Incommensurate Correlations Research Articles

Eleven competing phases in the Heisenberg-Gamma ( JΓ ) ladder

The spin-orbit generated Γ interaction is known to induce strong frustration and to be significant in realistic models of materials. To gain an understanding of the possible phases that can arise from this interaction, it is of considerable interest to focus on a limited part of parameter space in a quasi one-dimensional model where high precision numerical results can be obtained. Here we study the Heisenberg–Gamma ( JΓ ) ladder, determining the complete zero temperature phase diagram by analyzing the entanglement spectrum (ES) and energy susceptibility. A total of 11 different phases can be identified, among them the well known rung-singlet (RS) phase and 5 other phases, FM, FM-Z, FM-XY, AF and AF-Z, with conventional long-range magnetic order. The 3 ferromagnetic phases, FM, FM-Z and FM-XY simultaneously have non-zero scalar chirality. Two other phases, the antiferromagnetic Gamma (AΓ) and ferromagnetic Gamma (FΓ) phases, have previously been observed in the Kitaev–Gamma ladder, demonstrating that the AΓ-phase is a symmetry protected topological phase (SPT) protected by TR ×Rb symmetry, the product of time-reversal (TR) and π rotation around the b-axis ( Rb ), while the FΓ-phase is related to the RS phase through a local unitary transformation. The 3 remaining phases, ϒ, Ω and δ show no conventional order, a doubling of the ES and for the ϒ and Ω-phases a gap is clearly present. The δ-phase has a significantly smaller gap and displays incommensurate correlations, with a peak in the static structure factor, S(k) continuously shifting from k/π=2/3 to k=π . In the Ω-phase we find pronounced edge-states consistent with a SPT phase protected by the same TR ×Rb symmetry as the AΓ-phase. The precise nature of the ϒ and δ-phases is less clear.

Topological and quantum critical properties of the interacting Majorana chain model

We study Majorana chain with the shortest possible interaction term and in the presence of hopping alternation. When formulated in terms of spins the model corresponds to the transverse field Ising model with nearest-neighbor transverse and next-nearest-neighbor longitudinal repulsion. The phase diagram obtained with extensive DMRG simulations is very rich and contains six phases. Four gapped phases include paramagnetic, period-2 with broken translation symmetry, \mathbb{Z}_2ℤ2 with broken parity symmetry and the period-2-\mathbb{Z}_2ℤ2 phase with both symmetries broken. In addition there are two floating phases: gapless and critical Luttinger liquid with incommensurate correlations, and with an additional spontaneously broken \mathbb{Z}_2ℤ2 symmetry in one of them. By analyzing an extended phase diagram we demonstrate that, in contrast with a common belief, the Luttinger liquid phase along the self-dual critical line terminates at a weaker interaction strength than the end point of the Ising critical line that we find to be in the tri-critical Ising universality class. We also show that none of these two points is a Lifshitz point terminating the incommensurability. In addition, we analyzed topological properties through Majorana zero modes emergent in the two topological phases, with and without incommensurability. In the weak interaction regime, a self-consistent mean-field treatment provides a remarkable accuracy for the description of the spectral pairing and the parity switches induced by the interaction.

From SU(2)5 to SU(2)3 Wess-Zumino-Witten transitions in a frustrated spin- 52 chain

We investigate the properties of a frustrated spin-5/2 chain with next-nearest neighbor two and three-site interactions, with two questions in mind: the nature of the transition into the dimerized phase induced by the three-site interaction, and the possible presence of a critical floating phase at intermediate values of the next-nearest neighbor interaction. We provide strong evidence that the continuous transition into the dimerized phase, which has been found to be generically in the Wess-Zumino-Witten SU(2)$_{2S}$ universality class up to spin $S=2$, is SU(2)$_5$ only at two isolated points of the phase diagram, and that it is SU(2)$_3$ in between, in agreement with the presence of two relevant operators allowed by symmetry for SU(2)$_5$, and with the conservation of the parity of the level index along the renormalization flow between SU(2)$_k$ theories with different values of $k$. We also find that the dimerization induced by the next-nearest neighbor interaction is a three step process, with first a small partially dimerized phase followed by a broad critical floating phase with incommensurate correlations before the fully dimerized phase is reached. Implications for the iron oxide ${\mathrm{Bi}}_{3}{\mathrm{FeMo}}_{2}{\mathrm{O}}_{12}$ are briefly discussed.

