Haldane State Research Articles

Criticality at the Haldane-insulator charge-density-wave quantum phase transition

Exploiting the entanglement concept within a matrix-product-state based infinite density-matrix renormalization group approach, we show that the spin-density-wave and bond-order-wave ground states of the one-dimensional half-filled extended Hubbard model give way to a symmetry-protected topological Haldane state in case an additional alternating ferromagnetic spin interaction is added. In the Haldane insulator the lowest entanglement level features a characteristic twofold degeneracy. Increasing the ratio between nearest-neighbor and local Coulomb interaction $V/U$, the enhancement of the entanglement entropy, the variation of the charge, spin and neutral gaps, and the dynamical spin/density response signal a quantum phase transition to a charge-ordered state. Below a critical point, which belongs to the universality class of the tricritical Ising model with central charge 7/10, the model is critical with $c=1/2$ along the transition line. Above this point, the transition between the Haldane insulator and charge-density-wave phases becomes first order.

Competing valence-bond states of spin-32fermions on a strongly coupled ladder

We study the possible ground-state configurations of two strongly coupled chains of charge neutral spin-$\frac{3}{2}$ fermionic atoms interacting via short-range van der Waals interaction. The coupling between the two chains is realized by a relatively large hopping amplitude. Exploiting the fact that such a ladder configuration can be mapped to an effective one-band model, we analyze the emerging ground states of the system. We show that various spatially inhomogeneous states, valence-bond states, and plaquette states compete depending on the filling and the ratio of the interaction strengths in the singlet and quintet scattering channel. We find that a Luttinger liquid state is the ground state of the strongly coupled ladder in an extended region of the parameter space, and we also show that a topologically nontrivial charge Haldane state can emerge in the strongly coupled ladder at quarter and three-quarter fillings.

Time evolution of the haldane state after a quantum quench

In the framework of the Matrix Product States representation the effect of a sudden quench of the uniaxial anisotropy on the time evolution of the Haldane state has been investigated. The existence of the non-vanishing string correlations in the limit of a large distance in the Haldane phase has been verified. The overlap of the initial and time-evolved states, the so-called Loschmidt echo, has been investigated.

Haldane State Formed by Oxygen Molecules Encapsulated in Single-Walled Carbon Nanotubes

We report on the results of X-ray diffraction (XRD), magnetic susceptibility, and high-field magnetization measurements of oxygen molecules, which are unique magnetic homonuclear diatomic molecules with spin-1, encapsulated in single-walled carbon nanotubes (SWCNTs) with diameters of about 0.8 nm. Antiferromagnetic interactions between neighboring oxygen molecules are expected in SWCNTs, resulting in the formation of a spin-1 one-dimensional Heisenberg antiferromagnet, known as a Haldane magnet. The XRD pattern can be predicted accurately by considering the expected oxygen molecule alignment. The temperature evolution of the magnetic susceptibility and the high-field magnetization curve are typical of those for a Haldane magnet with spin-1. The results indicate that the Haldane state has been realized in a nanospaced material for the first time. This provides an alternative to the conventional condensed matter approach to forming quantum spin systems.

Magnetization and phase transition induced by circularly polarized laser in quantum magnets

We theoretically predict a nonequilibrium phase transition in quantum spin systems induced by a laser, which provides a purely quantum-mechanical way of coherently controlling magnetization. Namely, when a circularly polarized laser is applied to a spin system, the magnetic component of a laser is shown to induce a magnetization normal to the plane of polarization, leading to an ultrafast phase transition. We first demonstrate this phenomenon numerically for an $S=1$ antiferromagnetic Heisenberg spin chain, where a new state emerges with magnetization perpendicular to the polarization plane of the laser in place of the topologically ordered Haldane state. We then elucidate its physical mechanism by mapping the system to an effective static model. The theory also indicates that the phenomenon should occur in general quantum spin systems with a magnetic anisotropy. The required laser frequency is in the terahertz range, with the required intensity being within a prospective experimental feasibility.

