Heisenberg Ladder Research Articles

The Category of Ladders

In the first part of this paper we introduce the category of ladders and develop the underlying theory. A ladder consists of a sequence of vector spaces (Vn) and linear operators \({(A^+_n), (A^-_n)}\) acting between these vector spaces in ascending and descending direction. Unlike as in classical quantum mechanics ladders are defined as objects of a category, the corresponding notion of ladder homomorphisms allows to perform a mathematically structural and rigorous analysis of ladder theory. We allow dependence on n for spaces and operators in the ladder. The job in ladder theory is to find SIE-subladders, on which the Intrinsic Endomorphisms \({A^-_n A^+_n}\) and \({A^+_n A^-_n}\) act as Scalars αn. A fundamental ladder theorem will provide conditions on the (generalized) commutators or anticommutators assuring the existence of SIE-subladders. The second part contains examples of ladders from classical quantum mechanics, such as the Heisenberg ladder, the Dirac ladder the ladder for the Lie algebra \({{\bf sl}(2,\mathbb{C})}\). Whereas these classical examples are distinguished by constance of operators and spaces, we then show how the generalized ladder theory allows to handle deformations: h-discretization, q-discretization and “periodization”. Other examples come from orthogonal polynomials. The Legendre, Laguerre and Bessel ladder are presented. Ladder theory allows a certain “anticommutator factorization” of the relevant second order differential operators. In a final section we apply ladder theory to ladders that are at the same time complexes. This enables us to give a transparent structural proof of Hodge’s theorem. The idea of ladder operators and factorization is well-known in quantum mechanics and quantum field theory. The book of Shi-Hai Dong (Factorization method in quantum mechanics, Springer, Dordrecht, 2007) contains a good survey of applications in physics, it provides historical background and links to sources in the physics literature. Note that our approach is independent—not only with respect to notation—of that in Shi-Hai Dong (Factorization method in quantum mechanics, Springer, Dordrecht, 2007).

Adapted continuous unitary transformation to treat systems with quasi-particles of finite lifetime

An improved generator for continuous unitary transformations is introduced to describe systems with unstable quasi-particles. Its general properties are derived and discussed. To illustrate this approach we investigate the asymmetric antiferromagnetic spin-1/2 Heisenberg ladder, which allows for spontaneous triplon decay. We present results for the low-energy spectrum and the momentum resolved spectral density of this system. In particular, we show the resonance behavior of the decaying triplon explicitly.

Bond-Alternating Antiferromagnetic S = 1/2 Heisenberg Ladder with Ferromagnetic Diagonal Coupling

Using the density matrix renormalization group method, we determine the phase diagram of a frustrated bond-alternating S = 1/2 Heisenberg ladder with ferro-antiferromagnetic couplings at zero temperature. With the interactions between spins along the rungs set, we identify three spin-gapped phases (the Haldane phase, the singlet phase and the dimer phase) in the whole parameter range. The analysis of our data shows that two-leg spin bond-alternating ladders have a rich phase diagram if both rung and diagonal couplings are taken into account.

Dynamics in a two-leg spin ladder with a four-spin cyclic interaction

We study the two-leg Heisenberg ladder with four-spin cyclic interaction using the (dynamical) density-matrix renormalization-group method. We demonstrate the dependence of the low-lying excitations in the spin wave, staggered dimer order, and scalar-chirality order structure factors on the four-spin cyclic interaction. We find that the cyclic interaction enhances spin-spin correlations with wave vector around momentum $({q}_{x},{q}_{y})=(\frac{\ensuremath{\pi}}{2},0)$. Also, the presence of long-range order in the staggered dimer and scalar-chirality phases is confirmed by a $\ensuremath{\delta}$-function peak contribution of the structure factors at energy $\ensuremath{\omega}=0$.