Parallel spin stripes and their coexistence with superconducting ground states at optimal and high doping in La1.6−xNd0.4SrxCuO4

Three-dimensional, commensurate long-range magnetic order in La2CuO4 quickly evolves to quasi-two-dimensional, incommensurate correlations upon doping with mobile holes, and superconductivity follows for x as small as 0.05 in the La2−xSrx/BaxCuO4 family of superconductors. The onset of superconductivity in these systems is known to be coincident with a remarkable rotation of the incommensurate spin order from “diagonal stripes” below x = 0.05 to “parallel stripes” above. However, little is known about the spin correlations at optimal and high doping levels, around and beyond the proposed quantum critical point for the pseudogap phase, p*. Here, we present elastic and inelastic neutron scattering measurements on single crystals of La1.6−xNd0.4SrxCuO4 with x = 0.125, 0.19, 0.24, and 0.26 and show that two-dimensional, quasistatic, parallel spin stripes have an onset at temperatures such that the parallel spin stripe phase extends beyond p* and envelops the entirety of superconducting ground states in this system. We also show that the elastic order parameter for parallel spin stripes at optimum doping, x = 0.19, displays an inflection point at superconducting Tc, while the low-energy dynamic spectral weight of parallel stripe fluctuations grows with decreasing temperature and saturates below Tc.3 MoreReceived 2 September 2020Revised 12 March 2021Accepted 29 April 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023151Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasSuperconductivityPhysical SystemsCupratesHigh-temperature superconductorsTechniquesNeutron scatteringCondensed Matter, Materials & Applied Physics

Long-range entangled-plaquette states for critical and frustrated quantum systems on a lattice

We explore a variational Ansatz for lattice quantum systems---named long-range entangled-plaquette state---in which pairs of clusters of adjacent lattice sites are correlated at any distance. The explicit scale-free structure of correlations built in this wave function makes it fit to reproduce critical states with long-range entanglement. The use of complex weights in the Ansatz allows for an efficient optimization of nonpositive definite states in a fully variational fashion, namely without any additional bias (arising, e.g., from pre-imposed sign structures) beyond that imposed by the parametrization of the state coefficients. These two features render the Ansatz particularly appropriate for the study of quantum phase transitions in frustrated systems. Moreover, the Ansatz can be systematically improved by increasing the long-range plaquette size, as well as by the inclusion of even larger adjacent-site plaquettes. We validate our Ansatz and its optimization procedure in the case of the XX and Heisenberg chain, and further apply it to the case of a simple, yet paradigmatic model of frustration, namely the ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Heisenberg chain. For this model we provide clear evidence that our trial wave function faithfully describes both the short-range physics (particularly in terms of ground-state energy) and the long-range one expressed by the Luttinger exponent, and the central charge of the related conformal field theory, which govern the decay of correlations and the scaling of the entanglement entropy, respectively. Finally we successfully reproduce the incommensurate correlations developing in the system at strong frustration, as a result of the flexible representation of sign (phase) structures via complex weights.

Direct observation of incommensurate magnetism in Hubbard chains.

