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arXiv:2003.00554 · Physical review. B./Physical review. B

Roles of easy-plane and easy-axis XXZ anisotropy and bond alternation in a frustrated ferromagnetic spin- 12 chain

Hiroshi Ueda, Shigeki Onoda

Published Jun 30, 2020• 9 citations
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Abstract

The spin-$1/2$ Heisenberg chain with a ferromagnetic first-neighbor exchange coupling ${J}_{1}$ and an antiferromagnetic second-neighbor ${J}_{2}$ has a Haldane dimer ground state with an extremely small spin gap. Thus the ground state is readily altered by perturbations. Here, we investigate the effects of XXZ exchange magnetic anisotropy of both the easy-axis and easy-plane types and an alternation in ${J}_{1}$ on the ground state, the spin gap, and magnetic properties of the frustrated ferromagnetic spin-$1/2$ chain. It is found that there are two distinct dimerized spin-gap phases, in one of which the spin gap and the magnetic susceptibility are extremely small around the SU(2) symmetric case and in the other they are moderately large far away from the SU(2) symmetric case. A small alternation in the amplitude of ${J}_{1}$ rapidly shortens the pitch of spin correlations towards the four-spin periodicity, as in the limit of ${J}_{1}/{J}_{2}\ensuremath{\rightarrow}0$. These effects are not sufficient to quantitatively explain overall experimentally observed magnetic properties in the quasi-one-dimensional spin-gapped magnetoelectric cuprate ${\mathrm{Rb}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}$ that exhibits ferroelectricity stabilized by a magnetic field. Our results are also relevant to ${\mathrm{Cs}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}$, where the ferromagnetic intrachain and antiferromagnetic interchain order has recently been found, in a single chain level. We also reveal the nature of symmetry-protected topological phase transitions in the model by mapping onto effective spin-1 chain models.