Doubly degenerate entanglement spectrum and nonzero string order in the spin-1/2 Heisenberg–Ising chain

The ground-state properties and quantum phase transitions (QPTs) in spin-1/2 Heisenberg-Ising alternating chain has been investigated by the iTEBD algorithm. Four different ground-state phases, i.e., a ferromagnetic phase (FM), an antiferromagnetic phase (AF), a stripe phase (SP), and a disordered phase were distinguished. The disordered phase, which has nonzero string orders and the doubly degenerate entanglement spectrum, was observed as Heisenberg coupling JH>0.5. The disordered phase in such a model is found to belong to the same topological phase as the Haldane state. In the disordered phase, every two nearest-neighbor spin-1/2 spins connected by the Ising coupling behave like an integer (S=1) spin. Furthermore, the QPTs from the disordered phase to the AF and SP phases belong to the Ising universality class with central charges c=c¯=1/2.

Fidelity susceptibility and conductivity of the current in one-dimensional lattice models with open or periodic boundary conditions

We study, both numerically and analytically, the finite-size scaling of the fidelity susceptibility ${\ensuremath{\chi}}_{J}$ with respect to the charge or spin current in one-dimensional lattice models and relate it to the low-frequency behavior of the corresponding conductivity. It is shown that in gapless systems with open boundary conditions the leading dependence on the system size $L$ stems from the singular part of the conductivity and is quadratic, with a universal form ${\ensuremath{\chi}}_{J}=[7\ensuremath{\zeta}(3)/2{\ensuremath{\pi}}^{4}]K{L}^{2}$, where $K$ is the Luttinger liquid parameter and $\ensuremath{\zeta}(x)$ is the Riemann $\ensuremath{\zeta}$ function. In contrast to that for periodic boundary conditions the leading system size dependence is directly connected to the regular part of the conductivity and is subquadratic, ${\ensuremath{\chi}}_{J}\ensuremath{\propto}{L}^{\ensuremath{\gamma}}$, where the $K$-dependent exponent $\ensuremath{\gamma}$ is equal to 1 in most situations (as a side effect, this relation provides an alternative way to study the low-frequency behavior of the regular part of the conductivity). For open boundary conditions, we also study another current-related quantity, the fidelity susceptibility to the lattice tilt ${\ensuremath{\chi}}_{\mathcal{P}}$, and show that it scales as the quartic power of the system size, ${\ensuremath{\chi}}_{\mathcal{P}}=[31\ensuremath{\zeta}(5)/8{\ensuremath{\pi}}^{6}](K{L}^{4}/{u}^{2})$, where $u$ is the sound velocity. Thus, the ratio ${L}^{2}{\ensuremath{\chi}}_{J}/{\ensuremath{\chi}}_{\mathcal{P}}$ directly measures the sound velocity in open chains. The behavior of the current fidelity susceptibility in gapped phases is discussed, particularly in the topologically ordered Haldane state.

The exact ground state and the origin of the spin gap in the distorted mixed spin (1, 1/2) diamond chain

By using the method of exact diagonalization, we investigated the properties of the distorted mixed spin (1, 1/2) diamond chain with antiferromagnetic interactions along the rung and leg. The ground states of this model contain the sawtooth chain state and the rung dimer plus Haldane state. The research on the origin of the spin gap of the model discloses that there are three different types of spin excitations at different parameter regimes due to the competition among the interactions J1, J2 and J3.