The interplay between magnetism and doping is at the origin of exotic strongly correlated electronic phases and can lead to novel forms of magnetic ordering. One example is the emergence of incommensurate spin-density waves, which have wavevectors that do not belong to the reciprocal lattice. In one dimension this effect is a hallmark of Luttinger liquid theory, which also describes the low-energy physics of the Hubbard model1. Here we use a quantum simulator that uses ultracold fermions in an optical lattice2-8 to directly observe such incommensurate spin correlations in doped and spin-imbalanced Hubbard chains using fully spin- and density-resolved quantum gas microscopy. Doping is found to induce a linear change in the spin-density wavevector, in excellent agreement with predictions fromLuttinger theory. For non-zero polarization we observe a reduction in the wavevector with magnetization, as expected from the antiferromagnetic Heisenberg model in a magnetic field. We trace the microscopic-scale origin of these incommensurate correlations to holes, doublons (double occupancies) and excess spins, which act as delocalized domain walls for the antiferromagnetic order. In addition, by inducing interchain coupling we observe fundamentally different spin correlations around doublons and suppression of incommensurate magnetism at finite (low)temperature in the two-dimensional regime9. Our results demonstrate how access to the full counting statistics of all local degrees of freedom can be used to study fundamental phenomena in strongly correlated many-body physics.

Rigorous decoupling between edge states in frustrated spin chains and ladders

We investigate the occurrence of exact zero modes in one-dimensional quantum magnets of finite length that possess edge states. Building on conclusions first reached in the context of the spin-1/2 XY chain in a field, then for the spin-1 $J_1-J_2$ Heisenberg model, we show that the development of incommensurate correlations in the bulk invariably leads to oscillations in the sign of the coupling between edge states, hence to exact zero energy modes at the crossing points where the coupling between the edge states rigorously vanishes. This is true regardless of the origin of the frustration (e.g. next-nearest neighbor coupling or biquadratic coupling for the spin-1 chain), of the value of the bulk spin (we report on spin-1/2, spin-1 and spin-2 examples), and of the value of the edge-state emergent spin (spin-1/2 or spin-1).

Magnetic structure and dispersion relation of the S=12 quasi-one-dimensional Ising-like antiferromagnet BaCo2V2O8 in a transverse magnetic field

${\mathrm{BaCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$ consists of Co chains in which a ${\mathrm{Co}}^{2+}$ ion carries a fictitious spin $\frac{1}{2}$ with Ising anisotropy. We performed elastic and inelastic neutron scattering experiments in ${\mathrm{BaCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$ in a magnetic field perpendicular to the $c$ axis which is the chain direction. With applying magnetic field along the $a$ axis at 3.5 K, the antiferromagnetic order with the easy axis along the $c$ axis, observed in zero magnetic field, is completely suppressed at 8 T, while the magnetic field gradually induces an antiferromagnetic order with the spin component along the $b$ axis. We also studied magnetic excitations as a function of transverse magnetic field. The lower boundary of the spinon excitations splits gradually with increasing magnetic field. The overall feature of the magnetic excitation spectra in the magnetic field is reproduced by the theoretical calculation based on the spin $\frac{1}{2} XXZ$ antiferromagnetic chain model, which predicts that the dynamic magnetic structure factor of the spin component along the chain direction is enhanced and that along the field direction has clear incommensurate correlations.

Emergent incommensurate correlations in frustrated ferromagnetic spin-1 chains

We study the frustrated ferromagnetic spin-1 chains, where the ferromagnetic nearest-neighbor coupling competes with the antiferromagnetic next-nearest-neighbor coupling. We use the density matrix renormalization group to obtain the ground states. Through the analysis of spin-spin correlations we identify the double Haldane phase as well as the ferromagnetic phase. It is shown that the ferromagnetic coupling leads to incommensurate correlations in the double Haldane phase. Such short-range correlations transform continuously into the ferromagnetic instability at the transition to the ferromagnetic phase. We also compare the results with the spin-1/2 and classical spin systems, and discuss the string orders in the system.

