No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Monte-Carlo study of correlations in quantum spin chains at non-zero temperature
Abstract
Antiferromagnetic Heisenberg spin chains with various spin values (S=1/2,1,3/2,2,5/2) are studied numerically with the quantum Monte-Carlo method. Effective spin S chains are realized by ferromagnetically coupling n=2S antiferromagnetic spin chains with S=1/2. The temperature dependence of the uniform susceptibility, the staggered susceptibility, and the static structure factor
peak intensity are computed down to very low temperatures, . The correlation length at each temperature is deduced from numerical measurements of the instantaneous spin-spin correlation
function. At high temperatures, very good agreement with exact results for the classical spin chain is obtained independent
of the value of S. For the S=2 chain which has a gap , the correlation length and the uniform susceptibility in the temperature range are well predicted by the semi-classical theory of Damle and Sachdev.
... Due to this gap A, the magnetic correlation length stays practically constant for 0 < _ T < A, taking the value _(0) --6 (in lattice units). According to numerical calculation [2], 4.(T) is expected to decrease with increasing temperature for T> A. 4_(T) is a quantity which is difficult to probe experimentally, particularly at high temperature where inelastic neutron scattering measurements are hampered by the damping of the spin excitations [3]. On the other hand, the translational invariance inhibits the occurrence of a static distribution of local mag-netic moments, which could be detectable in principle by nuclear magnetic resonance (NMR) spectra. ...
... In Table 1, the decay length (T) of the magnetization, obtained from the data in Fig. 6 through Eq. (4), is compared with the correlation length 4 for a translationally invariant chain in the thermodynamic limit [2]. The good agreement between the two quantities indicates that the boundary magnetization can be used to derive the correlation length of the L = no system. ...
By means of89Y nuclear magnetic resonance (NMR) spectra direct evidence of staggered magnetization induced by a uniform field has been
obtained in the Heisenberg antiferromagneticS = 1 chain Y2BaNi1−xMgxO5. A correspondence between the resonance lines and the lattice positions is established, providing an image of the alternating
magnetic moments that develop around the Mg impurities at the chain boundaries. The amplitude of these moments is found to
decrease exponentially from the edges, with a characteristic decay distance equal to the magnetic correlation length numerically
evaluated for an infinite chain. While the magnetization pertaining to ions far from the boundaries behaves as in typical
gapped systems, the edge spins exhibit anS = 1/2 Curie-like deviation. These results promote the NMR approach to access the spin-spin correlation function in antiferro-magnetic
quantum spin systems at finite temperatures.
... The correlation length ξ is typically characterized as the length scale over which the magnetic spins can perceive one another. It can be extracted by considering the absolute value of the spin-pair correlations and modelling them with the Ornstein-Zernike equation [38]: ...
Frustrated magnetism in face-centred cubic (fcc) magnetic sublattices remains underexplored but holds considerable potential for exotic magnetic behaviour. Here we report on the crystal structure, magnetic and thermodynamic properties of the A-site-vacant double hydroxide perovskite MnSn(OH). Despite dominant antiferromagnetic interactions among Mn moments, evidenced by a negative Curie-Weiss temperature, the lack of a sharp thermodynamic transition down to 350mK implies the absence of long-range magnetic order. However, a broad hump in the specific heat at 1.6K suggests short-range correlations. Neutron diffraction at low temperatures confirms the presence of three-dimensional (3D) antiferromagnetic correlations, manifested as diffuse magnetic scattering with a correlation length {\AA} and magnetic propagation vectors and at 20mK.
... Our one-dimensional example is the ground state of an (S=1) 1-dimensional antiferromagnetic Heisenberg chain [64,65]. Recalling (8), we use the magnitude squared of the correlation function as an estimate for the mutual information. ...
We examine how to construct a spatial manifold and its geometry from the entanglement structure of an abstract quantum state in Hilbert space. Given a decomposition of Hilbert space into a tensor product of factors, we consider a class of "redundancy-constrained states" in that generalize the area-law behavior for entanglement entropy usually found in condensed-matter systems with gapped local Hamiltonians. Using mutual information to define a distance measure on the graph, we employ classical multidimensional scaling to extract the best-fit spatial dimensionality of the emergent geometry. We then show that entanglement perturbations on such emergent geometries naturally give rise to local modifications of spatial curvature which obey a (spatial) analog of Einstein's equation. The Hilbert space corresponding to a region of flat space is finite-dimensional and scales as the volume, though the entropy (and the maximum change thereof) scales like the area of the boundary. A version of the ER=EPR conjecture is recovered, in that perturbations that entangle distant parts of the emergent geometry generate a configuration that may be considered as a highly quantum wormhole.
... Our data are very well described by Eq. (3.14), with 0 = 0.2823 (16) and Λ = 22.7 (20), in the temperature range 0.002 ≤ ≤ 0.1 , These parameters differ from the ones reported in Refs. [75][76][77] where QMC was performed at higher temperature. ...
While classical spin systems in random networks have been intensively studied, much less is known about quantum magnets in random graphs. Here, we investigate interacting quantum spins on small-world networks, building on mean-field theory and extensive quantum Monte Carlo simulations. Starting from one-dimensional (1D) rings, we consider two situations: all-to-all interacting and long-range interactions randomly added. The effective infinite dimension of the lattice leads to a magnetic ordering at finite temperature with mean-field criticality. Nevertheless, in contrast to the classical case, we find two distinct power-law behaviors for versus the average strength of the extra couplings. This is controlled by a competition between a characteristic length scale of the random graph and the thermal correlation length of the underlying 1D system, thus challenging mean-field theories. We also investigate the fate of a gapped 1D spin chain against the small-world effect.
... For example, it is clear that since only neighboring sites are coupled in singlets, the correlation length, n, of the VBS state must be small and indeed n VBS % 0:9 (1 corresponds to the nearest neighbor), [19] while the correlations are more extended in the true ground state, n $ 6 as T ! 0: [22,23] Since the ground state wave function is composed of "elemental" spin singlets separated from higher energy states by the spin gap, the whole chain will also have the spin gap D H : This parameter in fact defines various physical properties of integer spin chains; for example, the magnetic susceptibility behaves as vðTÞ $ e ÀD H =T = ffiffiffi ffi T p , [24] while magnetic part of the specific heat C m ðTÞ $ e ÀD H =T =T 2=3 : [25] The simplest state, which would compete with the VBS state in AKLT model (b ¼ 1/3) or with the spin singlet ground state of a pure Heisenberg model (b ¼ 0), is the dimerized state characterized by alternation parameter d, see Equation (3) later on. For S ¼ 1 chain, this is the state, when both "elemental" r 1 and r 2 spins form valence bonds only with one of neighbors. ...
In 1983, F. Duncan M. Haldane predicted a singlet ground state for isolated integer-spin one-dimensional antiferromagnets with low single-ion anisotropy D. Since then, a lot of species containing chains of integer spin ions were tested to check the basic conjecture on an energy gap separating the continuum of the excited states from the ground state. As a result of these studies, it has been established that there are numerous states competing with the Haldane phase, namely long-range ordered, dimerized, and large-D phases. The long-range magnetic order takes place due to sufficiently strong exchange interactions between adjacent chains. Dimerization results from the alternation of the exchange interactions within the chains. Both uniaxial and rhombic single-ion anisotropies can suppress the Haldane phase, which is robust only until some critical values. The choice between the competing phases depends also on exchange anisotropy. Excellent reviews on the basic results obtained during the first 20 years of investigation of these phenomena provided solid background for the future studies. Here, we present some developments in this field obtained over the next two decades of research on spin-1 chain systems.
... The only clear and conclusive feature in the high temperature data associated with a magnetic transition is the feature at 313 ± 2 K. The broad local maximum in magnetization well above the transition temperature can be associated with low dimensional, or linear chain anisotropic Heisenberg anitiferromagnetism [19][20][21][22][23]. The fitted Curie Weiss temperatures for various axes were obtained to be θ 010 = - 815 K, θ 100 = -750 K, and θ 001 = -835 K. From the average slope of Curie Weiss plot, the effective moment of Mn is found to be ∼2.5µ ...
We present self flux growth and characterization of single crystalline AlMn2B2. It is an orthorhombic (space group Cmmm), layered material with a platelike morphology. The anisotropic bulk magnetization data, electrical transport, and B11 nuclear magnetic resonance (NMR) data revealed an antiferromagnetic (AFM) transition at 313±2K. In the magnetization data, there is also a broad local maximum significantly above the AFM transition that could be a signature of low-dimensional magnetic interactions in AlMn2B2.
