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arXiv:cond-mat/0105250 · Physical review. B, Condensed matter

Geometric frustration and magnetization plateaus in quantum spin and Bose-Hubbard models on tubes

Dmitry Green, Claudio Chamon

Published Mar 1, 2002• 2 citations
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Abstract

We study $\mathrm{XXZ}$ Heisenberg models on frustrated triangular lattices wrapped around a cylinder. In addition to having interesting magnetic phases, these models are equivalent to Bose-Hubbard models that describe the physical problem of adsorption of noble gases on the surface of carbon nanotubes. We find analytical results for the possible magnetization plateau values as a function of the wrapping vectors of the cylinder, which in general introduce extra geometric frustration besides the one due to the underlying triangular lattice. We show that for particular wrapping vectors $(N,0),$ which correspond to the zigzag nanotubes, there is a macroscopically degenerate ground state in the classical Ising limit. The Hilbert space for the degenerate states can be enumerated by a mapping first into a path in a square lattice wrapped around a cylinder (a Bratteli diagram), and then to free fermions interacting with a single ${\mathbf{Z}}_{N}$ degree of freedom. From this model we obtain the spectrum in the anisotropic Heisenberg limit, showing that it is gapless. The continuum limit is a $c=1$ conformal field theory with compactification radius $R=N$ set by the physical tube radius. This result cannot be checked against a Lieb-Schultz-Mattis argument, for the argument is inconclusive when applied to this problem. We show that the compactification radius quantization is exact in the projective ${J}_{\mathrm{xy}}{/J}_{z}\ensuremath{\ll}1$ limit, and that higher-order corrections reduce the value of R. The particular case of a $(N=2,0)$ tube, which corresponds to a two-leg ladder with cross links, is studied separately and shown to be gapped because the fermion-mapped problem contains superconducting pairing terms.