Two iridates, two models, and two approaches: A comparative study on magnetism in three-dimensional honeycomb materials

Two recent theoretical works studied the role of Kitaev interactions in the newly observed incommensurate magnetic order in the hyper-honeycomb ($\beta$-Li$_2$IrO$_3$) and stripy-honeycomb ($\gamma$-Li$_2$IrO$_3$) iridates. Each of these works analyzed a different model ($JK\Gamma$ versus coupled zigzag chain model) using a contrasting method (classical versus soft-spin analysis). The lack of commonality between these works precludes meaningful comparisons and a proper understanding of these unusual orderings. In this study, we complete the unfinished picture initiated by these two works by solving both models with both approaches for both 3D honeycomb iridates. Through comparisons between all combinations of models, techniques, and materials, we find that the bond-isotropic $JK\Gamma$ model consistently predicts the experimental phase of $\beta$-Li$_2$IrO$_3$ regardless of the method used, while the experimental phase of $\gamma$-Li$_2$IrO$_3$ can be generated by the soft-spin approach with eigenmode mixing irrespective of the model used. To gain further insights, we solve a 1D quantum spin-chain model related to both 3D models using the density matrix renormalization group method to form a benchmark. We discover that in the 1D model, incommensurate correlations in the classical and soft-spin analysis survive in the quantum limit only in the presence of the symmetric-off-diagonal exchange $\Gamma$ found in the $JK\Gamma$ model. The relevance of these results to the real materials are also discussed.

Frustration and multicriticality in the antiferromagnetic spin-1 chain

We study the spin $S=1$ Heisenberg chain, with nearest neighbor, next nearest neighbor ($\alpha$) and biquadratic ($\beta$) interactions using a combination of the density matrix renormalization group (DMRG), an analytic variational matrix product state wavefunction, and non-Abelian bosonization. We study the effect of frustration ($\alpha>0$) on the Haldane phase with $-1\leq \beta < 1$ which reveals a rich phase diagram. For $-1<\beta<\beta^\ast$, we establish the existence of a spontaneously dimerized phase for large $\alpha>\alpha_c$, separated from the Haldane phase by the critical line $\alpha_c(\beta)$ of second-order phase transitions connected to the Takhtajan--Babudjian integrable point $\alpha_c(\beta=-1)=0$. In the opposite regime, $\beta>\beta^\ast$, the transition from the Haldane phase becomes first-order into the next nearest neighbor (NNN) AKLT phase. Based on field theoretical arguments and DMRG calculations, we conjecture that these two regimes are separated by a multicritical point ($\beta^\ast, \alpha^\ast$) of a different universality class, described by the $SU(2)_4$ Wess--Zumino--Witten critical theory. From the DMRG calculations we estimate this multicritical point to lie in the range $-0.2<\beta^\ast<-0.15$ and $0.47<\alpha^\ast < 0.53$. We find that the dimerized and NNN-AKLT phases are separated by a line of first-order phase transitions that terminates at the multicritical point. Inside the Haldane phase, we show the existence of two incommensurate crossovers: the Lifshitz transition and the disorder transition of the first kind, marking incommensurate correlations in momentum and real space, respectively. We show these crossover lines stretch across the entire $(\beta,\alpha)$ phase diagram, merging into a single incommensurate-to-commensurate transition line for negative $\beta\lesssim \beta^\ast$ outside the Haldane phase.

Exact diagonalization study of the anisotropic triangular lattice Heisenberg model using twisted boundary conditions

The spin-$1/2$ anisotropic triangular model, which is believed to describe the materials ${\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}$ and ${\mathrm{Cs}}_{2}{\mathrm{CuBr}}_{4}$, among others, is dominated by incommensurate spiral physics and is thus extremely resistant to numerical analysis on small system sizes. In this paper, we use twisted boundary conditions and exact diagonalization techniques to study the ground state of this model as a function of exchange parameter strength. With these boundary conditions we are able to extract the inter- and intrachain ordering $q$ vectors for the $\frac{{J}^{\ensuremath{'}}}{J}&lt;1$ region finding very close agreement with recent DMRG results on much larger systems. Our results suggest a phase transition between a long-range incommensurate spiral ordered phase, and a more subtle phase with short-range spiral correlations with the $q$ vector describing the incommensurate correlations varying smoothly through the transition. In the latter phase correlations between next-nearest chains exhibit an extremely close competition between predominantly antiferromagnetic and ferromagnetic correlations. Further analysis suggests that the antiferromagnetic next-nearest chain correlations may be slightly stronger than the ferromagnetic ones. This difference is found to be slight but in line with previous renormalization group predictions of a collinear antiferromagnetic ordering in this region.