... The only clear and conclusive feature in the high temperature data associated with a magnetic transition is the feature at 313 ± 2 K. The broad local maximum in magnetization well above the transition temperature can be associated with low dimensional, or linear chain anisotropic Heisenberg anitiferromagnetism [19][20][21][22][23]. The fitted Curie Weiss temperatures for various axes were obtained to be θ 010 = - 815 K, θ 100 = -750 K, and θ 001 = -835 K. From the average slope of Curie Weiss plot, the effective moment of Mn is found to be ∼2.5µ ...
We present self flux growth and characterization of single crystalline AlMnB. It is an orthorhombic (space group Cmmm), layered material with a plate like morphology. The anisotropic bulk magnetization data, electrical transport and B nuclear magnetic resonance(NMR) data revealed an antiferromagnetic (AFM) transition at 313 2 K. In the magnetization data, there is also a broad local maximum significantly above the AFM transition that could be a signature of low dimensional magnetic interactions in AlMnB.
... The quasi one-dimensional (1D) magnetic system has received considerable attention due to the unique magnetic properties. It has been proved in theory that there is no magnetic order formed even at 0 K in a purely one-dimensional spin 1/2 chain system but there will be a Neel ordering state if a three-dimensional(3D) antiferromagnet is present due to the appearance of the inter-chain interactions [1][2][3][4] . Such a quasi 1D system with the week coupling between the magnetic chains will exhibit a crossover from 1D magnetic behavior [5][6][7][8][9][10] at high temperatures to a 3D ordered state at the low temperatures. ...
We performed susceptibility, magnetization, specific heat, and single crystal neutron diffraction measurements on single crystalline BaMnSiO. Based on the results, we revisited its spin structure with a more accurate solution and constructed a magnetic phase diagram with applied field along the b-axis, which contains a spin flop transition around 6 T. We also used susceptibility, magnetization, and specific heat results confirmed the ferrimagnetic-like magnetism in polycrystalline BaCoSiO. Furthermore, we performed LSDA + U calculations for the BaMSiO (M = Cu, Co, and Mn) system. Our discussions based on the comparison among the obtained magnetic exchange interactions suggest the different structures and electronic configurations are the reasons for the different magnetic properties among the system members.
... Our one-dimensional example is the ground state of an (S=1) 1-dimensional antiferromagnetic Heisenberg chain [60,61]. Recalling (8), we use the magnitude squared of the correlation function as an estimate for the mutual information. ...
We examine how to construct a spatial manifold and its geometry from the entanglement structure of an abstract quantum state in Hilbert space. Given a decomposition of Hilbert space into a tensor product of factors, we consider a class of "redundancy-constrained states" in that generalize the area-law behavior for entanglement entropy usually found in condensed-matter systems with gapped local Hamiltonians. Using mutual information to define a distance measure on the graph, we employ classical multidimensional scaling to extract the best-fit spatial dimensionality of the emergent geometry. We then show that entanglement perturbations on such emergent geometries naturally give rise to local modifications of spatial curvature which obey a (spatial) analog of Einstein's equation. The Hilbert space corresponding to a region of flat space is finite-dimensional and scales as the volume, though the entropy (and the maximum change thereof) scales like the area of the boundary. A version of the ER=EPR conjecture is recovered, in that perturbations that entangle distant parts of the emergent geometry generate a configuration that may be considered as a highly quantum wormhole.
... for which quantum Monte Carlo approach gives [23,24] a = 0.32, 0.30 and b = 5.9, 9.8. Results for random δJ = 0 shown in Fig. 1a clearly indicate that the increasing δJ reduces χ π and consequently leads to a systematic decrease of T N (for fixed J ⊥ and J) as shown in Fig. 1b. ...
The ordering of weakly coupled random antiferromagnetic S=1/2 chains, as
relevant for recent experimentally investigated spin chain materials, is
considered theoretically. The one-dimensional isotropic Heisenberg model with
random exchange interactions is treated numerically on finite chains with the
density-matrix renormalization-group approach as well as with the standard
renormalization analysis, both within the mean-field approximation for
interchain coupling . Results for the ordering temperature and
for the ordered moment are presented and are both reduced with the
increasing disorder agreeing with experimental observations. The most
pronounced effect of the random singlet concept appears to be a very large span
of local ordered moments, becoming wider with decreasing ,
consistent with SR experimental findings.
... To explain the observed temperature dependent magnetic susceptibility at a microscopic level, we consider a model in which impurities or structural imperfections break the quasi-1D magnetic chains into weakly interacting segments of finite size. We have calculated the susceptibility of finite spin-chain segments using quantum Monte Carlo simulations utilizing the loop cluster algorithm [17,18]. The only parameters in these calculations are the size of the chain segments, and the intrachain magnetic coupling J . ...
We report single-crystal growth and magnetic susceptibility and neutron
diffraction studies of the S=12 quasi-one-dimensional antiferromagnet
CaCu2O3. The structure of this material is similar
to that of the prototype two-leg spin-ladder compound
SrCu2O3. However, the Cu-O-Cu bond angle in the
ladder rungs in CaCu2O3 is equal to 123°, and
therefore the magnetic interaction along the rung is expected to be much
weaker in this material. At high temperatures, the magnetic
susceptibility of CaCu2O3 can be decomposed into a
contribution from one-dimensional antiferromagnetic chains or
finite-size chain segments together with a weak Curie contribution. The
intrachain magnetic exchange constant J||, determined from
the magnetic susceptibility measurements, is 2000+/-300 K.
CaCu2O3 undergoes a Néel transition at
TN=25 K with ordering wave vector of [0.429(5), 12, 12]. The
magnetic structure is incommensurate in the direction of the frustrated
interchain interaction. Weak commensurate magnetic Bragg peaks with the
reduced wave vector (12, 12, 12) are also observed below TN.
Application of a magnetic field induces a metamagnetic transition at
which the incommensurability of the magnetic structure is substantially
reduced. Above the transition field, the material possesses only
short-range magnetic order, and no well defined temperature-driven
transition is observed.
... The high-temperature bump is a typical behavior of the linear-chain Heisenberg antiferromagnet and could be predicted by the modified Bonner-Fisher model for the S=5/2 system with two assumptions: (1) The interchain interactions are small; (2) the manganese ions interact with each other along the individual chain equally, although there are three different Mn sites and three different interactions accordingly, as described in Sec.III-B. The susceptibility χ is then expressed as, 20,21 where ...
The structural and magnetic properties of BaMn2Si2O7 have been investigated.
The magnetic susceptibility and specific heat, measured using single crystals,
suggest that the quasi-one-dimensional magnetism originating from the loosely
coupled Mn2+ chain carrying S=5/2 is present at high temperatures, which is
similar to the other quasi-one-dimensional barium silicates, BaM2Si2O7 (M: Cu
and Co). The Neel temperature (TN=26 K) is high compared to the magnetic
interaction along the chain (J = -6 K). The neutron powder diffraction study
has revealed that the magnetic structure is long-ranged with antiferromagnetic
arrangement along the chain (c) direction and ferromagnetic arrangement along
the a and b axes. The detailed structural analysis suggests that the interchain
interaction via Mn-O-Mn bond along the a axis is relatively large, which makes
the system behave more two-dimensionally in the ac plane and enhances TN.
... While the classical susceptibilities level off at low temperatures, the QMC results decrease to the lowest temperatures. As expected, the differences between the QMC susceptibilities and the classical susceptibilities decrease with increasing spin value S. For high temperatures, as a rule of thumb, the classical susceptibility represents an acceptable approach above the susceptibility maxima, as has been seen in previous works [33,34]. ...
The temperature dependence of the spin susceptibilities of S = 1, [Formula: see text], 2, [Formula: see text] and [Formula: see text] Heisenberg antiferromagnetic 1D spins chains with nearest-neighbor coupling was simulated via quantum Monte Carlo calculations, within the reduced temperature range of 0.005 ≤ T* ≤ 100, and fitted to a Padé approximation with deviations between the simulated and fitted data of the same order of magnitude as or smaller than the quantum Monte Carlo simulation error. To demonstrate the practicality of our theoretical findings, we compare these results with the susceptibility of the well known 1D chain compound TMMC ([(CH(3))(4)N[MnCl(3)]], d(5), S = 5/2) and find that different intra-chain spin-exchange parameters result if we consider the data above and below the structural phase transition reported for TMMC at ∼126 K. The structural phase transition, which gives rise to an anomaly in the magnetic susceptibility, is independent of the magnetic field up to magnetic fields of 7 T. Additionally, we show that the S = 1 system NiTa(2)O(6) with tri-rutile crystal structure can be very well described as a Heisenberg S = 1 spin chain.
... Unfortunately, their algorithm is rather complicated (105 different graphs appear even in the S 1 case), and moreover the Trotter limit is not well defined in the algorithm. Another type of algorithm in the discrete-time representation was used in Ref. [7]. ...
