Haldane Phase Research Articles

Emerging topological bound states in Haldane model zigzag nanoribbons

Zigzag nanoribbons hosting the Haldane Chern insulator model are considered. In this context, a reentrant topological phase, characterized by the emergence of quasi zero dimensional in-gap states, is discussed. The bound states, which reside in the gap opened by the hybridization of the counter-propagating edge modes of the Haldane phase, are localized at the ends of the strip and are found to be robust against on-site disorder. These findings are supported by the behavior of the Zak phase over the parameter space, which exhibits jumps of π in correspondence to the phase transitions between the trivial and the non-trivial phases. The effective mass inversion leading to the jumps in the Zak phase is interpreted in a low energy framework. Setups with non-uniform parameters also show topological bound states via the Jackiw-Rebbi mechanism. All the properties reported are shown to be extremely sensitive to the strip width.

Measuring Nonlocal Brane Order with Error-Corrected Quantum Gas Microscopes

Exotic quantum many-body states, such as Haldane and spin liquid phases, can exhibit remarkable features like fractional excitations and non-Abelian statistics and offer new understandings of quantum entanglement in many-body quantum systems. These phases are classified by nonlocal correlators that can be directly measured in atomic analog quantum simulating platforms, such as optical lattices and Rydberg atom arrays. However, characterizing these phases in large systems is experimentally challenging because they are sensitive to local errors like atom loss, which suppress its signals exponentially. Additionally, protocols for systematically identifying and mitigating uncorrelated errors in analog quantum simulators are lacking. Here, we address these challenges by developing an error-correction method for large-scale neutral atom quantum simulators using optical lattices. Our error-correction method can distinguish correlated particle-hole pairs from uncorrelated holes in the Mott insulator. After removing the uncorrelated errors, we observe a dramatic improvement in the nonlocal parity correlator and find the perimeter scaling law. Furthermore, the error model provides a statistical estimation of fluctuations in site occupation, from which we measure the generalized brane correlator and confirm that it can be an order parameter for Mott insulators in two dimensions. Our work provides a promising avenue for investigating and characterizing exotic phases of matters in large-scale quantum simulators. Published by the American Physical Society 2024

Transition to the Haldane phase driven by electron-electron correlations

One of the most famous quantum systems with topological properties, the spin S=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{S}}}}}}}}=1$$\\end{document} antiferromagnetic Heisenberg chain, is well-known to display exotic S=1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{{{{{{\\mathcal{S}}}}}}}}=1/2$$\\end{document} edge states. However, this spin model has not been analyzed from the more general perspective of strongly correlated systems varying the electron-electron interaction strength. Here, we report the investigation of the emergence of the Haldane edge in a system of interacting electrons – the two-orbital Hubbard model—with increasing repulsion strength U and Hund interaction JH. We show that interactions not only form the magnetic moments but also form a topologically nontrivial fermionic many-body ground-state with zero-energy edge states. Specifically, upon increasing the strength of the Hubbard repulsion and Hund exchange, we identify a sharp transition point separating topologically trivial and nontrivial ground-states. Surprisingly, such a behaviour appears already at rather small values of the interaction, in a regime where the magnetic moments are barely developed.

Dynamically generated quadrupole polarization using Floquet adiabatic evolution

We investigate the nonequilibrium dynamics of the $S=1$ quantum spin chain subjected to a time-dependent external drive, where the driving frequency is adiabatically decreased as a function of time (``Floquet adiabatic evolution''). We show that when driving the rhombic anisotropy term (known as the ``two-axis countertwisting'' in the context of squeezed spin states) of a N\'eel antiferromagnet, we are able to induce an overall enhancement in the quadrupole polarization, while at the same time suppressing the staggered magnetization order. The system evolves into a new state with a net quadrupole moment and antiferroquadrupolar correlations. This state remains stable at long times once the driving frequency is kept constant. On the other hand, we find that we cannot achieve a quadrupole polarization for the symmetry-protected Haldane phase, which remains robust against such driving.