S-wave superconductivity in superconducting BaTi2Sb2O revealed by121/123Sb-NMR/nuclear quadrupole resonance measurements

We report the ${}^{121/123}$Sb-NMR/nuclear quadrupole resonance (NQR) measurements on the superconductor BaTi${}_{2}$Sb${}_{2}$O with a two-dimensional Ti${}_{2}$O square-net layer formed with Ti${}^{3+}$ (3${d}^{1}$). NQR measurements revealed that the in-plane four-fold symmetry is broken at the Sb site below ${T}_{\mathrm{A}}\ensuremath{\sim}40$ K, without an internal field appearing at the Sb site. These exclude a spin-density wave (SDW)/ charge density wave (CDW) ordering with incommensurate correlations, but can be understood with the commensurate CDW ordering at ${T}_{\mathrm{A}}$. The spin-lattice relaxation rate $1/{T}_{1}$, measured at the four-fold symmetry breaking site, decreases below superconducting (SC) transition temperature ${T}_{\mathrm{c}}$, indicative of the microscopic coexistence of superconductivity and the CDW/SDW phase below ${T}_{\mathrm{A}}$. Furthermore, $1/{T}_{1}$ of ${}^{121}$Sb-NQR shows a coherence peak just below ${T}_{\mathrm{c}}$ and decreases exponentially at low temperatures. These results are in sharp contrast with those in cuprate and iron-based superconductors, and strongly suggest that its SC symmetry is classified to an ordinary $s$-wave state.

From Incommensurate Correlations to Mesoscopic Spin Resonance inYbRh2Si2

Spin fluctuations are reported near the magnetic field-driven quantum critical point in YbRh(2)Si(2). On cooling, ferromagnetic fluctuations evolve into incommensurate correlations located at q(0) = ±(δ,δ), with δ = 0.14 ± 0.04 r.l.u. At low temperatures, an in-plane magnetic field induces a sharp intradoublet resonant excitation at an energy E(0) = gμ(B)μ(0)H with g = 3.8 ± 0.2. The intensity is localized at the zone center, indicating precession of spin density extending ξ = 6 ± 2 Å beyond the 4f site.

Incommensurate correlations in the anisotropic triangular Heisenberg lattice

We study the anisotropic spin-1/2 antiferromagnetic triangular Heisenberg lattice in two dimensions, seen as a set of chains with couplings $J$ (${J}^{\ensuremath{'}}$) along (in-between) chains, respectively. Our focus is on the incommensurate correlation that emerges in this system in a wide parameter range due to the intrinsic frustration of the spins. We study this system with traditional density matrix renormalization group using cylindrical boundary conditions to least constrain possible incommensurate order. Despite that the limit of essentially decoupled chains ${J}^{\ensuremath{'}}/J\ensuremath{\lesssim}0.5$ is not very accessible numerically, it appears that the spin-spin correlations remain incommensurate for any finite $0&lt;{J}^{\ensuremath{'}}&lt;{J}_{{}^{{}_{\mathrm{C}}}}^{\ensuremath{'}}$, where ${J}_{{}^{{}_{\mathrm{C}}}}^{\ensuremath{'}}/J&gt;1$. The incommensurate wave vector ${q}_{J}$, however, approaches the commensurate value corresponding to the antiferromagnetic correlation of a single chain very rapidly with decreasing ${J}^{\ensuremath{'}}/J$, roughly as ${q}_{J}\ensuremath{\sim}\ensuremath{\pi}\ensuremath{-}{c}_{1}{({J}^{\ensuremath{'}}/J)}^{n}{e}^{\ensuremath{-}{c}_{2}J/{J}^{\ensuremath{'}}}$.