Reasonable Faith by John Haldane

BOOK REVIEWS 141 of serious theological objections from both ecumenical partners and Catholic theologians, and ready to explain the principles and perspectives that govern his own position, Matthew Levering reconstitutes a solid theological account of the hierarchical or ministerial priesthood in the Catholic Church. Levering’s theological colleagues owe him a profound debt of gratitude for providing such a masterful synthesis. SARA BUTLER University of St. Mary of the Lake Mundelein, Illinois Reasonable Faith. By JOHN HALDANE. London and New York: Routledge Press, 2010. Pp. 201. $39.99 (paper). ISBN: 978-0-415-43025-8. This collection of essays by John Haldane, Professor of Philosophy at the University of St. Andrews in Scotland, is something of a companion piece to his earlier collection of essays, Faithful Reason (2004). While the earlier volume focuses on Catholic topics viewed through a philosophical lens, this current collection might be thought of as a series of essays on philosophical topics seen through a Catholic, and more specifically Thomist, lens. Even more specifically, Haldane approaches Thomas from a perspective that has been dubbed “analytical Thomism,” which, in its broadest sense, is characterized by its commitment both to the thought of Thomas Aquinas and to the insights of Anglo-American philosophy. Indeed, despite the suspicions of some Thomists that “analytic philosophy” is synonymous with “anti-metaphysical logic-chopping,” Haldane sees a profound sympathy between these two traditions, grounded in what he calls their shared “empiricalist” (as distinct from “empiricist”) approach, which holds that “all natural knowledge is acquired through experience (broadly conceived of) or formed by reflection upon it” (11). Thomas and Aristotle exemplify this empiricalist approach, but so too do Thomas Reid, C. S. Peirce, John Henry Newman, Ludwig Wittgenstein, G. E. M. Anscombe, Peter Geach, P. F. Strawson, Hillary Putnam, and Nicholas Rescher, to mention some of the figures who appear in Haldane’s essays. Ultimately the unity of this collection rests upon the way in which it displays, through its own philosophical performance, the mutual benefits gained by Thomism and analytic philosophy from their encounter with each other. For Haldane, the strength of the analytic tradition is in “the construction and dissection of arguments” (12). Anyone who has come away from an encounter with Heidegger or Derrida or Deleuze, dazzled by the verbal pyrotechnics but wondering whether any actual argument has been made or refuted, is likely to BOOK REVIEWS 142 find analytic clarity refreshing. Moreover, a Thomist will recognize in analytic philosophy the same love of precision and distinction, as well as attentiveness to language, that characterizes Thomas’s own writings. At the same time, Haldane states quite forthrightly the two chief deficiencies of analytic philosophy as this is usually practiced. First, analytic philosophers can be quite ahistorical in their approach, seeming to favor the crystalline world of logical analysis over the messiness of the history of philosophy. This weakness, I would note, is one that analytic philosophy shares with some forms of Thomism. Second, many analytic philosophers treat the construction and dissection of arguments as an end in itself, ignoring or actively eschewing any larger purpose to the philosophical project. This latter weakness seems to be of particular concern to Haldane, and it is here that he sees analytic philosophy as having the most to gain from an encounter with Thomas Aquinas, for whom the making of arguments and the drawing of distinctions is never an end but always a means of attaining an evergreater love of wisdom. Though Haldane says that he has sought “to weave the chapters into . . . a continuous narrative,” the book seems to hang together fairly loosely. The essays are grouped together in two parts. In good Thomist fashion, the prima pars, under the heading “Reason, Faith and God,” includes essays on God and metaphysics, while secundapars,under the heading “Reason, Faith and the Soul,” includes essays in the broad area of philosophical anthropology. Though most of the essays are interesting in their own way, they definitely retain the marks of their original contexts and show a certain diversity in both form and content. For example, chapter 2, “A Thomist Metaphysics,” was originally written as a contribution to a general work on metaphysics and, as is appropriate...

How to distinguish the Haldane/Large-D state and the intermediate-D state in an S = 2 quantum spin chain with the XXZ and on-site anisotropies

We numerically investigate the ground-state phase diagram of an S = 2 quantum spin chain with the XXZ and on-site anisotropies described by ℋ = Σj(SjxSj+ 1x +SjySj + 1y + Δ Sjz Sj+1z) + D Σj(Sjz)2, where Δ denotes the XXZ anisotropy parameter of the nearest-neighbor interactions and D the on-site anisotropy parameter. We restrict ourselves to the Δ > 0 and D > 0 case for simplicity. Our main purpose is to obtain the definite conclusion whether there exists or not the intermediate-D (ID) phase, which was proposed by Oshikawa in 1992 and has been believed to be absent since the DMRG studies in the latter half of 1990's. In the phase diagram with Δ > 0 and D > 0 there appear the XY state, the Haldane state, the ID state, the large-D (LD) state and the Néel state. In the analysis of the numerical data it is important to distinguish three gapped states; the Haldane state, the ID state and the LD state. We give a physical and intuitive explanation for our level spectroscopy method how to distinguish these three phases.

