Dynamical properties of systems of localized electron spins and systems of itinerant electrons at zero temperature

Shu Zhang

Published Jan 1, 1996• 0 citations

Abstract

This dissertation reports the results of three loosely connected research projects. Each project employs the recursion method in combination with recently developed techniques of continued-fraction analysis for the investigation of the dynamical properties at zero temperature of a particular quantum many-body system, including models of localized electron spins and models of strongly correlated itinerant electrons. We have studied these four models: two-dimensional (2D) s = 1/2 Heisenberg antiferromagnet (square lattice), the 2D s = 1/2 Heisenberg model with competing nearest-neighbor ($J\\sb1$) and next-nearest neighbor ($J\\sb2$) interactions, the 1D t-J model, and the 1D t-$J\\sb{z}$ model.

DOI: 10.23860/diss-4114

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Dynamical properties of systems of localized electron spins and systems of itinerant electrons at zero temperature

Shu Zhang

Published Jan 1, 1996• 0 citations

Publisher

Abstract

J\\sb1

) and next-nearest neighbor (

J\\sb2

) interactions, the 1D t-J model, and the 1D t-

J\\sb{z}

model.

DOI: 10.23860/diss-4114

Related papers

Dynamics of one-dimensional quantum spin systems with competing interactions at zero temperature