Interplay between field-induced and frustration-induced quantum criticalities in the frustrated two-leg Heisenberg ladder

The antiferromagnetic Heisenberg two-leg ladder in the presence of frustration and an external magnetic field is a system that is characterized by two sorts of quantum criticalities, not only one. One criticality is the consequence of intrinsic frustration, and the other one is a result of the external magnetic field. So the behaviour of each of them in the presence of the other deserves to be studied. Using the Jordan–Wigner transformation in dimensions higher than one and the bond-mean-field theory we examine the interplay between the field-induced and frustration-induced quantum criticalities in this system. The present work could constitute a prototype for those systems showing multiple, perhaps sometimes competing, quantum criticalities. We calculate several physical quantities such as magnetization and spin susceptibility as functions of field and temperature.PACS Nos.: 75.10.Jm 73.43.Nq 75.10.Pq 75.10.Dg

Field-induced multiple-point quantum criticality in the three-leg Heisenberg ladder

Field-induced quantum (zero temperature) criticality described using parameters that are not associated with a symmetry-breaking long-range order is found in the spin-$\frac{1}{2}$ antiferromagnetic three-leg Heisenberg ladder. These parameters represent the spin bond order, which is a consequence of the probability for spins on adjacent sites to be bound in a singletlike arrangement. They underwent two phase transitions, one at the lower critical field ${h}_{c1}$ and the other at the upper critical field ${h}_{c2}$. The field dependence of these bond parameters and of the magnetization is calculated for several values of the rung coupling in all regimes. This yields the coupling field phase diagram. It is found that the rung coupling must exceed a threshold value for the plateau, at one third of the saturation magnetization, to appear. The results obtained here within the bond mean-field theory compare well with existing numerical data.

Quantum and classical criticalities in the frustrated two-leg Heisenberg ladder

This paper is about the frustration-induced criticality in the antiferromagnetic Heisenberg model on the two-leg ladder with exchange interactions along the chains, rungs, and diagonals, and also about the effect of thermal fluctuations on this criticality. The method used is the bond-mean-field theory, which is based on the Jordan–Wigner transformation in dimensions higher than one. In this paper, we will summarize the main results presented in a talk, and report on new results about the couplings and temperature dependences of the spin susceptibility.PACS Nos.: 75.10.Jm, 75.10.Dg, 75.50.Ee

Magnetic Phase Diagram of Spin-1/2 Two-Leg Ladder with Four-Spin Ring Exchanges

We study a spin-1/2 two-leg Heisenberg ladder with four-spin ring exchanges under a magnetic field. We introduce an exact duality transformation, which is an extension of the spin-chirality duality developed previously and yields a new self-dual surface in the parameter space. We then determine the magnetic phase diagram using the numerical approaches of the density-matrix renormalization-group and exact diagonalization methods. We demonstrate the appearance of a magnetization plateau and a Tomonaga–Luttinger liquid with a dominant vector-chirality quasi-long-range order for a wide parameter regime of strong ring exchange. A “nematic” phase, in which magnons form bound pairs and magnon-pairing correlation functions dominate, is also identified.

Ising Phases of Heisenberg Ladders in a Magnetic Field

We show that Dzyaloshinskii-Moriya (DM) interactions can substantially modify the phase diagram of spin-1/2 Heisenberg ladders in a magnetic field provided they compete with exchange. For nonfrustrated ladders, they induce a local magnetization along the DM vector that turns the gapless intermediate phase into an Ising phase with broken translational symmetry, while for frustrated ladders, they extend the Ising order of the half-integer plateau to the surrounding gapless phases of the purely Heisenberg case. Implications for experimental ladder and dimer systems are discussed.

Effect of temperature on quantum criticality in the frustrated two-leg Heisenberg ladder

The antiferromagnetic Heisenberg model on the two-leg ladder with exchange interactions along the chains, rungs, and diagonals is studied using the Jordan-Wigner transformation and bond-mean-field theory. The inclusion of all three couplings introduces frustration to the system and depending on their relative strengths the ladder can adopt one of three possible magnetically disordered gapped states. The phase diagram found in this mean-field approach is in very good agreement with the one calculated by Weihong et al. [Phys. Rev. B 57, 11439 (1998)] using the Lanczos exact diagonalization method. By analyzing the ground-state energy, we study quantum criticality when the coupling parameters are varied at zero temperature. We study the effect of temperature on the phase boundaries and find that the system shows thermally induced criticality for some values of the rung and diagonal coupling constants. All the phase transitions encountered in this system occur between disordered phases and are all caused by frustration.