DOI
10.1103/physrevb.101.224439

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The spin-1/21/21/2 Heisenberg chain with a ferromagnetic first-neighbor exchange coupling J1{J}_{1}J1​ and an antiferromagnetic second-neighbor J2{J}_{2}J2​ has a Haldane dimer ground state with an extremely small spin gap. Thus the ground state is readily altered by perturbations. Here, we investigate the effects of XXZ exchange magnetic anisotropy of both the easy-axis and easy-plane types and an alternation in J1{J}_{1}J1​ on the ground state, the spin gap, and magnetic properties of the frustrated ferromagnetic spin-1/21/21/2 chain. It is found that there are two distinct dimerized spin-gap phases, in one of which the spin gap and the magnetic susceptibility are extremely small around the SU(2) symmetric case and in the other they are moderately large far away from the SU(2) symmetric case. A small alternation in the amplitude of J1{J}_{1}J1​ rapidly shortens the pitch of spin correlations towards the four-spin periodicity, as in the limit of J1/J2\ensuremath→0{J}_{1}/{J}_{2}\ensuremath{\rightarrow}0J1​/J2​\ensuremath→0. These effects are not sufficient to quantitatively explain overall experimentally observed magnetic properties in the quasi-one-dimensional spin-gapped magnetoelectric cuprate Rb2Cu2Mo3O12{\mathrm{Rb}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}Rb2​Cu2​Mo3​O12​ that exhibits ferroelectricity stabilized by a magnetic field. Our results are also relevant to Cs2Cu2Mo3O12{\mathrm{Cs}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}Cs2​Cu2​Mo3​O12​, where the ferromagnetic intrachain and antiferromagnetic interchain order has recently been found, in a single chain level. We also reveal the nature of symmetry-protected topological phase transitions in the model by mapping onto effective spin-1 chain models.
The ground-state phase diagram of a spin-12\frac{1}{2}21​ XXZ chain with competing ferromagnetic nearest-neighbor (J1<0{J}_{1}<0J1​<0) and antiferromagnetic second-neighbor (J2>0{J}_{2}>0J2​>0) exchange couplings is studied by means of the infinite time evolving block decimation algorithm and effective field theories. For the SU(2)-symmetric (Heisenberg) case, we show that the nonmagnetic phase in the range $\ensuremath{-}4<{J}_{1}/{J}_{2}<0$ has a small but finite ferromagnetic dimer order. We argue that this spontaneous dimer order is associated with effective spin-1 degrees of freedom on dimerized bonds, which collectively form a valence bond solid state as in the spin-1 antiferromagnetic Heisenberg chain (the Haldane spin chain). We thus call this phase the Haldane dimer phase. With easy-plane anisotropy, the model exhibits a variety of phases including the vector chiral phase with gapless excitations and the even-parity dimer and N\'eel phases with gapped excitations, in addition to the Haldane dimer phase. Furthermore, we show the existence of gapped phases with coexisting orders in narrow regions that intervene between the gapless chiral phase and any one of Haldane dimer, even-parity dimer, and N\'eel phases. Possible implications for quasi-one-dimensional edge-sharing cuprates are discussed.
A frustrated spin-1/21/21/2 XXZ chain model comprising a ferromagnetic nearest-neighbor coupling with the bond alternation, ${J}_{1}(1\ifmmode\pm\else\textpm\fi{}\ensuremath{\delta})<0$, and an antiferromagnetic second-neighbor exchange coupling J2>0{J}_{2}>0J2​>0 is studied at zero and weak magnetic fields by means of density-matrix renormalization-group calculations of order parameters, correlation functions, and the entanglement entropy, as well as an Abelian bosonization analysis. At zero magnetic field, the bond alternation $\ensuremath{\delta}>0$ suppresses the gapless phase characterized by a vector-chiral (VC) long-range order (LRO) and a quasi-LRO of an incommensurate spin spiral, whereas this phase occupies a large region in the space of J1/J2{J}_{1}/{J}_{2}J1​/J2​ and the easy-plane exchange anisotropy for $\ensuremath{\delta}=0$ [S. Furukawa et al., Phys. Rev. Lett. 105, 257205 (2010)]. Then, four gapped phases are found to appear as the exchange anisotropy varies from the SU(2)-symmetric case to the U(1)-symmetric case: the Haldane dimer (D+{}_{+}+​) phase with the same sign of the x,yx,yx,y- and zzz-component dimer order parameters, two VC dimer (VCD+{}_{+}+​/VCD${}_{\ensuremath{-}}$) phases with the sign of the zzz-component dimer order parameter being unaltered/reversed, and the even-parity dimer (D${}_{\ensuremath{-}}$) phase. At small magnetic fields, a field-induced ring-exchange interaction, which is proportional to a staggered scalar chirality and a magnetic flux penetrating the associated triangle, drives a transition from the D${}_{\ensuremath{-}}$ phase into a VC-Neel-dimer (VCND) phase, but not from the D+{}_{+}+​ phase. This VCND phase is stable up to the large magnetic field at which the Zeeman term closes the spin gap. A possible relevance to Rb2{}_{2}2​Cu2{}_{2}2​Mo3{}_{3}3​O12{}_{12}12​ is discussed.