A loop cluster algorithm, which can be applied to quantum spin systems
with arbitrary size of spins, is proposed. We also present a generalized
Swendsen-Wang algorithm for general-S classical Ising models.
... However, before going into the dynamics, it would be important to establish the validity of the present approach by considering static properties. This was done earlier in Ref. 26, in the zero field limit. ...
We develop a semi-classical approximation to electron spin resonance in
quantum spin systems, based on the rotor or non-linear sigma model. The
classical time evolution is studied using molec- ular dynamics while random
initial conditions are sampled using classical Monte Carlo methods. Although
the approximation may be especially powerful in two dimensions, we apply it
here to one- dimensional systems of large spin at intermediate temperatures, in
the presence of staggered and uniform magnetic fields. We first test the
validity of the semi-classical approximation by comparing the magnetization to
quantum Monte Carlo results on S = 2 chains. Then we calculate the ESR
spectrum, finding broad coexisting paramagnetic and spin wave resonances.
The frustrated spin-chain (FSC) systems exhibit exotic ground states and distinct quantum phase transitions. The S=12 FSC is known to exhibit the Kosterlitz-Thouless transition from a commensurate gapless phase to a fully dimerized gapped phase upon the ratio of next-nearest-neighbor to nearest-neighbor coupling (α=J2J1) being tuned. On the other hand, the S=52 FSC system is known to show transitions from a commensurate gapless phase to partially dimerized and incommensurate floating phases [Chepiga, Affleck, and Mila, Phys. Rev. B 105, 174402 (2022)]. While a large region of the floating phase has been theoretically predicted for the S=52 FSC model when α>0.43, it is yet to be explored experimentally. Here, we have investigated a compound K3Fe(MoO4)2(Mo2O7), having well-separated S=52 FSCs. The electronic structure calculations show that the α=J2J1 is close to 0.9, being similar to another FSC compound Bi3FeMo2O12 (α≈1.1). No magnetic long-range order is found down to 0.09 K, despite the relatively sizable Curie-Weiss temperature θCW=−18K. The magnetic heat capacity shows the power-law behavior, indicating that the compound exhibits gapless excitations. Based on the experimental results and the theoretical calculations employed by density functional theory, we argue that the titled system is a possible candidate for exhibiting the floating phase.
While classical spin systems in random networks have been intensively studied, much less is known about quantum magnets in random graphs. Here, we investigate interacting quantum spins on small-world networks, building on mean-field theory and extensive quantum Monte Carlo simulations. Starting from one-dimensional (1D) rings, we consider two situations: all-to-all interacting and long-range interactions randomly added. The effective infinite dimension of the lattice leads to a magnetic ordering at finite temperature Tc with mean-field criticality. Nevertheless, in contrast to the classical case, we find two distinct power-law behaviors for Tc versus the average strength of the extra couplings. This is controlled by a competition between a characteristic length scale of the random graph and the thermal correlation length of the underlying 1D system, thus challenging mean-field theories. We also investigate the fate of a gapped 1D spin chain against the small-world effect.
We investigated the magnetic properties of the low-dimensional BaM2Si2O7 (M = Cu, Co, and Mn) system using both experimental measurements and theoretical calculations. Magnetization, specific heat, and single crystal neutron diffraction measurements have been performed on single crystal BaMn2Si2O7. The spin structure was determined and a magnetic phase diagram with applied field along the b axis was constructed, which contains a spin flop transition around 6 T. Magnetization and specific heat measurements confirmed the presence of weak ferromagnetism in BaCo2Si2O7. Furthermore, we performed local-spin density approximation with on-site Coulomb energy (LSDA+U) calculations for the BaM2Si2O7 (M = Cu, Co, and Mn) system. Based on the first-principles calculations, the origin of the magnetic differences of the three materials is discussed.
A large-scale parallel loop cluster quantum Monte Carlo simulation is presented. On 24,576 nodes of the K computer, one loop cluster Monte Carlo update of the world-line configuration of the S=1∕2 antiferromagnetic Heisenberg chain with 2.6×10⁶ spins at inverse temperature 3.1×10⁵ is executed in about 8.62 s, in which global union-find cluster identification on a graph of about 1.1 trillion vertices and edges is performed. By combining the nonlocal global updates and the large-scale parallelization, we have virtually achieved about 10¹³-fold speed-up from the conventional local update Monte Carlo simulation performed on a single core. We have estimated successfully the antiferromagnetic correlation length and the magnitude of the first excitation gap of the S=4 antiferromagnetic Heisenberg chain for the first time as ξ=1.040(7)×10⁴ and Δ=7.99(5)×10⁻⁴, respectively.
A large-scale parallel loop cluster quantum Monte Carlo simulation is presented. On 24,576 nodes of the K computer, one loop cluster Monte Carlo update of the world-line configuration of the S=1/2 antiferromagnetic Heisenberg chain with spins at inverse temperature is executed in about 8.62 seconds, in which global union-find cluster identification on a graph of about 1.1 trillion vertices and edges is performed. By combining the nonlocal global updates and the large-scale parallelization, we have virtually achieved about -fold speed-up from the conventional local update Monte Carlo simulation performed on a single core. We have estimated successfully the antiferromagnetic correlation length and the magnitude of the first excitation gap of the S=4 antiferromagnetic Heisenberg chain for the first time as and , respectively.
Jury : Henri Alloul, Claude Berthier, Philippe Bourges, Marc Gabay, Maurice Rice, Alain Sacuto
IntroductionNon-critical-scaling: the Other Solutions of the Scaling ModelUniversality Classes and Lower Critical DimensionalityPhase Transition in Layered CompoundsDescription of Ferromagnetic Heisenberg Chains Application to Ferromagnetic Sψ= 1 ChainsApplication to the Spin-1 Haldane ChainConclusion
References Application to Ferromagnetic Sψ= 1 Chains
Inhomogeneity in the ground state is an intriguing, emergent phenomenon in
magnetism. Recently, it has been observed in the magnetostructural channel of
the geometrically frustrated -NaMnO, for the first time in the
absence of active charge degrees of freedom. Here we report an in-depth
numerical and local-probe experimental study of the isostructural sister
compound CuMnO that emphasizes and provides an explanation for the crucial
differences between the two systems. The experimentally verified, much more
homogeneous, ground state of the stoichiometric CuMnO is attributed to the
reduced magnetoelastic competition between the counteracting magnetic-exchange
and elastic-energy contributions. The comparison of the two systems
additionally highlights the role of disorder and allows an understanding of the
puzzling phenomenon of phase separation in uniform antiferromagnets.
Describing the physical properties of quantum materials near critical points with long-range many-body quantum entanglement, this book introduces readers to the basic theory of quantum phases, their phase transitions and their observable properties. This second edition begins with a new section suitable for an introductory course on quantum phase transitions, assuming no prior knowledge of quantum field theory. It also contains several new chapters to cover important recent advances, such as the Fermi gas near unitarity, Dirac fermions, Fermi liquids and their phase transitions, quantum magnetism, and solvable models obtained from string theory. After introducing the basic theory, it moves on to a detailed description of the canonical quantum-critical phase diagram at non-zero temperatures. Finally, a variety of more complex models are explored. This book is ideal for graduate students and researchers in condensed matter physics and particle and string theory.
We investigate both numerically and analytically the behavior of a spin-1 antiferromagnetic isotropic Heisenberg chain in an external uniform magnetic field. Extensive DMRG studies of chains up to N = 80 sites extend previous analyses and obtain the zero-temperature correlation functions. We argue that, despite the presence of a magnetic field, the model is still well described by an O(3) nonlinear σ-model. A saddle-point analysis, that also includes Gaussian fluctuations, is developed for the latter both at zero and finite temperatures. Its predictions are compared with those of the numerical analysis and with the existing experimental literature.
Based on the method considering spin and spatial symmetry, numerical calculations of the spin-level spectra have been performed for n-nuclear cyclic clusters with S
i
= 3/2 (n ≤ 11) and S
i
= 2 (n ≤ 10). The theoretical curves of the magnetic susceptibility, the magnetic contribution to the heat capacity, the internal energy, and the entropy as a function of temperature have been obtained. The theoretical curves of the magnetic susceptibility and the magnetic contribution to the heat capacity have been extrapolated to n → ∞ with a controlled accuracy.