Condensate-Free Superfluid Induced by the Frustrated Proximity Effect

Since the discovery of superfluidity in 4He and Landau's phenomenological theory, the relationship between Bose condensation and superfluidity has been intensely debated. 4He is known by now to be both superfluid and condensed at low temperature, and more generally, in dimension D≥2, all superfluid bosonic models realized in experiments are condensed in their ground state, the most recent example being provided by ultracold bosonic atoms trapped in an optical lattice. In this Letter, it is shown that a 2D gas of bosons which is not condensed at T=0 can be achieved by populating a layer through a frustrated proximity effect from a superfluid reservoir. This condensate-free bosonic fluid is further shown to be a superfluid with incommensurate correlations.

Incommensurate spin Luttinger liquid phase in a model for the spin-Peierls materials TiOBr and TiOCl

In the present work we aim to characterize the lattice configurations and the magnetic behavior in the incommensurate phase of spin-Peierls systems. This phase emerges when the magnetic exchange interaction is coupled to the distortions of an underlying triangular lattice and has its experimental realization in the quasi-one-dimensional compound family TiO$X$ ($X=\text{Cl}$, Br). With a simple model of spin-$1/2$ chains inserted in a planar triangular geometry which couples them elastically, we are able to obtain the uniform-incommensurate and incommensurate-dimerized phase transitions seen in these compounds. Moreover, we follow the evolution of the wave vector of the distortions with temperature inside the incommensurate phase. Finally, we predict gapless spin excitations for the intermediate phase of TiO$X$ compounds along with incommensurate spin-spin correlations, offering to experimentalists a candidate to observe an exotic Luttinger liquidlike behavior.

Neutron scattering study of the competing magnetic correlations inLa0.85Sr0.15CoO3

The nature of the competing ferromagnetic and incommensurate spin correlations in the spin-glass phase of ${\text{La}}_{0.85}{\text{Sr}}_{0.15}{\text{CoO}}_{3}$ has been investigated by various neutron scattering techniques. Spin-polarized scattering indicates that the observed incommensurate peaks are dominantly magnetic in nature. Magnetic field experiments show that a field applied perpendicular to the short-range ordering wave vector destroys the incommensurate correlations and induces long-range ferromagnetic order. However, even for fields up to 7 T, short-range ferromagnetic correlations still coexist with the long-range ordered regions.

Exotic Finite-Temperature Phase Diagram for Weakly CoupledS= 1/2 XXZ Chain in a Magnetic Field

We investigate the ordered state in S = 1/2 XXZ spin chains with weak interchain couplings in a magnetic field by using quantum Monte Carlo simulations based on the directed-loop algorithm. The H – T phase diagram is illustrated for a square lattice and we find that an exotic spin-liquid state with quasi long-range incommensurate correlations in both the spin-chain and interchain directions may appear in the low-magnetization region without a finite-temperature transition. We also show that the incommensurate long-range-ordered state is stable for a cubic lattice.

Charge Fluctuations in Geometrically Frustrated Charge Ordering System

Effects of geometrical frustration in low-dimensional charge ordering systems are theoretically studied, mainly focusing on dynamical properties. We treat extended Hubbard models at quarter-filling, where the frustration arises from competing charge ordered patterns favored by different intersite Coulomb interactions, which are effective models for various charge transfer-type molecular conductors and transition metal oxides. Two different lattice structures are considered: (a) one-dimensional chain with intersite Coulomb interaction of nearest neighbor V_1 and that of next-nearest neighbor V_2, and (b) two-dimensional square lattice with V_1 along the squares and V_2 along one of the diagonals. From previous studies, charge ordered insulating states are known to be unstable in the frustrated region, i.e., V_1 \simeq 2V_2 for case (a) and V_1 \simeq V_2 for case (b), resulting in a robust metallic phase even when the interaction strenghs are strong. By applying the Lanczos exact diagonalization to finite-size clusters, we have found that fluctuations of different charge order patterns exist in the frustration-induced metallic phase, showing up as characteristic low energy modes in dynamical correlation functions. Comparison of such features between the two models are discussed, whose difference will be ascribed to the dimensionality effect. We also point out incommensurate correlation in the charge sector due to the frustration, found in one-dimensional clusters.