We study the ground-state phase diagram of a one-dimensional spin-12\frac{1}{2}21​ easy-plane XXZ model with a ferromagnetic nearest-neighbor (NN) coupling J1{J}_{1}J1​ and a competing next-nearest-neighbor (NNN) antiferromagnetic coupling J2{J}_{2}J2​ in the parameter range 0<J2/∣J1∣<0.40<{J}_{2}/|{J}_{1}|<0.40<J2​/∣J1​∣<0.4. When ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}1/4$, the model is in a Tomonaga-Luttinger liquid phase which is adiabatically connected to the critical phase of the XXZ model of J2=0{J}_{2}=0J2​=0. On the basis of the effective (sine-Gordon) theory and numerical analyses of low-lying energy levels of finite-size systems, we show that the NNN coupling induces phase transitions from the Tomonaga-Luttinger liquid to gapped phases with either N\'eel or dimer order. Interestingly, these two types of ordered phases appear alternately as the easy-plane anisotropy is changed toward the isotropic limit. The appearance of the antiferromagnetic (N\'eel) order in this model is remarkable, as it is strongly unfavored by both the easy-plane ferromagnetic NN coupling and the antiferromagnetic NNN coupling in the classical-spin picture. We argue that emergent trimer degrees of freedom play a crucial role in the formation of the N\'eel order.
A frustrated spin-12\frac{1}{2}21​ XXZ zigzag chain relevant to Rb2Cu2Mo3O12{\mathrm{Rb}}_{2}{\mathrm{Cu}}_{2}{\mathrm{Mo}}_{3}{\mathrm{O}}_{12}Rb2​Cu2​Mo3​O12​ is revisited in the light of symmetry-protected topological (SPT) phases. Using a density-matrix renormalization group method for infinite systems, we identify projective representations for four distinct time-reversal invariant SPT phases; two parity-symmetric dimer phases near the Heisenberg and XX limits and two parity-broken vector-chiral (VC) dimer phases in between. A small bond alternation in the nearest-neighbor ferromagnetic exchange coupling induces a direct SPT transition between the two distinct VC dimer phases. It is also found numerically that two Berezinskii-Kosterlitz-Thouless transitions, which occur from the gapless to the two distinct gapped VC phases in the case of $\ensuremath{\delta}=0$, meet each other in the case of $\ensuremath{\delta}>0$ at a Gaussian criticality of the same Tomonaga-Luttinger parameter value as in the SU(2)-symmetric case.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
Exact diagonalization of finite spin-1/2 chains with periodic boundary conditions is applied to the ground state (GS) of chains with ferromagnetic (F) exchange J1<0{J}_{1}<0J1​<0 between first neighbors, antiferromagnetic (AF) exchange ${J}_{2}=\ensuremath{\alpha}{J}_{1}>0$ between second neighbors, and axial anisotropy $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$. In zero field, the GS is in the Sz=0{S}_{z}=0Sz​=0 sector for the relevant parameters and is doubly degenerate at multiple points ${\ensuremath{\gamma}}_{m}=({\ensuremath{\alpha}}_{m},{\ensuremath{\Delta}}_{m})$ in the $\ensuremath{\alpha}$, $\ensuremath{\Delta}$ plane. Degeneracy under inversion at sites or spin-parity or both leads, respectively, to a bond order wave (BOW), to staggered magnetization, or to vector chiral (VC) order. Exact results up to N=28N=28N=28 spins directly yield order parameters and spin correlation functions whose weak NNN dependencies allow inferences about infinite chains. The high-spin GS at J2=0{J}_{2}=0J2​=0 changes discontinuously at ${\ensuremath{\gamma}}_{1}=(\ensuremath{-}1/4,1)$ to a singlet in the isotropic ($\ensuremath{\Delta}=1$) chain. The transition from high to low spin $S(\ensuremath{\alpha},\ensuremath{\Delta})$ is continuous for $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{B}=0.95\ifmmode\pm\else\textpm\fi{}0.01$ on the degeneracy line ${\ensuremath{\alpha}}_{1}(\ensuremath{\Delta})$. The GS has staggered magnetization between ${\ensuremath{\Delta}}_{A}=0.72$ and ${\ensuremath{\Delta}}_{B}$ and a BOW for $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{A}$. When both inversion and spin-parity are reversed at ${\ensuremath{\gamma}}_{m}$, the correlation functions C(p)C(p)C(p) for spins separated by ppp sites are identical. C(p)C(p)C(p) minima are shifted by $\ensuremath{\pi}/2$ from the minima of VC order parameters at separation ppp, consistent with right- and left-handed helices along the zzz axis and spins in the xyxyxy plane. Degenerate GSs of finite chains are related to quantum phase diagrams of extended $\ensuremath{\alpha}$, $\ensuremath{\Delta}$ chains, with good agreement for order parameters along the line ${\ensuremath{\alpha}}_{1}(\ensuremath{\Delta})$. Degenerate GSs limit a VC phase to intermediate $\ensuremath{\alpha}$ and $\ensuremath{\Delta}$ where $S(\ensuremath{\alpha},\ensuremath{\Delta})$ varies rapidly but continuously, in contrast to many-body treatments in which VC phases extend over a larger range of parameters.