ВЛИЯНИЕ ВНЕШНЕГО МАГНИТНОГО ПОЛЯ НА МАГНИТНЫЕ СВОЙСТВА КВАЗИОДНОМЕРНЫХ ХАЛДЕЙНОВСКИХ МАГНЕТИКОВ (Y1-XNDX)2BANIO5 (X = 0.25 И 0.04)

In the compounds of the (Y1-xNdx)2BaNiO5 family, the paradoxical coexistence of the Haldane phase and spin waves is realized. The magnetic ordering in the system is caused by the interaction of Nd3+ ions through spin fluctuations of a nickel chain, which remains internally disordered. Nd3+ magnetic moments do not deviate from the c axis of the crystal even in the presence of an external magnetic field. In this paper, for compounds (Y1-xNdx)2BaNiO5 with x = 0.25 and x = 0.04, an experimental study of the field dependence of the magnetization M(B) and the temperature dependence of the heat capacity C( T ), measured in fields B = 0; 2 and 5 T. For compounds with x = 0.25 on the dependence M(B) measured at T = 4.2 K, an anomaly was detected indicating a metamagnetic transition. Changes in the parameters of the magnetic interaction between the magnetic moments Nd along the c axis of the crystal leads to the periorientation of the magnetic moments of all Nd3+ ions along the direction of the external field B||c. For the connection with x = 0.04, no anomalies were detected on the dependence M(B), the magnetic moments of both neodymium sublattices lie along the direction of the applied magnetic field. The presence of a Schottky anomaly on the dependence C( T ) at B =0 indicates the existence of an internal magnetic field acting on the Nd3+ ion from the side of the nickel subsystem, and leading to the splitting of the main Kramers doublet of the Nd3+ ion. For a connection with x = 0.04, the Schottky anomaly shifts towards higher temperatures with an increase in the external magnetic field, and for a connection with x = 0.25, the Schottky anomaly shifts towards low temperatures only in the field B = 5 T. The displacement of the Schottky anomaly in the presence of a field is due to the ratio between the magnitude of the applied magnetic field and the component of the internal magnetic field fields с along the c axis of the crystal. For connection with x = 0.25 , the magnetic moments of the two neodymium sublattices are oppositely directed, and the Schottky anomaly of the 1st sublattice shifts towards higher temperatures, and the Schottky anomaly of the 2nd sublattice shifts towards lower temperatures. For connection with x = 0.04 с The magnetic moments of the two neodymium sublattices are directed towards the applied field, and the Schottky anomaly of both sublattices shifts towards higher temperatures, which is consistent with the experimen

Duality, criticality, anomaly, and topology in quantum spin-1 chains

In quantum spin-1 chains, there is a nonlocal unitary transformation known as the Kennedy-Tasaki transformation ${U}_{\mathrm{KT}}$, which defines a duality between the Haldane phase and the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry-breaking phase. In this paper, we find that ${U}_{\mathrm{KT}}$ also defines a duality between a topological Ising critical phase and a trivial Ising critical phase, which provides a ``hidden symmetry breaking'' interpretation of the topological criticality. Moreover, since the duality relates different phases of matter, we argue that a model with self-duality (i.e., invariant under ${U}_{\mathrm{KT}}$) is natural to be at a critical or multicritical point. We study concrete examples to demonstrate this argument. In particular, when $H$ is the Hamiltonian of the spin-1 antiferromagnetic Heisenberg chain, we prove that the self-dual model $H+{U}_{\mathrm{KT}}H{U}_{\mathrm{KT}}$ is exactly equivalent to a gapless spin-$1/2$ XY chain, which also implies an emergent quantum anomaly. On the other hand, we show that the topological and trivial Ising critical phases that are dual to each other meet at a multicritical point which is indeed self-dual.

Critical phenomena around the SU(3) symmetric tricritical point of a spin-1 chain

We investigate critical phenomena of a spin-1 chain in the vicinity of the SU(3) symmetric critical point, which we already specified in a previous study [Mashiko and Nomura, Phys. Rev. B 104, 155405 (2021)]. We numerically diagonalize a Hamiltonian combining the bilinear-biquadratic Hamiltonian with the trimer Hamiltonian. We then discuss the numerical results based on the conformal field theory and the renormalization group. As a result, we first verify that the critical point found in our previous study is the tricritical point among the Haldane phase, the trimer phase, and the the trimer-liquid (TL) phase. Second, with regard to the TL-trimer transition and the TL-Haldane transition, we find that the critical phenomena around this tricritical point belong to the Berezinskii-Kosterlitz-Thouless-like universality class. Third, we find the boundary between the Haldane phase and the trimer phase, which is illustrated by the massive self-dual sine-Gordon model.