Certain apparently simple antiferromagnets show a broad range of fluctuating behaviour at temperatures below the mean field interaction energy, generally due to particular lattice properties such as low dimensionality or geometric frustration. In such systems competition between qualitatively different ground states is common. A ‘spin-liquid’ ground state, characterised in the ideal case by a total absence of elastic response and possibly by a gap for magnetic excitations can occur while, due to various reasons including non-magnetic dilution, a tendency to ordering can be recovered. A common feature originating from the competing ground states of these unconventional spin systems is the appearance of a fluctuation energy scale well below the typical nearest neighbour interactions, even in the absence of an associated development of long-range spatial order. In this paper we outline the qualitative differences of such a magnetic response with respect to the standard paramagnet-critical regime-ordered state sequence. We examine representative examples of these type of studies on some antiferromagnetic chain compounds and frustrated antiferromagnets. In all these cases we demonstrate the importance of energy resolved observation in understanding the ordered, elastic diffuse and quasi-elastic components of the magnetic scattering.
Based on the previously developed method considering spin and point symmetry, numerical calculations of spin-level spectra
have been performed for n-nuclear cyclic clusters with S
i = 5/2 (n ≤ 9), S
i = 3 (n ≤ 8), and S
i = 7/2 (n ≤ 8). The theoretical curves of the magnetic susceptibility, the magnetic contribution to the heat capacity, and the internal
energy as a function of temperature have been obtained. The theoretical curves of the magnetic susceptibility and magnetic
contribution to the heat capacity have been extrapolated to n → ∞ with a controlled accuracy. These curves are compared to available data.
Study of an impurity-driven phase transition into a magnetically ordered state in the spin-liquid Haldane chain compound PbNi2V2O8 is presented. Both macroscopic magnetization as well as 15V nuclear magnetic resonance (NMR) measurements reveal that the magnetic nature of dopants has a crucial role in determining the stability of the induced long-range magnetic order. In the case of non-magnetic (Mg2+ doping at the Ni2+ sites (S = 1), a metamagnetic transition is observed in relatively low magnetic fields. On the other hand, the magnetic order in magnetically (Co2+) doped compounds survives at much higher magnetic fields and temperatures. We attribute this feature to a significant anisotropic impurity host magnetic interaction. The NMR measurements confirm that the staggered magnetic moments liberated next to the impurity sites, are the reason for the magnetic ordering. In addition, differences in the broadening of NMR spectra and the increase of nuclear spin-lattice relaxation in doped samples indicate an impurity-dependent character for the dominant electron-spin correlations; the latter begin to develop at rather high temperatures with respect to the antiferromagnetic phase-transition temperature.
We present the effect of Zn (S=0) and Cu (S=1/2) substitution at the Ni site of S=1 Haldane chain compound Y2BaNiO5. 89Y nuclear-magnetic resonance (NMR) allows us to measure the local magnetic susceptibility at different distances from the defects. The 89Y NMR spectrum consists of one central peak and several less intense satellite peaks. The central peak represents the chain sites far from the defect. Its shift measures the uniform susceptibility, which displays a Haldane gap Delta≈100 K and it corresponds to an antiferromagnetic (AF) coupling J≈260 K between the nearest neighbor Ni spins. Zn or Cu substitution does not affect the Haldane gap. The satellites, which are evenly distributed on the two sides of the central peak, probe the antiferromagnetic staggered magnetization near the substituted site. The spatial variation of the induced magnetization is found to decay exponentially from the impurity for both Zn and Cu for T>50 K. Its extension is found identical for both impurities and corresponds accurately to the correlation length xi(T) determined by Monte Carlo simulations for the pure compound. In the case of nonmagnetic Zn, the temperature dependence of the induced magnetization is consistent with a Curie law with an ``effective'' spin S=0.4 on each side of Zn. This staggered effect is quantitatively well accounted for in all the explored range by quantum Monte Carlo (QMC) computations of the spinless-defect-induced magnetism. In the case of magnetic Cu, the similarity of the induced magnetism to the Zn case implies a weak coupling of the Cu spin to the nearest-neighbor Ni spins. The slight reduction of about 20-30 % of the induced polarization with respect to Zn is reproduced by QMC computations by considering an antiferromagnetic coupling of strength J'=0.1J-0.2J between the S=1/2 Cu spin and nearest-neighbor Ni spin. Macroscopic susceptibility measurements confirm these results as they display a clear Curie contribution due to the impurities nearly proportional to their concentration. This contribution is indeed close to that of two spin half for Zn substitution. The Curie contribution is smaller in the Cu case, which confirms that the coupling between Cu and near-neighbor Ni is antiferromagnetic.
A comprehensive study of the impurity-induced magnetism in nonmagnetically (Mg2+) and magnetically (Co2+) doped PbNi2V2O8 compounds is undertaken by both macroscopic dc magnetization and local-probe electron spin resonance (ESR) techniques. The magnetic coupling between impurity-liberated spins is estimated from the linewidth of the low-temperature ESR signal in Mg-doped samples. In the case of magnetic cobalt dopants the impurity-host magnetic exchange is evaluated from the Co-induced contribution to the linewidth in the paramagnetic phase. The experimentally observed severe broadening of the ESR lines in the magnetically doped compounds is attributed to a rapid spin-lattice relaxation of the Co2+ ions. The exchange parameters obtained from the ESR analysis offer a satisfactory explanation of the impurity-induced contribution to the low-temperature magnetization in doped samples.
We present a study of the dynamic structure factor of the antiferromagnetic spin-1/2 Heisenberg chain at finite temperatures and finite magnetic fields. Using quantum Monte Carlo based on the stochastic series expansion and maximum entropy methods, we evaluate the longitudinal and the transverse dynamic structure factors from vanishing magnetic fields up to and above the threshold Bc for ferromagnetic saturation, as well as for high and for intermediate temperatures. We study the field-induced redistribution of spectral weight contrasting longitudinal versus transverse excitations. At finite fields below saturation incommensurate low-energy modes are found consistent with zero-temperature Bethe ansatz. The crossover between the field-induced ferromagnet above Bc and the Luttinger liquid below Bc is analyzed in terms of the transverse spin dynamics. Evaluating sum rules we assess the quality of the analytic continuation and demonstrate excellent consistency of the maximum entropy results.
We present a static, model-independent, experimental determination of the spin-spin correlation length xi in a quantum spin system. This is achieved in the doped Haldane (i.e., S = 1 Heisenberg antiferromagnetic) chain Y2BaNi1-xMgxO5 by 89Y NMR imaging of the staggered magnetization induced around the Mg impurities (i.e., chain boundaries) by a magnetic field. The magnitude of this magnetization is found to decay exponentially, with xi equal to the theoretical prediction for an infinite S = 1 chain and the staggered magnetic moment at the edge site showing the Curie behavior of an effective S = 1/2 spin.
Using the density-matrix-renormalization-group technique, we calculate numerically the low-energy excitation spectrum and magnetization curve of the spin-1 antiferromagnetic chain in a staggered magnetic field, which is expected to describe the physics of the R2BaNiO5 (R!=Y) family below the Néel temperature of the magnetic rare-earth (R) sublattice. These results are valid in the entire range of the staggered field, and agree with those given by the nonlinear sigma model study for small fields, but differ from the latter for large fields. They are consistent with the available experimental data. The correlation functions for this model are also calculated. The transverse correlations display the anticipated exponential decay with shorter correlation length, while the longitudinal correlations show explicitly the induced staggered magnetization.
Using the density-matrix renormalization-group (DMRG) technique, we carry out a large scale numerical calculation for the S=2 antiferromagnetic Heisenberg (AFH) chain. We obtain the magnetization curve of S=2 chain near the critical field. The corrections to the magnetization due to the magnon-magnon interaction are the same order for S=2 and S=1 chains. By comparing the experimental magnetization with our calculation, we find that the gap obtained experimentally is about 0.08J which is close to the accurate DMRG estimation for the S=2 AFH chain. Numerical errors in our large scale calculation are also discussed.
The phenomenological expression χT∕(Ng2μB2∕k)=C1n exp(−W1n∕T)+C2n exp(−W2n∕T) describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins S=1, whose size n is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For n large, the coefficients of the expression converge towards C1=0.125, W1=0.451J, C2=0.564, W2=1.793J (J is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie–Weiss temperature, the correlation length at T=0 and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of Y2BaNiO5 and of Ni(C2H8N2)2NO2ClO4 (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.
Motivated by the indication of a new critical theory for the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the square lattice, we reinvestigate the phase transition of this model induced by dimerization using first-principle Monte Carlo simulations. We focus on studying the finite-size scaling of ρs12L and ρs22L, where L stands for the spatial box size used in the simulations and ρsi, with i∈{1,2}, is the spin-stiffness in the i-direction. Remarkably, while we observe a large correction to scaling for the observable ρs12L, the data for ρs22L exhibit a good scaling behavior without any indication of a large correction. As a consequence, we are able to obtain a numerical value for the critical exponent ν, which is consistent with the known O(3) result with moderate computational effort. Further, we additionally carry out an unconventional finite-size scaling analysis with which we assume that the ratio of the spatial winding numbers squared is fixed through all simulations. The theoretical correctness of our idea is argued and its validity is confirmed. Using this unconventional finite-size scaling method, even from ρs1L, which receives the most serious correction among the observables considered in this study, we are able to arrive at a value for ν consistent with the expected O(3) value. A detailed investigation to compare these two finite-size scaling methods should be performed.