Yongmin Yu

This dissertation reports three projects on the zero-temperature dynamics of some prominent one-dimensional (1D) quantum spin systems. A common feature of all three model systems is the presence of competing interactions. Each project employs the recursion method in combination with recently developed techniques of continued-fraction analysis. The first model investigated is the 1D s = 1 XXZ model with an additional uniaxial single-site term. The results for the in-plane and out-of-plane spin dynamic structure factors $S\\sb{\\mu\\mu}(q,\\omega),\\ \\mu = x,\\ z$ are found to bear the characteristic signatures of several phase transitions between the Haldane phase, the Neel phase, the singlet phase, and two different critical phases. The second model investigated is the 1D s = 1/2 nearest-neighbor (nn) XXZ model with an additional isotropic next-nearest-neighbor (nnn) coupling. The focus here is on the dynamic structure factors $S\\sb{zz}(q,\\omega)$ and $S\\sb{DD}(q,\\omega$ ), which describe (for q = $\\pi$ ) the fluctuations of the Neel and dimer order parameters, respectively. By means of a weak-coupling continued-fraction analysis, the dependence on the exchange constants of the infrared exponent, the renormalized bandwidth of spinon excitations, and the spectral-weight distributions of S $\\sb{zz}(\\pi,\\omega$ ) and $S\\sb{DD}(\\pi,\\omega$ ) in the weak-coupling regime of the spin-fluid phase are calculated. For some parameter values, a discrete branch of excitations above the spinon continuum is found. The discrete states contribute to $S\\sb{zz}(q,\\omega$ ) but not to $S\\sb{DD}(q,\\omega$ ). A strong-coupling continued-fraction analysis is employed for the study of the dynamically relevant dispersions and spectral-weight distributions of $S\\sb{zz}(q,\\omega$ ) and $S\\sb{DD}(q,\\omega$ ) in the strong-coupling regime of the spin-fluid phase and in the dimer or Neel ordered phases. Special attention is given to the case where the pure dimer ground state is realized exactly. The third model investigated is the 1D s = 1 XXX model with bilinear and biquadratic exchange. Here the study is focused on the special coupling ratio where the valence-bond-solid ground state is realized. The excitation spectra which are dynamically relevant for the dynamic spill and dimer structure factors, $S\\sb{zz}(q,\\omega$ ) and $S\\sb{DD}(q,\\omega$ ), are investigated via two different types of continued-fraction analysis. The spectral weight in $S\\sb{ZZ}(q,\\omega$ ) is found to be carried predominantly by a single branch of quasi-particles and a continuum of two-particle scattering states thereof, whereas the spectral weight in $S\\sb{DD}(q,\\omega$ ) is dominated by the continuum alone, which shrinks to a single mode at $q = \\pi$ .This dissertation reports three projects on the zero-temperature dynamics of some prominent one-dimensional (1D) quantum spin systems. A common feature of all three model systems is the presence of competing interactions. Each project employs the recursion method in combination with recently developed techniques of continued-fraction analysis. The first model investigated is the 1D s = 1 XXZ model with an additional uniaxial single-site term. The results for the in-plane and out-of-plane spin dynamic structure factors $S\\sb{\\mu\\mu}(q,\\omega),\\ \\mu = x,\\ z$ are found to bear the characteristic signatures of several phase transitions between the Haldane phase, the Neel phase, the singlet phase, and two different critical phases. The second model investigated is the 1D s = 1/2 nearest-neighbor (nn) XXZ model with an additional isotropic next-nearest-neighbor (nnn) coupling. The focus here is on the dynamic structure factors $S\\sb{zz}(q,\\omega)$ and $S\\sb{DD}(q,\\omega$), which describe (for q = $\\pi$) the fluctuations of the Neel and dimer order parameters, respectively. By means of a weak-coupling continued-fraction analysis, the dependence on the exchange constants of the infrared exponent, the renormalized bandwidth of spinon excitations, and the spectral-weight distributions of S$\\sb{zz}(\\pi,\\omega$) and $S\\sb{DD}(\\pi,\\omega$) in the weak-coupling regime of the spin-fluid phase are calculated. For some parameter values, a discrete branch of excitations above the spinon continuum is found. The discrete states contribute to $S\\sb{zz}(q,\\omega$) but not to $S\\sb{DD}(q,\\omega$). A strong-coupling continued-fraction analysis is employed for the study of the dynamically relevant dispersions and spectral-weight distributions of $S\\sb{zz}(q,\\omega$) and $S\\sb{DD}(q,\\omega$) in the strong-coupling regime of the spin-fluid phase and in the dimer or Neel ordered phases. Special attention is given to the case where the pure dimer ground state is realized exactly. The third model investigated is the 1D s = 1 XXX model with bilinear and biquadratic exchange. Here the study is focused on the special coupling ratio where the valence-bond-solid ground state is realized. The excitation spectra which are dynamically relevant for the dynamic spill and dimer structure factors, $S\\sb{zz}(q,\\omega$) and $S\\sb{DD}(q,\\omega$), are investigated via two different types of continued-fraction analysis. The spectral weight in $S\\sb{ZZ}(q,\\omega$) is found to be carried predominantly by a single branch of quasi-particles and a continuum of two-particle scattering states thereof, whereas the spectral weight in $S\\sb{DD}(q,\\omega$) is dominated by the continuum alone, which shrinks to a single mode at $q = \\pi$.