Mean-Field Theory for Heisenberg Zigzag Ladder: Ground State Energy and Spontaneous Symmetry Breaking

The spin-1/2 zig-zag Heisenberg ladder (J1 − J2 model) is considered. A new representation for the model is found and a saddle point approximation over the spin-liquid order parameter \({\left\langle {\vec{\sigma }_{{{{n - 1}} }} {\left( {\vec{\sigma }_{n} \times \vec{\sigma }_{{n + 1}} } \right)}} \right\rangle }\) is performed. Corresponding effective action is derived and analytically analyzed. We observe the presence of phase transitions at values J2/J1 = 0.230971 and J2/J1 = 1/2.

Field-induced quantum criticality in low-dimensional Heisenberg spin systems

We study the quantum critical behavior in the antiferromagnetic Heisenberg chain and two-leg Heisenberg ladder resulting from the application of an external magnetic field. In each of these systems a finite-temperature crossover line between two different ferromagnetic phases ends with a quantum critical point at zero temperature. Using the bond-mean-field theory, we calculate the field dependence of the magnetization and the mean-field spin bond parameters in both systems. For the Heisenberg chain, we recover the existing exact results and show in addition that the saturation of the zero-temperature magnetization at the field ${h}_{c}=2J$ is accompanied by a quantum phase transition, where the bond parameter vanishes. Here $J$ is the exchange coupling constant along the chain. For the two-leg ladder, we also recover the known results, like the two magnetization plateaus, and show that at the upper critical field, which corresponds to the appearance of the saturation magnetization plateau, the chain and rung spin bond parameters vanish. The identification of the order parameters that govern the field-induced quantum criticality in the systems we study here constitutes an original contribution. Because no long-range order, which breaks symmetry, characterizes the bond order, the latter could be a proposal for the so-called hidden order. We calculate analytically the bond parameters in both systems as functions of the field in the low- and high-field limits at zero temperature. At nonzero temperatures, the calculation of the magnetization and bond parameters is carried out by solving the mean-field equations numerically.

Susceptibilities and spin gaps of weakly coupled spin ladders

We calculate the uniform and staggered susceptibilities of two-chain spin-1/2 Heisenberg ladders using Monte-Carlo simulations. We show that the gap extracted from the uniform susceptibility and the saturation value of the staggered susceptibility are independent of the sign of the inter-chain coupling J_perp in the asymptotic limit |J_perp|/J -> 0. Furthermore, we examine the existence of logarithmic corrections to the linear scaling of the gap with |J_perp|.

Static and Dynamic Properties of Antiferromagnetic Heisenberg Ladders: Fermionic versus Bosonic Approaches

In terms of spinless fermions via the Jordan-Wigner transformation along a snake-like path and spin waves modified so as to restore the sublattice symmetry, we investigate static and dynamic properties of two-leg antiferromagnetic Heisenberg ladders. The specific heat is finely reproduced by the spinless fermions, whereas the magnetic susceptibility is well described by the modified spin waves. The nuclear spin-lattice relaxation rate is discussed in detail with particular emphasis on its novel field dependence.

Spinless fermions versus spin waves in describing antiferromagnetic Heisenberg ladders

We investigate magnetic properties of the spin − 1 2 two-leg Heisenberg ladder antiferromagnet developing two formulations, one of which starts from the Jordan–Wigner transformation along an elaborately ordered path, and the other of which modifies the conventional spin-wave theory so as to restore the sublattice symmetry. The static and dynamic quantities are calculated and a novel field dependence of the nuclear spin-lattice relaxation rate, which is peculiar to one-dimensional spin-gapped antiferromagnets, is pointed out in particular.