We study integrated density of states (DOS) and thermodynamic properties for the S = 1=2 quantum spin icosidodecahedron, using a numerical method for calculating an eigenvalue distribution function. For the ideal quantum spin icosidodecahedron with next-nearest neighbor interactions, we find that there exist many singlet states before the first triplet state, which is an evidence of the resonating valence bond (RVB) state, and the experimental susceptibilities of Mo72V30 measured by Müller et al. and Botar et al. cannot be reproduced within the ideal model. Because V4+ ions are on vertices of a slightly distorted icosidodecahedron in Mo72V30, we study the distorted version of the model and find that agreement between theoretical and experimental susceptibilities is improved. For this distorted model, there exist about ten singlet states before the first triplet state.
The correlations of currents flowing in mesoscopic rings induced by both classical and
quantum correlations of photons have been considered. The system of two and three rings
in the presence of three types of non-classical radiation, factorizable (uncorrelated),
separable (classically correlated) and entangled (quantum mechanically correlated), has
been studied. The results show that entangled photons can produce entangled electrons.
Mixed-spin systems composed of quantum spin chains with gapped magnetic excitations interacting through `auxiliary' magnetic ions show an effective separation between low-frequency classical and high-frequency quantum spin correlations. This phenomenon is realized in a family of rare-earth nickelates. Studies of these materials enable experimental measurements of some previously inaccessible fundamental properties of quantum spin chains to be made. The non-trivial intrinsic dynamics of the auxiliary spins gives rise to peculiar excitations of a mixed nature, and in certain cases may play a key role in long-range magnetic ordering.
We report analytical calculations of the Helmholtz free energy of non-integrable anisotropic quantum XXZ chains in the high-temperature regime for several values of the spin S. Single-ion anisotropy and interaction with an external magnetic field are taken into account. The seven lowest-order terms in the high-temperature expansion of Helmholtz free energy are obtained. Our results contribute to the existing literature on high-temperature expansions and numerical studies of those models by discussing the effects of anisotropy upon their high-temperature thermodynamic properties, such as the average energy per site, the specific heat and magnetic susceptibility.
Borehole radar, both reflection and tomography methods, and GPR surveys were conducted at a granite quarry to delineate rock inhomogeneities, including major fractures, and to estimate the fracture density. The borehole reflection survey used a directional antenna to get the spatial orientations of the reflectors. The center frequency was 20 MHz for the borehole radar reflection and tomography surveys and 100 MHz for GPR. Through the interpretation of borehole reflection data using dipole and directional antennas, as well as surface GPR images, we could determine the orientation of the major fractures in three dimensions. Portions of the tomography travel time curves exhibited velocity anisotropy, which is uncommon in granite. Comparing the tomography data with Televiewer images showed that the anisotropy effect in this area is closely related to the alignment of fine fissures. From the borehole radar, tomography, and GPR images as well as the distribution of anisotropy, we conclude that the area bounded by the two fractures, MF2 and MF5, has the freshest granite in the surveyed area. This case study demonstrates that a combination of surface, borehole reflection, and tomographic radar surveys can provide the three-dimensional distribution of major fractures and an estimate of fracture density.
Recent neutron scattering experiments on CsNiCl3 reveal some features that are not well described by the standard nonlinear σ model, nor by numerical simulations, for isolated S
= 1 spin chains. In particular, in real systems at the antiferromagnetic point of the Brillouin zone, the intensity of the continuum
of multiparticle excitations, at T
= 6 K, is about 5 times greater than predicted. Also, the spin gap is higher and the correlation length is smaller than predicted.
We propose a theoretical scenario where the interchain interaction is approximated by an effective staggered magnetic field,
and that yields a correct prediction for the observed quantities.
Phase transitions (PT) are typical collective phenomena occurring in many-body systems with interparticle interactions. A
rather general description of the PT’s and of the accompanying “critical dynamics” can be given in the framework of Landau-type
statistical theory, which includes more specific theories such as the Weiss mean field theory for magnetic systems and the
Van der Waals theory for fluids. This approach is suited for second order or slightly first-order PT3.
We make a new proposal to describe the very low temperature susceptibility of the doped Haldane gap compound Y2BaNi1-xZnxO5. We propose a new mean field model relevant for this compound. The ground state of this mean field model is unconventional
because antiferromagnetism coexists with random dimers. We present new susceptibility experiments at very low temperature.
We obtain a Curie-Weiss susceptibility χ(
T
) ∼
C
/(Θ +
T
) as expected for antiferromagnetic correlations but we do not obtain a direct signature of antiferromagnetic long range order.
We explain how to obtain the “impurity” susceptibility
(
T
) by subtracting the Haldane gap contribution to the total susceptibility. In the temperature range [1 K, 300 K] the experimental
data are well fitted by T
(
T
) =
C
imp
1 +
T
imp
/
T
. In the temperature range [100 mK, 1 K] the experimental data are well fitted by T
(
T
) =
A
ln(
T
/
T
c
), where T
c
increases with x. This fit suggests the existence of a finite Néel temperature which is however too small to be probed directly in our experiments.
We also obtain a maximum in the temperature dependence of the ac-susceptibility
(
T
) which suggests the existence of antiferromagnetic correlations at very low temperature.
D-theory is an alternative non-perturbative approach to quantum field theory formulated in terms of discrete quantized variables instead of classical fields. Classical scalar fields are replaced by generalized quantum spins and classical gauge fields are replaced by quantum links. The classical fields of a d-dimensional quantum field theory reappear as low-energy effective degrees of freedom of the discrete variables, provided the (d+1)-dimensional D-theory is massless. When the extent of the extra Euclidean dimension becomes small in units of the correlation length, an ordinary d-dimensional quantum field theory emerges by dimensional reduction. The D-theory formulation of scalar field theories with various global symmetries and of gauge theories with various gauge groups is constructed explicitly and the mechanism of dimensional reduction is investigated.
We demonstrate that the âwormâ algorithm allows very effective and
precise quantum Monte Carlo (QMC) simulations of spin systems in
a magnetic field, and its autocorrelation time is rather insensitive
to the value of H at low temperature. Magnetization curves for the
s=1/2 and s=1 chains are presented and compared with existing Bethe
ansatz and exact diagonalization results. From the Green-function
analysis we deduce the magnon spectra in the s=1 system, and directly
establish the ârelativisticâ form E(p)=(Î2+v2p2)1/2 of the dispersion
law.
We compute the entropy of antiferromagnetic quantum spin-s chains in an
external magnetic field using exact diagonalization and Quantum Monte Carlo
simulations. The magnetocaloric effect, i.e., temperature variations during
adiabatic field changes, can be derived from the isentropes. First, we focus on
the example of the spin-s=1 chain and show that one can cool by closing the
Haldane gap with a magnetic field. We then move to quantum spin-s chains and
demonstrate linear scaling with s close to the saturation field. In passing,
we propose a new method to compute many low-lying excited states using the
Lanczos recursion.
Combining high-temperature expansions with the recursion method and quantum Monte Carlo simulations with the maximum entropy method, we study the dynamics of the spin-1/2 Heisenberg chain at temperatures above and below the coupling J. By comparing the two sets of calculations, their relative strengths are assessed. At high temperatures, we find that there is a low-frequency peak in the momentum-integrated dynamic structure factor, due to diffusive long-wavelength modes. This peak is rapidly suppressed as the temperature is lowered below J. Calculation of the complete dynamic structure factor S(k,ω) shows how the spectral features associated with the two-spinon continuum develop at low temperatures. We extract the nuclear spin-lattice relaxation rate 1/T1 from the ω→0 limit, and compare with recent experimental results for Sr2CuO3 and CuGeO3. We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility χ(k). We confirm the asymptotic low-temperature forms S(π)∼[ln (T)]3/2 and χ(π)∼T-1[ln (T)]1/2, expected from previous theoretical studies.
Quasielastic magnetic neutron scattering from the linear-chain antiferromagnet NMn is reported. The system is found to exhibit planes of critical scattering perpendicular to the Mn chains from >40\ifmmode^\circ\else\textdegree\fi{}K down to 1.1\ifmmode^\circ\else\textdegree\fi{}K. Both the spatial and thermal variation of the scattering can be quantitatively accounted for at all temperatures using Fisher's theory for the classical Heisenberg linear chain.