Ground-state phase diagram of S = 2 quantum spin chain with the XXZ and on-site anisotropies

By use of mainly numerical methods, we investigate the ground-state phase diagram of an S = 2 quantum spin chain with the XXZ and on-site anisotropies described by ℋ = Σj(SjxSj+1x + SyjSyj+1 + ΔSzjSzj+1 + D Σj(Szj)2, where Δ denotes the XXZ anisotropy parameter of the nearest-neighbor interactions and D the on-site anisotropy parameter. In the phase diagram with Δ > 0 and D > 0 there appear the XY phase, the Haldane/large-D (LD) phase, the intermediate-D (ID) phase, and the Néel phase. Among them the ID phase was proposed by Oshikawa in 1992 and has been considered to be absent since the DMRG studies in the latter half of 1990's. And also, the Haldane state and the LD state belong to the same phase, which should be named Haldane/LD phase.

Gaps tunable by electrostatic gates in strained graphene

We show that when the pseudomagnetic fields created by long-wavelength deformations are appropriately coupled with a scalar electric potential, a significant energy gap can emerge due to the formation of a Haldane state. Ramifications of this physical effect are examined through the study of various strain geometries commonly seen in experiments, such as strain superlattices and wrinkled suspended graphene. Of particular technological importance, we consider setups where this gap can be tunable through electrostatic gates, allowing for the design of electronic devices not realizable with other materials.

Haldane, Large-D, and Intermediate-DStates in anS= 2 Quantum Spin Chain with On-Site andXXZAnisotropies

Using mainly numerical methods, we investigate the ground-state phase diagram of the S=2 quantum spin chain described by $H = \sum_j (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y + \Delta S_j^z S_{j+1}^z) + D \sum_j (S_j^z)^2$, where $\Delta$ denotes the $XXZ$ anisotropy parameter of the nearest-neighbor interactions and $D$ the on-site anisotropy parameter. We restrict ourselves to the case with $\Delta \ge 0$ and $D \ge 0$ for simplicity. Each of the phase boundary lines is determined by the level spectroscopy or the phenomenological renormalization analysis of numerical results of exact-diagonalization calculations. The resulting phase diagram on the $\Delta$-$D$ plane consists of four phases; the XY 1 phase, the Haldane/large-$D$ phase, the intermediate-$D$ phase and the N\'eel phase. The remarkable natures of the phase diagram are: (1) the Haldane state and the large-$D$ state belong to the same phase; (2) there exists the intermediate-$D$ phase which was predicted by Oshikawa in 1992; (3) the shape of the phase diagram on the $\Delta$-$D$ plane is different from that believed so far. We note that this is the first report of the observation of the intermediate-$D$ phase.

Exact spin-cluster ground states in a mixed diamond chain

The mixed diamond chain is a frustrated Heisenberg chain composed of successive diamond-shaped units with two kinds of spins of magnitudes S and S/2 (S: integer). Ratio $lambda$ of two exchange parameters controls the strength of frustration. With varying $lambda$, the Haldane state and several spin cluster states appear as the ground state. A spin cluster state is a tensor product of exact local eigenstates of cluster spins. We prove that a spin cluster state is the ground state in a finite interval of $lambda$. For S=1, we numerically determine the total phase diagram consisting of five phases.

Erratum: “Critical Properties of Spin-1 Antiferromagnetic Heisenberg Chains with Bond Alternation and Uniaxial Single-Ion-Type Anisotropy”

One dimensional S =1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D ≥0. The ground states of this model contain the Haldane state, large- D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.