Low-Dimensional Metal-Organic Magnets as a Route toward the S = 2 Haldane Phase.

Metal-organic magnets (MOMs), modular magnetic materials where metal atoms are connected by organic linkers, are promising candidates for next-generation quantum technologies. MOMs readily form low-dimensional structures and so are ideal systems to realize physical examples of key quantum models, including the Haldane phase, where a topological excitation gap occurs in integer-spin antiferromagnetic (AFM) chains. Thus, far the Haldane phase has only been identified for S = 1, with S ≥ 2 still unrealized because the larger spin imposes more stringent requirements on the magnetic interactions. Here, we report the structure and magnetic properties of CrCl2(pym) (pym = pyrimidine), a new quasi-1D S = 2 AFM MOM. We show, using X-ray and neutron diffraction, bulk property measurements, density-functional theory calculations, and inelastic neutron spectroscopy (INS), that CrCl2(pym) consists of AFM CrCl2 spin chains (J1 = -1.13(4) meV) which are weakly ferromagnetically coupled through bridging pym (J2 = 0.10(2) meV), with easy-axis anisotropy (D = -0.15(3) meV). We find that, although small compared to J1, these additional interactions are sufficient to prevent observation of the Haldane phase in this material. Nevertheless, the proximity to the Haldane phase together with the modularity of MOMs suggests that layered Cr(II) MOMs are a promising family to search for the elusive S = 2 Haldane phase.

Commensurate and incommensurate Haldane phases for a spin-1 bilinear–biquadratic model

Commensurate and incommensurate Haldane phases for a spin-1 bilinear-biquadratic model are investigated using an infinite matrix product state algorithm. The bipartite entanglement entropy can detect a transition point between the two phases. In both phases, the entanglement spectrum shows double degeneracy. We calculate the nonlocal order parameter of the bond-centered inversion in both phases, which rapidly approaches a saturation value of −1 as the segment length increases. The nonlocal order parameter of the bond-centered inversion with a saturation value −1 and the nonzero value string order indicate that the Haldane phase is a symmetry-protected topological phase. To distinguish the commensurate and incommensurate Haldane phases, the transversal spin correlation and corresponding momentum distribution of the structure factor are analyzed. As a result, the transversal spin correlations exhibit different decay forms in both phases.

Magnetic phases of spatially modulated spin-1 chains in Rydberg excitons: Classical and quantum simulations.

In this work, we study the magnetic phases of a spatially modulated chain of spin-1 Rydberg excitons. Using the Density Matrix Renormalization Group (DMRG) technique, we study various magnetic and topologically nontrivial phases using both single-particle properties, such as local magnetization and quantum entropy, and many-body ones, such as pair-wise Néel and long-range string correlations. In particular, we investigate the emergence and robustness of the Haldane phase, a topological phase of anti-ferromagnetic spin-1 chains. Furthermore, we devise a hybrid quantum algorithm employing restricted Boltzmann machine to simulate the ground state of such a system that shows very good agreement with the results of exact diagonalization and DMRG.

Frustrated mixed-spin ladders: Evidence for a bond-order wave phase between rung-singlet and Haldane phases

In frustrated spin ladders the interplay of frustration and correlations leads to the familiar Haldane (H) and rung-singlet (RS) phases. The nature of the transition between these two phases is still under debate. In this paper we tackle this issue using tools of quantum information theory. We consider frustrated mixed-spin-(1, 1/2) ladders with antiferromagnetic leg, rung and diagonal couplings, and calculate various quantities, such as the entanglement entropy (EE), the Schmidt gap, and the level degeneracy of the entanglement spectrum (ES). We use two numerical techniques, the infinite time-evolving block decimation (iTEBD) and the density matrix renormalization group (DMRG). We demonstrate that there exists an intermediate phase in which the ES levels do not exhibit the characteristic degeneracies of the H and RS phases. To understand the underlying physics in this phase, we investigate short-range spin correlations along legs, rungs and diagonals and show that in this intermediate phase long-wavelength modulations occur, akin to bond order waves.