We have investigated the spin dynamics of the nearly ideal one-dimensional S=1 antiferromagnet Ni(C2H8N2)2NO2ClO4 (NENP) by inelastic neutron scattering. The measurements have been performed as a function of both the wave vector (in the vicinity of the antiferromagnetic point q=π) and the magnetic field, for field configurations parallel (up to 5 T) or perpendicular to the chain axis (up to 10 T). Our experimental results at low temperature are in good quantitative agreement with most theoretical predictions (both analytical and numerical) established for the spin-1 Haldane-gap system in presence of finite orthorhombic anisotropy: nonmagnetic singlet ground state, three gaps in the excitation spectrum corresponding to the excited triplet (Δy0≊1.05 meV, Δx0≊1.23 meV, Δz0≊2.5 meV), finite correlation lengths at T=0 (ξxy/d≊8,ξz/d≊4) and unconventional field dependencies. The evolution with field of the various modes and their respective polarization have been determined experimentally and are compared to recent field-theory treatments of the S=1 anisotropic antiferromagnetic chain together with numerical calculations on finite-size systems. We present results concerning the field dependencies of the correlation lengths, both in field parallel and perpendicular to the chain axis, which are not fully understood from the existing theories.
Magnetic susceptibility and inelastic neutron scattering experiments have been performed in the nearly ideal one-dimensional Heisenberg antiferromagnet with spin one, Ni(C2H8N2)2NO2ClO4. The experimental results are consistent with the recent theoretical predictions for a quantum energy gap between the ground state and the first excited states.
The crystal structure of Y2BaNiO5 is characterized by the existence of isolated linear magnetic chains. Nickel is located within compressed oxygen octahedra sharing two opposite corners to form (NiO5)n chains along the a-axis separated by Y and Ba atoms. Within a large temperature range, the magnetic susceptibility can be fitted using a one dimensional S = 1 Heisenberg model. The best fit is obtained for J/k ≈ −285 K (Tmax ≈ 410 K). The three dimensional long range magnetic ordering has not been observed by neutron diffraction down to 1.8 K, implying an inter-intrachain coupling ratio J'/J ⪡ 10−2. Inelastic neutron scattering experiments have given evidences for a singlet ground state and two gaps in the excitation spectrum at energies Δxy ≈ 8.5 meV and Δz ≈ 16 meV. Our experimental data are interpreted quantitatively within the framework of the Haldane conjecture for the S = 1 antiferromagnetic chain.
Low-lying energy states of an S=1/2 ladder model are investigated by applying the numerical diagonalization method to finite clusters. This model has the antiferromagnetic intrachain coupling J (J>0) and the ferromagnetic interchain one -lambdaJ (lambda>0). Both the inverse correlation length xi-1(lambda) and an energy gap are shown to be finite at least for lambda>=0.05, the former of which is shown to approach the same value as that of the S=1 antiferromagnetic Heisenberg chain with increasing lambda. A generation mechanism of the gap is also discussed in terms of a simple model by using the Lieb-Mattis theorem.
The magnetic susceptibility of the quasi-linear chain Heisenberg antiferromagnet (2,-bipyridine)trichloromanganese(III), MnCl_{3}(bipy), has been measured from 1.8 to 300 K with the magnetic field, H, parallel and perpendicular to the chains. The analyzed data yield and K. The magnetization, M, has been studied at 30 mK and 1.4 K in H up to 16 T. No evidence of long-range order is observed. Depending on crystal orientation, at 30 mK until a critical field is achieved ( and $H_{c\bot} = 1.8\pm 0.2 T), where M increases continuously as H is increased. These results are interpreted as evidence of a Haldane gap. Comment: 11 pages, 4 figures
Neutron scattering from copper benzoate, Cu(C6D5COO)2 3D2O, provides the first direct experimental evidence for field-dependent incommensurate low energy modes in a one-dimensional spin S = 1/2 antiferromagnet. Soft modes occur for wavevectors q=\pi +- dq(H) where dq(H) ~ 2 \pi M(H)/g\mu_B as predicted by Bethe ansatz and spinon descriptions of the S = 1/2 chain. Unexpected was a field-induced energy gap , where as determined from specific heat measurements. At H = 7 T (g\mu_B H/J = 0.52), the magnitude of the gap varies from 0.06 - 0.3 J depending on the orientation of the applied field. Comment: 11 pages, 5 postscript figures, LaTeX, Submitted to PRL 3/31/97, e-mail comments to dhr@pha.jhu.edu
An effective low-energy description for multi-leg spin-1/2 Heisenberg ladders with an odd number of legs is proposed. Using a newly developed Monte Carlo loop algorithm and exact diagonalization techniques, the uniform and staggered magnetic susceptibility and the entropy are calculated for ladders with 1, 3, and 5 legs. These systems show a low-temperature scaling behavior similar to spin-1/2 chains with longer ranged unfrustrated exchange interactions. The spinon velocity does not change as the number of legs increases, but the energy scale parameter decreases markedly. Comment: 4 pages and 5 figures
Using the density-matrix renormalization-group technique we study the long-wavelength properties of the spin nearest-neighbor Heisenberg chain. We obtain an accurate value for the spin velocity v\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3.87\ifmmode\pm\else\textpm\fi{}0.02, in agreement with experiment. Our results show conclusively that the model belongs to the same universality class as the Heisenberg chain, with a conformal central charge and critical exponent \eta.
One-dimensional antiferromagnets have exotic disordered ground states. As was first argued by Haldane (1983), there is an excitation gap for integer, but not half-integer, spin. The authors review the arguments for this behaviour based on field-theory mappings, the Lieb-Schultz-Mattis theorem, exactly solvable models, finite-chain diagonalisation and real experiments.
An action-angle representation of spin variables is used to relate the large-S Heisenberg antiferromagnet to the O(3) nonlinear sigma model quantum field theory, with precise equivalence for integral S. A variant theory is found for half-integral S. Dynamic mass generation by the Néel magnon is predicted.
Magnetic susceptibility is measured on the one-dimensional spin 1/2 antiferromagnets SrâCuOâ and SrCuOâ using single crystals. Large, nearly contamination-free single crystals enable us to measure the intrinsic spin susceptibility over a wide temperature range between 5 and 800K. The results are in excellent agreement with the recent calculation by Eggert, Affleck, and Takahashi, and the exchange interaction energy {ital J} is estimated to be 2200±200 and 2100±200K, respectively. In SrâCuOâ, an isotropic susceptibility drop is observed below about 20K, which is also consistent with the result of this rigorous calculation. {copyright} {ital 1996 The American Physical Society.}
The quasielastic magnetic scattering from K2NiF4 over the temperature range from 97.2°K to at least 200°K is found to correspond to reciprocal lattice rods rather than points thus giving the first concrete evidence of the two-dimensional character of K2NiF4. At 97.1°K the crystal undergoes an extremely sharp phase transition to long-range order in three dimensions. From 97.0°K to 5°K the sublattice magnetization follows a (97.1-T)beta law with beta=0.15.
CsVCl3 is composed of one-dimensional antiferromagnetic chains which are coupled weakly antiferromagnetically in the basal hexagonal plane. In order to clarify the anomalous properties hitherto observed, an inelastic neutron scattering has been studied. We observed good one-dimensionality of J'/J{=}2.7× 10-4. But, though S{=}3/2, the spin wave dispersion along the chain disagrees with the classical picture. The excitation spectra along the triangular lattice plane are unusual. The excitation is dominated by a dispersionless mode of strong intensity. Compared to this mode, the conventional acoustic spin wave branch seems to be of minor excitation at lower temperatures than the Néel point.
Not just biological phenomena but most physical phenomena as well become drastically altered or impossible in spaces of fewer than three dimensions. As is well known, the cooperative behavior of many‐body systems depends very strongly on the coordination of the closely interacting particles and therefore on the number of spatial dimensions. As we shall see, not only does an examination of physics in fewer than three dimensions illuminate the familiar world of three dimensions, there are also real systems whose structure effectively reduces their dimensionality and for which, therefore, low‐dimensional physics is a prerequisite. As a consequence, a considerable effort has been expended in explaining the properties of lower‐dimensional and, especially, linear systems. During the last decade there has been a veritable explosion in the “one‐dimensional” literature. In this article we would like to discuss in the context of magnetism some of the motivating factors for this explosion and the progress that has been achieved so far. Crystals whose magnetic ions are arranged in separated chains have magnetic properties that indicate nearly ideal one‐dimensional rather than three‐dimensional behavior.