DOI
10.1103/physrevb.65.104431

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We study XXZ\mathrm{XXZ}XXZ Heisenberg models on frustrated triangular lattices wrapped around a cylinder. In addition to having interesting magnetic phases, these models are equivalent to Bose-Hubbard models that describe the physical problem of adsorption of noble gases on the surface of carbon nanotubes. We find analytical results for the possible magnetization plateau values as a function of the wrapping vectors of the cylinder, which in general introduce extra geometric frustration besides the one due to the underlying triangular lattice. We show that for particular wrapping vectors (N,0),(N,0),(N,0), which correspond to the zigzag nanotubes, there is a macroscopically degenerate ground state in the classical Ising limit. The Hilbert space for the degenerate states can be enumerated by a mapping first into a path in a square lattice wrapped around a cylinder (a Bratteli diagram), and then to free fermions interacting with a single ZN{\mathbf{Z}}_{N}ZN​ degree of freedom. From this model we obtain the spectrum in the anisotropic Heisenberg limit, showing that it is gapless. The continuum limit is a c=1c=1c=1 conformal field theory with compactification radius R=NR=NR=N set by the physical tube radius. This result cannot be checked against a Lieb-Schultz-Mattis argument, for the argument is inconclusive when applied to this problem. We show that the compactification radius quantization is exact in the projective Jxy/Jz\ensuremath≪1{J}_{\mathrm{xy}}{/J}_{z}\ensuremath{\ll}1Jxy​/Jz​\ensuremath≪1 limit, and that higher-order corrections reduce the value of R. The particular case of a (N=2,0)(N=2,0)(N=2,0) tube, which corresponds to a two-leg ladder with cross links, is studied separately and shown to be gapped because the fermion-mapped problem contains superconducting pairing terms.
We formulate the problem of adsorption onto the surface of a carbon nanotube as a lattice gas on a triangular lattice wrapped around a cylinder. This model is equivalent to an XXZ Heisenberg quantum spin tube. We find density plateau structures for armchair, zigzag, and chiral nanotubes. The zigzags are special and have extensive zero temperature entropy plateaus in the classical limit. Quantum effects lift the degeneracy, leaving gapless excitations described by a c = 1 conformal field theory with compactification radius quantized by the tube circumference.
This thesis is dedicated to the analytical and numerical study of quantum entanglement and nonlocality in strongly correlated quantum spin tube models in connection with various thermodynamic properties and magnetization processes. More specifically, the study builds on the analysis of the behavior of these physical quantities within a particular spinning network that includes competitive interactions and quantum fluctuations known as a three-leg spins ladder or triangular spin tube, due to the periodic boundary conditions imposed on the three spins that form the Heisenberg steps of this network. In systems with this structure described above, magnetization plateaus are formed due to the competition imposed by the geometric frustration that the network imposes on the spins and the applied external field. Such competition is the mechanism that defines the orientation of the magnetic domains that the network will present, which within specific regions of the network parameters space create regions where there is a discontinuous response of local magnetization to external field variations, such regions are known as magnetization plateaus. To evaluate the frustrated spin- ½ Ising-Heisenberg triangular tube consisting of a spin tube formed by triangular unit cells with 3 spins-½ interconnected by Heisenberg couplings between neighboring first spins and the pairs of neighboring first spins belonging to the different cells have their interaction intermediated through classical Ising couplings, an exact analytical procedure, known as a transfer-matrix technique, was used to obtain the analytical solutions for various thermodynamic properties of the model, including magnetization, specific heat and magnetization profiles. These quantities were evaluated as a function of temperature, external magnetic field and the relevant couplings. However, it was necessary to apply a different methodology to obtain thermodynamic measurements for the other model proposed in this thesis: the frustrated spin- ½ Heisenberg triangular tube, which consists of a spin tube with similar structure to the previous one, except that the pairs of the first neighboring spins belonging to the different cells interact through Heisenberg-type quantum connections. In this case, a numerical procedure known as the quantum density matrix renormalization group method, known as DMRG, is used to obtain the ground state diagrams, magnetization processes, and bipartite quantum entanglement of the Heisenberg triangular tubes as a function of the external magnetic field and the relevant couplings. In both models, the magnetization curves exhibit discontinuous magnetization jumps and intermediate plateaus due to discontinuous phase transitions induced by the external field, These transitions are also reflected in the exotic behavior of some thermodynamic quantities of the models, such as the presence of a striking low temperature peak for the specific heat curve and a highlighted magnetocaloric effect.
We use a combination of numerical density matrix renormalization group calculations and several analytical approaches to comprehensively study a simplified model for a spatially anisotropic spin-12\frac{1}{2}21​ triangular lattice Heisenberg antiferromagnet: the three-leg triangular spin tube (TST). The model is described by three Heisenberg chains, with exchange constant JJJ, coupled antiferromagnetically with exchange constant ${J}^{\ensuremath{'}}$ along the diagonals of the ladder system, with periodic boundary conditions in the shorter direction. Here, we determine the full phase diagram of this model as a function of both spatial anisotropy (between the isotropic and decoupled chain limits) and magnetic field. We find a rich phase diagram, which is remarkably dominated by quantum states: the phase corresponding to the classical ground state appears only in an exceedingly small region. Among the dominant phases generated by quantum effects are commensurate and incommensurate coplanar quasiordered states, which appear in the vicinity of the isotropic region for most fields, and in the high-field region for most anisotropies. The coplanar states, while not classical ground states, can at least be understood semiclassically. Even more strikingly, the largest region of phase space is occupied by a spin density wave phase, which has incommensurate collinear correlations along the field. This phase has no semiclassical analog, and may be ascribed to enhanced one-dimensional fluctuations due to frustration. Cutting across the phase diagram is a magnetization plateau, with a gap to all excitations and ``up-up-down'' spin order, with a quantized magnetization equal to 13\frac{1}{3}31​ of the saturation value. In the TST, this plateau extends almost but not quite to the decoupled chains limit. Most of the above features are expected to carry over to the two-dimensional system, which we also discuss. At low field, a dimerized phase appears, which is particular to the one-dimensional nature of the TST, and which can be understood from quantum Berry phase arguments.
Recent developments of theoretical studies on spin nanotubes are reviewed, especially focusing on the S = 1/2 three-leg spin tube. In contrast to the three-leg spin ladder, the tube has a spin gap in the case of the regular-triangle unit cell when the rung interaction is sufficiently large. The effective theory based on the Hubbard Hamiltonian indicates a quantum phase transition to a gapless spin liquid due to the lattice distortion to an isosceles triangle. This is also supported by the numerical diagonalization and the density matrix renormalization group analyses. Furthermore, combining analytical and numerical approaches, we reveal several novel magnetic-field-induced phenomena: Néel, dimer, chiral and/or inhomogeneous orders, a new mechanism for the magnetization plateau formation, and others. The recently synthesized spin tube materials are also briefly introduced.
We have formulated a twist operator argument for the geometrically frustrated quantum spin systems on the kagome and triangular lattices, thereby extending the application of the Lieb-Schultz-Mattis and Oshikawa-Yamanaka-Affleck theorems to these systems. The equivalent large gauge transformation for the geometrically frustrated lattice differs from that for non-frustrated systems due to the existence of multiple sublattices in the unit cell and non-orthogonal basis vectors. Our study for the S = 1/2 kagome Heisenberg antiferromagnet at zero external magnetic field gives a criterion for the existence of a two-fold degenerate ground state with a finite excitation gap and fractionalized excitations. At finite field, we predict various plateaux at fractional magnetisation, in analogy with integer and fractional quantum Hall states of the primary sequence. These plateaux correspond to gapped quantum liquid ground states with a fixed number of singlets and spinons in the unit cell. A similar analysis for the triangular lattice predicts a single fractional magnetization plateau at 1/3. Our results are in broad agreement with numerical and experimental studies.
We develop a real space quantum renormalization group (QRG) to explore a frustrated anisotropic four-leg spin-1/2 nanotube in the thermodynamic limit. We obtain the phase diagram, fixed points, critical points, the scaling of coupling constants and magnetization curves. Our investigation points out that, in the case of strong leg coupling, the diagonal frustrating interaction is marginal under QRG transformations and does not affect the universality class of the model. Remarkably, the renormalization equations express that the spin nanotube prepared in the strong leg coupling case goes to the strong plaquette coupling limit (weakly interacting plaquettes). Subsequently, in the limit of weakly interacting plaquettes, the model is mapped onto a 1D spin-1/2 XXZ chain in a longitudinal magnetic field under QRG transformation. Furthermore, the effective Hamiltonian of the spin nanotube inspires both first and second order phase transitions accompanied by the fractional magnetization plateaus. Our results show that the anisotropy changes the magnetization curve and the phase transition points, significantly. Finally, we report the numerical exact diagonalization results to compare the ground state phase diagram with our analytical visions.