Bulk and surface critical behavior of a quantum Heisenberg antiferromagnet on two-dimensional coupled diagonal ladders

Using Quantum Monte Carlo simulations, we study the spin-1/2 Heisenberg model on a two-dimensional lattice formed by coupling diagonal ladders. The model hosts an antiferromagnetic N\'eel phase, a rung singlet product phase, and a topological none trivial Haldane phase, separated by two quantum phase transitions. We show that the two quantum critical points are all in the three-dimensional O(3) universality class. The properties of the two gapped phases, including the finite-size behavior of the string orders in the Haldane phase, are studied. We show that the surface formed by the ladders ends is gapless, while the surface exposed along the ladders is gapful, in the Haldane phase. Conversely, in the gapped rung singlet phase, the former surface is gapped, and the latter is gapless. We demonstrate that, although mechanisms of the two gapless modes are different, nonordinary surface critical behaviors are realized at both critical points on the gapless surfaces exposed by simply cutting bonds without fine-tuning the surface coupling required to reach a multicritical point in classical models. We also show that, on the gapped surfaces, the surface critical behaviors are in the ordinary class.

Ground-state phase diagram of a spin- 12 frustrated XXZ ladder

We study the ground-state phase diagram of a spin-$\frac{1}{2}$ frustrated XXZ ladder, in which two antiferromagnetic chains are coupled by competing rung and diagonal interactions, $J_\perp$ and $J_\times$. Previous studies on the isotropic model have revealed that a fluctuation-induced effective dimer attraction between the legs stabilizes the columnar dimer (CD) phase in the highly frustrated regime $J_\perp\approx 2J_\times$, especially for ferromagnetic $J_{\perp, \times}<0$. By means of effective field theory and numerical analyses, we extend this analysis to the XXZ model, and obtain a rich phase diagram. The diagram includes four gapped featureless phases with no symmetry breaking: the rung singlet (RS) and Haldane phases as well as their twisted variants, the RS* and Haldane* phases, which are all distinct in the presence of certain symmetries. Significantly, the Haldane-CD transition point in the isotropic model turns out to be a crossing point of two transition lines in the XXZ model, and the stripe N\'eel and RS* phases appear between these lines. This indicates a nontrivial interplay between the effective dimer attraction and the exchange anisotropy. In the easy-plane regime, the four featureless phases and two critical phases are found to compete in a complex manner depending on the signs of $J_{\perp, \times}$.

Towards lattice-gas description of low-temperature properties above the Haldane and cluster-based Haldane ground states of a mixed spin-(1,1/2) Heisenberg octahedral chain.

The rich ground-state phase diagram of the mixed spin-(1,1/2) Heisenberg octahedral chain was previously elaborated from effective mixed-spin Heisenberg chains, which were derived by employing a local conservation of a total spin on square plaquettes of an octahedral chain. Here we present a comprehensive analysis of the thermodynamic properties of this model. In the highly frustrated parameter region the lowest-energy eigenstates of the mixed-spin Heisenberg octahedral chain belong to flat bands, which allow a precise description of low-temperature magnetic properties within the localized-magnon approach exploiting a classical lattice-gas model of hard-core monomers. The present article provides a more comprehensive version of the localized-magnon approach, which extends the range of its validity down to a less frustrated parameter region involving the Haldane and cluster-based Haldane ground states. A comparison between results of the developed localized-magnon theory and accurate numerical methods such as full exact diagonalization and finite-temperature Lanczos technique convincingly evidence that the low-temperature magnetic properties above the Haldane and the cluster-based Haldane ground states can be extracted from a classical lattice-gas model of hard-core monomers and dimers, which is additionally supplemented by a hard-core particle spanned over the whole lattice representing the gapped Haldane phase.

Pair-density-wave superconductor from doping Haldane chain and rung-singlet ladder

We report the numerical discovery of a pair-density-wave (PDW) superconductor from doping either (i) a spin-one Haldane chain or (ii) a two-leg ladder in the rung singlet phase in which the doped charges occupy a single leg. We model these systems using a generalized Kondo model. The itinerant electrons are correlated and described by the $t-J$ model, and are further coupled to a spin $1/2$ Heisenberg model through the Kondo coupling $J_K$. When the density of electrons $x$ is one, the Mott insulator is in a Haldane phase or in a rung singlet phase depending on whether $J_K$ is negative or positive. Upon doping, a pair-density-wave with $\mathbf Q=\pi$ can emerge for both signs of $J_K$. In the $J_K \rightarrow -\infty$ limit, the model reduces to the recently proposed type II t-J model and we observes a continuous transition between the PDW superconductor and an unconventional Luttinger liquid phase with doping. We also identify a composite order parameter for the superconductor, which can be understood as a Cooper pair formed by two nearby fermionic spin-polarons. Our model and the predicted PDW phase can be experimentally realized by doping S=1 chains formed by Ni$^{2+}$ in a solid state system or a two-leg ladder of fermionic cold atoms with a potential bias between legs, which preferentially dopes carriers into a single leg.