Magnetic susceptibility measurements for Sr2CuO3±δ were made from 2 to 800 K, and a strong dependence upon oxygen content (δ) was observed. Samples synthesized under oxygen, followed by various nitrogen treatments, exhibited markedly different Curie-Weiss-type terms, and we discuss possible origins for this behavior. High-temperature magnetic susceptibility measurements for the sample with the smallest Curie-Weiss-type term clearly show the increase with temperature expected from the Bonner-Fisher model for a spin-1/2 one-dimensional (1D) Heisenberg antiferromagnet. This is a direct experimental observation of 1D magnetic behavior in this system. The in-chain superexchange coupling constant, as determined by a fit to the Bonner-Fisher model, is ‖J‖/kB≊1300-200+100 K, comparable to the values observed in the two-dimensional layered cuprates. Estimates of the interchain magnetic interaction indicate this material may be the best realization of a 1D spin-1/2 Heisenberg antiferromagnet reported to date. Low-temperature neutron and synchrotron x-ray powder-diffraction studies of Sr2CuO3 show that the low-temperature structure of this system has Immm space-group symmetry, the same structure reported at room temperature, indicating that this material, in contrast to La2CuO4, does not undergo any structural transformations upon cooling. The absence of crystallographic distortions precludes a magnetic anisotropy contribution from a Dzyaloshinsky-Moriya interaction, implying that Sr2CuO3 should be a nearly ideal spin-1/2 antiferromagnetic Heisenberg chain compound, in agreement with the magnetic susceptibility results. A search for the presence of long-range three-dimensional antiferromagnetic order by magnetic neutron powder diffraction at temperatures as low as 1.5 K was not successful, although we estimate an upper limit for the size of the ordered moment which could have been detected to be ∼0.1μB per Cu2+ ion.
The quasi-elastic magnetic scattering from the planar antiferromagnets K2NiF4, Rb2MnF4, Rb2FeF4 has been studied over a wide range of temperatures both above and below the phase transition. In all three compounds the diffuse scattering above the phase transition takes the form of a ridge rather than a peak, thus giving the first concrete evidence for the two-dimensional nature of the magnetism. At TN (97.1, 38.4, 56.3°K, respectively), the crystals undergo sharp phase transitions to long-range order (LRO) in three dimensions. For 0.002≤1-T/TN≤0.1, the sublattice magnetizations in K2NiF4 and Rb2MnF4 follow a (TN-T)β law with β=0.14 and 0.16, respectively. Rb2MnF4 is found to have two distinct magnetic phases, both with identical ordering within the planes but with different stacking arrangements of the spins between planes; both phases are found to have identical TN's and β's to within the experimental accuracy of 0.1°K. The sublattice magnetization in Rb2FeF4 has a rather more complicated behavior, apparently due to magnetostrictive effects. Finally, in the ordered phase in each compound, the three-dimensional magnetic Bragg peaks are accompanied by "diffuse" scattering which is completely two-dimensional in form. These results are discussed in terms of a model in which the phase transition is viewed as being essentially two-dimensional in character, the three-dimensional ordering simply following as a necessary consequence of the onset of LRO with the planes. The systems therefore should have distinct two- and three-dimensional critical regions. The three-dimensional region apparently was not experimentally accessible with 0.1°K temperature control in K2NiF4 and Rb2MnF4, indicating that in these compounds |T/TN-1|3≤2×10-3.
We compute the single-particle spectral density, susceptibility near the Kohn anomaly, and pair propagator for a one-dimensional interacting-electron gas. With an attractive interaction, the pair propagator is divergent in the zero-temperature limit and the Kohn singularity is removed. For repulsive interactions, the Kohn singularity is stronger than the free-particle case and the pair propagator is finite. The low-temperature behavior of the interacting system is not consistent with the usual Ginzburg-Landau functional because the frequency, temperature, and momentum dependences are characterized by power-law behavior with the exponent dependent on the interaction strength. Similarly, the energy dependence of the single-particle spectral density obeys a power law whose exponent depends on the interaction and exhibits no quasiparticle character. Our calculations are exact for the Luttinger or Tomonaga model of the one-dimensional interacting system.
We have carried out a neutron scattering investigation of the static structure factorS(q
2D
) (q
2D
is the in-plane wave vector) in the two-dimensional spinS=1/2 square-lattice Heisenberg antiferromagnet Sr2CuO2Cl2. For the spin correlation length we find quantitative agreement with Monte Carlo results over a wide range of temperature. The combined Sr2CuO2Cl2-Monte Carlo data, which cover the length scale from 1 to 200 lattice constants, are predicted without adjustable parameteres by renormalized classical theory for the quantum nonlinear sigma model. For the structure factor peakS(0), on the other hand, we findS(0)
2 for the reduced temperature range 0.16T/2
s
S(0)T
2
2. This discrepancy has important implications for the interpretation of many derivative quantities such as NMR relaxation rates. In the ordered phase, we have measured the temperature dependence of the out-of-plane spin-wave gap. Its low-temperature value of 5.0 meV corresponds to an XY anisotropyJ
XY
/J=1.410–4. From measurements of the sublattice mangetization we obtain =0.220.01 for the order parameter exponent. This may either reflect tricricality as in La2CuO4, or it may indicate finite-size two-dimensional XY behavior as suggested by Bramwell and Holdsworth. As in theS=1 system K2NiF4, the gap energy in Sr2CuO2Cl2 scales linearly with the order parameter up to the Nel temperature. We also reanalyze static structure factor data for K2NiF4 using the exact low temperature result for the correlation length of Hasenfratz and Niedermayer and including the Ising anisotropy explicitly. Excellent agreement between experiment and theory is obtained for the correlation length, albeit with the spin-stiffness
s
reduced by 20% from the spin-wave value. As in Sr2CuO2Cl2 we find thatS(0)
2 for the reduced temperature range 0.22T/2
s
We review recent studies of the static and dynamic magnetic fluctuations in La2 − xSrxCuO4. In La2CuO4(TN = 325K) the instantaneous two-dimensional (2D) spin correlations have been studied over the temperature range 340K < T ≤ 820K. We find quantitative agreement for the correlation length over the complete temperature range with no adjustable parameters with predictions for the 2D quantum non-linear sigma model in the renormalized classical regime. We have carried out bulk magnetization measurements in La1.96Sr0.04CuO4 which has neither antiferromagnetic long range order nor superconductivity. All of the features that characterize a canonical spin glass transition have been seen: irreversibility, remnant magnetization and scaling behavior. Finally, neutron inelastic scattering experiments have been performed on homogeneous single crystals of La1.85Sr0.15CuO4 with Tc = 37.3K (at onset), higher than any previously studied single crystals. The temperature dependence of the low energy incommensurate peak intensity at (π(1 ± δ), π) and (π, π(1 ± δ)) exhibits a pronounced maximum near 3Tc. In contrast to the results reported on lower Tc crystals, the intensity for energies below 3.5 meV dramatically decreases as the temperature decreases below Tc, vanishing into the background below ~ 15K. The behavior is consistent with predictions based on a dx2 − y2 superconducting order parameter.
We present a new type of cluster algorithm that strongly reduces critical
slowing down in simulations of vertex models. Since the clusters
are closed paths of bonds, we call it the loop algorithm. The basic
steps in constructing a cluster are the breakup and the freezing
of vertices. We concentrate on the case of the F model, which is
a subset of the six-vertex model exhibiting a Kosterlitz-Thouless
transition. The loop algorithm is also applicable to simulations
of other vertex models and of one- and two-dimensional quantum spin
systems.
To make the transition from the quasi-long-range order in a chain
of antiferromagnetically coupled S = 1/2 spins to the true long-range
order that occurs in a plane, one can assemble chains to make ladders
of increasing width. Surprisingly, this crossover between one and
two dimensions is not at all smooth. Ladders with an even number
of legs have purely short-range magnetic order and a finite energy
gap to all magnetic excitations. Predictions of this ground state
have now been verified experimentally. Holes doped into these ladders
are predicted to pair and possibly superconduct.
We have performed Monte Carlo calculations of the energies of several low-lying energy states of one-dimensional, spin-1 Heisenberg antiferromagnets with linear sizes up to n=32. Our results support Haldane’s prediction that a gap exists in the excitation spectrum for n→∞. .AE
We calculate correlation functions of a one-dimensional spin S=1 antiferromagnetic Heisenberg model by the large-cluster-decomposition Monte Carlo method. We find that the correlation functions are well approximated by modified Bessel functions. This result supports Haldane's conjecture, and the correlation length agrees with that obtained from spin-wave theory using elementary excitation data.
The S=1 isotropic antiferromagnetic Heisenberg chain is studied by exact diagonalizations using the Lanczös algorithm. Energy gaps, structure factors at k=pi, and staggered susceptibilities at T=0 are calculated for finite rings up to N=16, and extrapolated to an infinite system using Shanks' transformation. The estimated energy gap is 0.411+/-0.001, which agrees with the result of Monte Carlo calculations by Nightingale and Blöte. Further, it is found that a finite-size correction decays exponentially and the decay constant corresponds with the correlation length, which is about 5.