Magnetic properties of the Haldane-gap material [Ni(C2H8N2)2NO2](BF4)

Results of magnetization and high-field electron spin resonance (ESR) studies of the new spin-1 Haldane-chain material [Ni(C2H8N2)2NO2](BF4) (NENB) are reported. A definite signature of the Haldane state in NENB was obtained. From the analysis of the frequency–field dependence of magnetic excitations in NENB, the spin-Hamiltonian parameters were calculated, yielding Δ/kB=17.4 K, g∥=2.14, D/kB=7.5 K and |E/kB|=0.7 K for the Haldane gap, g factor and the crystal-field anisotropy constants, respectively. The presence of fractional S=1/2 chain-end states, revealed by ESR and magnetization measurements, is found to be responsible for spin-glass freezing effects. In addition, extra states in the excitation spectrum of NENB have been observed in the vicinity of the Haldane gap; their origin is discussed.

Half-ordered state in the anisotropic Haldane-gap antiferromagnetNi(C5D14N2)2N3(PF6)

Neutron diffraction experiments performed on the Haldane gap material NDMAP in high magnetic fields applied at an angle to the principal anisotropy axes reveal two consecutive field-induced phase transitions. The low-field phase is the gapped Haldane state, while at high fields the system exhibits 3-dimensional long-range Neel order. In a peculiar phase found at intermediate fields only half of all the spin chains participate in the long-range ordering, while the other half remains disordered and gapped.

Electron Spin Resonance in Ferromagnetic-Dominant Ferromagnetic–Antiferromagnetic Alternating Heisenberg Chains (4-BzpipdH)CuCl3

Ferromagnetic–antiferromagnetic (F–AF) alternating Heisenberg chains with S 1⁄4 1=2 form the Haldane state because a pair of ferromagnetically coupled S 1⁄4 1=2 spins (S1 and S2) behaves as a Ŝ 1⁄4 1 spin which is defined as Ŝ S1 þ S2. In a high enough temperature T region, however, the spins behave as individual S 1⁄4 1=2 spins because the spin correlation is absent or very weak. Therefore, a crossover of the spin state from S 1⁄4 1=2 to Ŝ 1⁄4 1 is expected when T decreases and the spin correlation develops. We already found such crossover of the spin state in (CH3)2CHNH3CuCl3 (abbreviated as IPACuCl3), which consists of a ferromagnetic-dominant F–AF chains and no magnetic long-range order appears at a finite T . From the magnetic susceptibility ðTÞ measurements, two kinds of exchange interactions were estimated to be JF=k 1⁄4 þ54:1K and JAF=k 1⁄4 23:5K when the Hamiltonian is expressed as

Magnetic phase transition induced by bond randomness in mixed alternating Heisenberg chains(CH3)2CHNH3Cu(ClxBr1−x)3observed by magnetization process

Mixed magnetic systems $({\mathrm{CH}}_{3}{)}_{2}{\mathrm{CHNH}}_{3}\mathrm{Cu}({\mathrm{Cl}}_{x}{\mathrm{Br}}_{1\ensuremath{-}x}{)}_{3}$ are good candidates for experimental studies of the effect of bond randomness in both the Haldane state and the singlet dimer state. We investigated the magnetization process in these mixed compounds with various values of $x (0l~xl1)$ under an external field up to 41 T at 1.7 K. As a result, the existence of a finite energy gap was confirmed for $0.87l~xl~1,$ while a typical spin flop transition, which indicates absence of the energy gap, was detected for $0.44lxl0.87.$ From the variation of the spin flop transition field with x, the expected value of spins $〈{S}_{i}^{z}〉$ was found to reach its maximum at $x\ensuremath{\simeq}0.71,$ the point at which bond randomness is highest. For $0l~xl~0.44,$ on the other hand, the magnetization curves showed indistinct increases, which made it difficult to ascertain whether the energy gap exists.

First-order spin-Peierls transition in S=1 antiferromagnetic Heisenberg chains

We investigate a one-dimensional S=1 antiferromagnetic Heisenberg model coupled to a lattice distortion by a quantum Monte Carlo method. The instability to a dimerized chain depends on the value of the spin–phonon coupling, unlike the case of S=1/2. We find the first-order transition between the uniform Haldane state and the dimerized state. Two non-magnetic states coexist in the chain and a domain-wall structure is observed. In the coexistent state, we find that the length of the uniform Haldane region shrinks, which causes an asymmetrical bond distortion in the dimerized region.