unit cell of 1/2. In the region of small interactions J2,{J}_{2},J2​, the mixed spin sawtooth chain maps on to an effective isotropic spin model representing two weakly interacting and frustrated spin-12\frac{1}{2}21​ Heisenberg chains composed of spin-12\frac{1}{2}21​ sites at odd and even vertices, respectively. Finally, we study the phenomenon of a macroscopic magnetization jump which occurs if a magnetic field is applied with a value close to the saturation field for J2=2.{J}_{2}=2.J2​=2.
The antiferromagnetic sawtooth chain is a prototypical example of a frustrated spin system with vertex-sharing triangles, giving rise to complex quantum states. Depending on the interaction parameters, this system has three phases, of which the gapless noncollinear phase (for strongly coupled basal spins and loosely attached apical spins) has received little theoretical attention so far. In this work, we comprehensively investigate the properties of the noncollinear phase using large-scale tensor network computations which exploit the full SU(2) symmetry of the underlying Heisenberg model. We study the ground state both for finite systems using the density-matrix renormalization group (DMRG) as well as for infinite chains via the variational uniform matrix-product state (VUMPS) formalism. Finite temperatures and correlation functions are tackled via imaginary or real time evolutions, which we implement using the time-dependent variational principle (TDVP). We find that the noncollinear phase is characterized by a low-momentum peak and a diffuse tail for the apex-apex correlations. Deep into the phase, the pattern sharpens into a peak indicating a 90 ∘ spiral. The apical spins are soft and highly susceptible to external perturbations; they give rise to a large number of gapless magnetic states that are polarized by weak fields and cause a long low-temperature tail in the specific heat. The dynamic spin-structure factor exhibits additive contributions from a two-spinon continuum (excitations of the basal chain) and a gapless peak at k = π / 2 (excitations of the apical spins). Small temperatures excite the gapless states and smear the spectral weight of the k = π / 2 peak out into a homogeneous flat-band structure. Our results are relevant, e.g., for the material atacamite Cu 2 Cl ( OH ) 3 in high magnetic fields.
Simulating strongly correlated systems with incommensurate order poses significant challenges for traditional finite-size-based approaches. Confining such a phase to a finite-size geometry can induce spurious frustration, with spin spirals in frustrated magnets being a typical example. Here, we introduce an Ansatz based on infinite projected entangled-pair states which overcomes these limitations and enables the direct search for the optimal spiral in the thermodynamic limit, with a computational cost that is independent of the spiral's wavelength. Leveraging this method, we simulate the Heisenberg model on the anisotropic triangular lattice, which interpolates between the square and isotropic triangular lattice limits. Besides accurately reproducing the magnetically ordered phases with arbitrary wavelength, the simulations reveal a quantum spin liquid phase emerging between the Néel and spin spiral phases.
We study the ground-state properties of weakly frustrated Heisenberg ferrimagnetic chains with nearest- and next-nearest-neighbor antiferromagnetic exchange interactions and two types of alternating sublattice spins S1>S2,{S}_{1}>{S}_{2},S1​>S2​, using 1/S1/S1/S spin-wave expansions, density-matrix renormalization group, and exact diagonalization techniques. It is argued that the zero-point spin fluctuations completely destroy the classical commensurate-incommensurate continuous transition. Instead, the long-range ferrimagnetic state disappears through a discontinuous transition to a singlet state at a larger value of the frustration parameter. In the ferrimagnetic phase we find a disorder point marking the onset of incommensurate real-space short-range spin-spin correlations.
Using the exact diagonalization technique, we study the properties of the ground state of a spin- antiferromagnetic Heisenberg model for a zigzag polymer chain with side radicals connected to the even sites. We consider the nearest-neighbour exchange J and the next-nearest-neighbour exchange ?J along the main chain, and J1 between the even site on the main chain and the radical site. For small ? the ground state is ferrimagnetic. For ?>?c 1, the ground state is a spiral phase, which is characterized by a peak of the static structure factor S(q) locating at an incommensurate value qmax. For ?>?c 2, the ground state is antiferromagnetic. With increasing J1, ?c 1 decreases while ?c 2 has a maximum at about J1 = 0.5. For very small J1 and ? = 0.5, the spin configuration on the main chain is a product of nearest-neighbour singlets. In the antiferromagnetic phase, if J1 is large enough the even site and the radical site form a singlet with exchange-decoupling from the odd site while the odd sites approximately form an antiferromagnetic chain.