Realizing the symmetry-protected Haldane phase in Fermi–Hubbard ladders

Topology in quantum many-body systems has profoundly changed our understanding of quantum phases of matter. The model that has played an instrumental role in elucidating these effects is the antiferromagnetic spin-1 Haldane chain1,2. Its ground state is a disordered state, with symmetry-protected fourfold-degenerate edge states due to fractional spin excitations. In the bulk, it is characterized by vanishing two-point spin correlations, gapped excitations and a characteristic non-local order parameter3,4. More recently it has been understood that the Haldane chain forms a specific example of a more general classification scheme of symmetry-protected topological phases of matter, which is based on ideas connected to quantum information and entanglement5–7. Here, we realize a finite-temperature version of such a topological Haldane phase with Fermi–Hubbard ladders in an ultracold-atom quantum simulator. We directly reveal both edge and bulk properties of the system through the use of single-site and particle-resolved measurements, as well as non-local correlation functions. Continuously changing the Hubbard interaction strength of the system enables us to investigate the robustness of the phase to charge (density) fluctuations far from the regime of the Heisenberg model, using a novel correlator.

Dynamics for the Haldane phase in the bilinear-biquadratic model

The bilinear-biquadratic model is a promising candidate to study spin-1 systems and to design quantum simulators based on its underlying Hamiltonian. The variety of different phases contains among other valuable and exotic phases the Haldane phase. We study the Kibble-Zurek physics of linear quenches into the Haldane phase. We outline ideal quench protocols to minimize defects in the final state while exploiting different linear quench protocols via the uniaxial or interaction term. Furthermore, we look at the fate of the string order when quenching from a topologically nontrivial phase to a trivial phase. Our studies show this depends significantly on the path chosen for quenching; for example, we discover quenches from N\'eel to Haldane phase which reach a string order greater than their ground state counterparts for the initial or final state at intermediate quench times.

Hubbard model for spin-1 Haldane chains

The Haldane phase for antiferromagnetic spin-1 chains is a celebrated topological state of matter, featuring gapped excitations and fractional spin-1/2 edge states. Here, we provide numerical evidence that this phase can be realized with a Hubbard model at half filling, where each $s=1$ spin is stored in a four-site fermionic structure. We find that the noninteracting limit of our proposed model describes a one-dimensional topological insulator, and we conjecture it to be adiabatically connected to the Haldane phase. Our work suggests a route to build spin-1 networks using Hubbard model quantum simulators.

Genuine Tripartite Entanglement as a Probe of Quantum Phase Transitions in a Spin‐1 Heisenberg Chain with Single‐Ion Anisotropy

AbstractThe quantum phase transitions of spin‐1 Heisenberg chains with an easy‐axis anisotropy Δ and a uniaxial single‐ion anisotropy D are studied using a multipartite entanglement approach. The genuine tripartite entanglement between the spin blocks, measured by the tripartite qutrit hyperdeterminant, is calculated within the quantum renormalization group method. Using this approach, the phase boundaries between the topological Haldane, large‐D, and anti‐ferromagnetic Néel phases are determined in the half plane with . When the size of the spin blocks increases, the genuine tripartite entanglement between the blocks exhibits a nonzero plateau in the topological Haldane phase, and experiences abrupt drops at both the phase boundaries between the Haldane–large‐D and Haldane–Néel phases, which suggests the usage of genuine multipartite entanglement as a probe of topological phases in spin systems.

Bilinear-biquadratic spin-1 model in the Haldane and dimerized phases

We study the low-lying spectrum of the bilinear-biquadratic Heisenberg model in the dimerized and Haldane phases using a tensor renormalization method. At the critical point $\ensuremath{\theta}=\ensuremath{-}\ensuremath{\pi}/4$, the finite-size spectrum predicted by the Wess-Zumino-Witten (WZW) model can only partly be confirmed. We find a singlet-singlet gap which does not fit into the WZW systematics. The results obtained are compared to Bethe ansatz, exact diagonalization, and density-matrix renormalization group calculations for specific parameters.