High order perturbation theory for spectral densities of multi-particle excitations: $\mathsf{S = \frac{1}{2}}$ two-leg Heisenberg ladder

We present a high order perturbation approach to quantitatively calculate spectral densities in three distinct steps starting from the model Hamiltonian and the observables of interest. The approach is based on the perturbative continuous unitary transformation introduced previously. It is conceived to work particularly well in models allowing a clear identification of the elementary excitations above the ground state. These are then viewed as quasi-particles above the vacuum. The article focuses on the technical aspects and includes a discussion of series extrapolation schemes. The strength of the method is demonstrated for S=1/2 two-leg Heisenberg ladders, for which results are presented.

LINKED CLUSTER SERIES EXPANSIONS FOR TWO-PARTICLE STATES IN QUANTUM LATTICE MODELS

We have developed strong-coupling series expansion methods to study the two-particle spectra in quantum lattice models. The properties of bound states and multiparticle excitations can reveal important information about the dynamics of a given model. At the heart of this method lies the calculation of an effective Hamiltonian in the two-particle subspace. We use an orthogonal transformation to perform this block diagonalising, and find that maintaining orthogonality is crucial for cases where the ground state and the two-particle subspace have identical quantum numbers. The two-particle Schrödinger equation is solved by using a finite lattice approach in coordinate space or an integral equation in momentum space. These methods allow us to determine precisely the low-lying excitation spectra and dispersion relations for the two-particle bound states. The method has been tested for the (1+1) D transverse Ising model, and applied to the two-leg spin-1/2 Heisenberg ladder. We study the coherence lengths of the bound states, and how they merge with the two-particle continuum. Finally, these techniques are applied to the frustrated alternating Heisenberg chain, which has been of considerable recent interest due to its relevance to spin-Peierls systems such as CuGeO 3. Starting from a limit corresponding to weakly-coupled dimers, we develop high-order series expansions for the effective Hamiltonian in the two-particle subspace. In the regime of strong dimerisation, various properties of the singlet and triplet bound states, and the quintet antibound states, can be accurately calculated. We also study the behaviour as the external bond alternation vanishes, and the way in which the bound states of triplet dimer excitations make the transition to a soliton-antisoliton continuum.

Magnetization plateaux in Bethe ansatz solvable spin-Sladders

We examine the properties of the Bethe Ansatz solvable two- and three-leg spin-$S$ ladders. These models include Heisenberg rung interactions of arbitrary strength and thus capture the physics of the spin-$S$ Heisenberg ladders for strong rung coupling. The discrete values derived for the magnetization plateaux are seen to fit with the general prediction based on the Lieb-Schultz- Mattis theorem. We examine the magnetic phase diagram of the spin-1 ladder in detail and find an extended magnetization plateau at the fractional value $<M > = {1/2}$ in agreement with the experimental observation for the spin-1 ladder compound BIP-TENO.

Phase Diagram of the Spin-\(\frac{1}{2}\) Bond-Alternating Two-Leg Ladder

We study phase transition between two topologically different gapped states in the spin-1/2 bond-alternating two-leg Heisenberg ladder by investigating string order parameters and the level-crossing method based on the exact diagonalization. We determine the phase diagram on the plane of the bond-alternation δ and the rung-leg coupling ratio J ⊥ / J , and conclude that the boundary is given as J ⊥ / J ∝δ 2/3 in the weak coupling region J ⊥ / J , δ≪1.

Ground-state phase diagram of a spin-12frustrated three-leg antiferromagnetic Heisenberg ladder

Using the density matrix renormalization group method we have studied the ground-state phase diagram of a spin-1/2 frustrated three-leg antiferromagnetic Heisenberg ladder, and identified two phases corresponding to the spin-3/2 chain and the standard spin ladder. Varying the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ leads to a quantum phase transition from one ground state to the other, but does not alter the gapless nature of the excitation spectra. On the phase boundary, the degenerate ground states form a chiral spin liquid with a central charge $c=2.$ The critical value of ${J}_{\ifmmode\times\else\texttimes\fi{}}$ is insensitive to the number of legs in the ladder, suggesting that a quantum phase transition may occur at ${J}_{\ifmmode\times\else\texttimes\fi{}}=0.6$ for the isotropic spin-1/2 antiferromagnetic Heisenberg model on the square lattice.