We investigate the correlation length of the one-dimensional S=1/2 Heisenberg antiferromagnet by the thermal Bethe-ansatz method proposed by Koma. The numerical result shows that the correlation length diverges as xi~T-1 at low temperatures with logarithmic corrections. This logarithmic correction can be explained by conformal field theory.
We present results of a numerical renormalization-group study of the isotropic S=1 Heisenberg chain. The density-matrix renormalization-group techniques used allow us to calculate a variety of properties of the chain with unprecedented accuracy. The ground-state energy per site of the infinite chain is found to be e0~=-1.401 484 038 971(4). Open-ended S=1 chains have effective S=1/2 spins on each end, with exponential decay of the local spin moment away from the ends, with decay length xi~=6.03(1). The spin-spin correlation function also decays exponentially, and although the correlation length cannot be measured as accurately as the open-end decay length, it appears that the two lengths are identical. The string correlation function shows long-range order, with g(∞)~=-0.374 325 096(2). The excitation energy of the first excited state, a state with one magnon with momentum q=pi, is the Haldane gap, which we find to be Delta~=0.410 50(2). We find many low-lying excited states, including one- and two-magnon states, for several different chain lengths. The magnons have spin S=1, so the two-magnon states are singlets (S=0), triplets (S=1), and quintuplets (S=2). For magnons with momenta near pi, the magnon-magnon interaction in the triplet channel is shown to be attractive, while in the singlet and quintuplet channels it is repulsive.
We show that while the exactly integrable spin-(3/2 Heisenberg antiferromagnet is a representation of the nonlinear sigma model with nontrivial Wess-Zumino term, the generic massless antiferromagnet has long-wavelength behavior equivalent to spin (1/2. This is demonstrated by comparing the predictions of conformal symmetry and rotational symmetry to the gaps calculated for finite chains. The leading corrections are eliminated leading to stable estimates of the conformal central charge and exponents for the spin-correlation functions.
Well-defined spin-wave excitations were observed by using pulsed neutrons from CsVCl3, an S = 3/2 Heisenberg antiferromagnetic chain. The observed spin-wave dispersion relation shows an enhancement of the excitation energy by a factor of 1.26, which must be compared with the quantum renormalization factor of pi/2 for an S = 1/2 system. The linewidth of the spectrum at 40 K was broad and independent of q. The q-independent linewidth is the first experimental observation, which was predicted by rigorous theories for a classical chain at sufficiently low temperature.
A world-line Monte Carlo approach is proposed to calculate the elementary excitations of quantum systems. The lowest energies as a function of momentum are successfully extracted from an imaginary-time correlation function evaluated at a sufficiently low temperature. The method is applied to the antiferromagnetic Heisenberg chains of S = 12, 1, 32, and 2 with length up to 64. The spectra for S = 12 and 1 coincide well with the result by des Cloizeaux and Pearson and with the result by Takahashi, respectively. The spectra for S = 32 and 2 turn out to be massless and massive, respectively. The S = 2 Haldane gap is estimated to be 0.049+/-0.018.
We study antiferromagnetic spin--1/2 Heisenberg ladders, comprised of chains () with ratio of inter-- to intra--chain couplings. From measurements of the correlation function we deduce the correlation length . For even , the static structure factor exhibits a peak at a temperature below the corresponding spin gap. Results for isotropically coupled ladders () are compared to those for the single chain and the square lattice. For , the correlation function of the two--chain ladder is in excellent agreement with analytic results from conformal field theory, and exhibits simple scaling behavior. Comment: 4 pages, 5 EPS figures, submitted to Phys. Rev. Lett
We present the theory of nonzero temperature (T) spin dynamics and transport in one-dimensional Heisenberg antiferromagnets with an energy gap . For , we develop a semiclassical picture of thermally excited particles. Multiple inelastic collisions between the particles are crucial, and are described by a two-particle S-matrix which has a super-universal form at low momenta. This is established by computations on the O(3) -model, and strong and weak coupling expansions (the latter using a Majorana fermion representation) for the two-leg S=1/2 Heisenberg antiferromagnetic ladder. As an aside, we note that the strong-coupling calculation reveals a S=1, two particle bound state which leads to the presence of a second peak in the T=0 inelastic neutron scattering (INS) cross-section for a range of values of momentum transfer. We obtain exact, or numerically exact, universal expressions for the thermal broadening of the quasi-particle peak in the INS cross-section, for the magnetization transport, and for the field dependence of the NMR relaxation rate of the effective semiclassical model: these are expected to be asymptotically exact for the quantum antiferromagnets. The results for are compared with the experimental findings of Takigawa et al and the agreement is quite good. In the regime we argue that a complementary description in terms of semiclassical waves applies, and give some exact results for the thermodynamics and dynamics.
Analytic expressions for the correlation length temperature dependences are given for antiferromagnetic spin-1/2 Heisenberg ladders using a finite-size non-linear sigma-model approach. These calculations rely on identifying three successive crossover regimes as a function of temperature. In each of these regimes, precise and controlled approximations are formulated. The analytical results are found to be in excellent agreement with Monte Carlo simulations for the Heisenberg Hamiltonian. Comment: 5 pages LaTeX using RevTeX, 3 encapsulated postscript figures
We have investigated Haldane's conjecture for the S=2 isotropic antiferromagnetic quantum spin chain with nearest-neighbor exchange J. Using a density matrix renormalization group algorithm for chains up to L=350 spins, we find in the thermodynamic limit a finite spin gap of Delta = 0.085(5)J and a finite spin-spin correlation length xi = 49(1) lattice spacings. We establish the ground state energy per bond to be E_0=-4.761248(1)J. We show that the ground state has a hidden topological order that is revealed in a nonlocal string correlation function. This means that the physics of the S=2 chain can be captured by a valence-bond solid description. We also observe effective free spin-1 states at the ends of an open S=2 chain. Comment: 6 pages, LaTeX 2.09, 3 PostScript figures
- S R White
- D A Huse
S.R. White and D.A. Huse, Phys. Rev. B 48, 3844 (1993).
- See R J For
- G Birgeneau
- Shirane
For reviews, see R.J. Birgeneau and G. Shirane, Phys. Today 31(12), 32 (1978); Physica B 86-88, 639 (1977).
- K Nomura
- M Yamada
K. Nomura and M. Yamada, Phys. Rev. B 43, 8217 (1991).
- H J Schulz
H.J. Schulz, Phys. Rev. B 34, 6372 (1986), and references
therein.
- Usp
Usp. Math.
Nauk. 37, 3 (1982); E. Witten, Commun. Math. Phys. 92,
455 (1984).
- E S Sørensen
- I Affleck
E.S. Sørensen and I. Affleck, Phys. Rev. B 49, 15771
(1994).
- B Keimer
- N Belk
- R J Birgeneau
- A Cassanho
- C Y Chen
- M Greven
- M A Kastner
- A Aharony
- Y Endoh
- R W Erwin
- G Shirane
B. Keimer, N. Belk, R.J. Birgeneau, A. Cassanho, C.Y.
Chen, M. Greven, M.A. Kastner, A. Aharony, Y. Endoh,
R.W. Erwin, and G. Shirane, Phys. Rev. B 46, 14034 (1992);
For a reveiw, see I. Affleck
For a reveiw, see I. Affleck, J. Phys. Condens. Matter 1,
3047 (1989).
- H Kadowaki
- K Hirakawa
- K Ubukoshi
- J S Itoh
- Y Endoh
- K Kakurai
- H Tanaka
H. Kadowaki, K. Hirakawa, and K. Ubukoshi, J. Phys. Soc.
Japan 52, 1799 (1983); S. Itoh, Y. Endoh, K. Kakurai, and
H. Tanaka, Phys. Rev. Lett. 74, 2375 (1995).
- G E Granroth
- M W Meisel
- M Chaparala
- T Jolicoeur
- B H Ward
- D R Talham
G.E. Granroth, M.W. Meisel, M. Chaparala, T. Jolicoeur,
B.H. Ward, and D.R. Talham, Phys. Rev. Lett. 77, 1616
(1996).
- H A Bethe
H.A. Bethe, Z. Phys. 71, 205 (1931).
- G Lee
- S Shirane
- B O Wakimoto
- K Wells
- Yamada
Lee, G. Shirane, S. Wakimoto, B.O. Wells, and K. Yamada,
J. Phys. Chem. Solids 56, 1913 (1995).
- B Frischmuth
- S Hass
- G Sierra
- T M Rice
B. Frischmuth, S. Hass, G. Sierra, and T.M. Rice, Phys.
Rev. B 55, R3340 (1997), and references therein.
- T Sakai
- M Takahashi
T. Sakai and M. Takahashi, Phys. Rev. B 42, 1090 (1990).