Journal of Media Literacy EducationJan 1, 19960 citations

Phase transitions in classical and quantum spin systems

Songbo Jin

Classical and quantum spin systems are widely used models in both experimental and theoretical condensed matter physics. In many materials, the electronic interactions can be difficult to model exactly. However, in some insulators (Mott insulators), the magnetic (spin) interactions can be captured well with spin-only models. Several models are studied in this thesis. First, I report the solution of a long-standing issue in a classical frustrated spin model (i.e., where the quantum effects are neglected and the complexity is due to competing interactions): the nature of the thermal phase transition to a stripe state in the two dimensional (2D) J1-J2 model. Here J1 and J2 are nearest- and next-nearest-neighbor couplings. Monte Carlo simulations with single-spin updates are used for the calculations, and an extended-ensemble method, a generalization of the Wang-Landau algorithm, is also developed and tested. I focus on the study of "weak universality" behavior (continuously varying critical exponents, with one of the exponents staying fixed), for which I show a correspondence with a known class of conformal field theories with charge c = 1. Next, moving to quantum spins, to shed light on magnetic systems studied experimentally and to investigate new types of quantum states of interest in developing theories of quantum magnetism, I study the S = 1/2 Heisenberg model on the 2D square lattice with added six-spin interactions (the so-called J-Q3 model) as well as a set of 3D quantum antiferromagnets on dimerized lattices. Here I use the stochastic series expansion quantum Monte Carlo method. In the study of the J-Q3 model, I report on a similar weak-universality behavior as in the classical J1-J2 model, but with a mapping to a different known class of c = 1 conformal field theories. The critical behavior of the system again shows continuously changing exponents, in a way which corresponds to a gradual weakening of Z4 symmetry-breaking to an emergent U(1) symmetry. In the study of dimerized antiferromagnets, I report on a universal behavior of the Néel temperature TN, which can be related to ground state parameters independently of the microscopic interaction details in several different models.Classical and quantum spin systems are widely used models in both experimental and theoretical condensed matter physics. In many materials, the electronic interactions can be difficult to model exactly. However, in some insulators (Mott insulators), the magnetic (spin) interactions can be captured well with spin-only models. Several models are studied in this thesis. First, I report the solution of a long-standing issue in a classical frustrated spin model (i.e., where the quantum effects are neglected and the complexity is due to competing interactions): the nature of the thermal phase transition to a stripe state in the two dimensional (2D) J1-J2 model. Here J1 and J2 are nearest- and next-nearest-neighbor couplings. Monte Carlo simulations with single-spin updates are used for the calculations, and an extended-ensemble method, a generalization of the Wang-Landau algorithm, is also developed and tested. I focus on the study of "weak universality" behavior (continuously varying critical exponents, with one of the exponents staying fixed), for which I show a correspondence with a known class of conformal field theories with charge c = 1. Next, moving to quantum spins, to shed light on magnetic systems studied experimentally and to investigate new types of quantum states of interest in developing theories of quantum magnetism, I study the S = 1/2 Heisenberg model on the 2D square lattice with added six-spin interactions (the so-called J-Q3 model) as well as a set of 3D quantum antiferromagnets on dimerized lattices. Here I use the stochastic series expansion quantum Monte Carlo method. In the study of the J-Q3 model, I report on a similar weak-universality behavior as in the classical J1-J2 model, but with a mapping to a different known class of c = 1 conformal field theories. The critical behavior of the system again shows continuously changing exponents, in a way which corresponds to a gradual weakening of Z4 symmetry-breaking to an emergent U(1) symmetry. In the study of dimerized antiferromagnets, I report on a universal behavior of the Néel temperature TN, which can be related to ground state parameters independently of the microscopic interaction details in several different models.

Jan 1, 20130 citations

Finite-temperature dynamics of low-dimensional quantum systems with DMRG methods

Alexander C. Tiegel

This thesis is concerned with the numerical study of one-dimensional (1D) spin-1/2 quantum magnets and related method development. Its focus is on the calculation of dynamical spin correlation functions both at zero and finite temperature. This is motivated by the accessibility of dynamical quantities in experiments such as inelastic neutron scattering (INS) and electron spin resonance (ESR). The numerical methods used in this thesis are based on extensions of the density-matrix renormalization group (DMRG) and are formulated in the framework of matrix product states (MPS). While zero-temperature dynamical correlation functions are computed with existing MPS frequency-domain methods, an MPS frequency-domain approach for their calculation at finite temperature is developed in this thesis. The new method combines the Liouville-space formulation of the dynamics with a moment expansion of the dynamical correlation function. The majority of the results are obtained via MPS-based Chebyshev expansions. These numerical techniques are applied to two different model systems describing real materials. The first one is the material copper pyrimidine dinitrate (Cu-PM) which is modeled by a 1D spin-1/2 Heisenberg antiferromagnet with Dzyaloshinskii-Moriya interactions. The spin dynamics of this model is studied in an applied magnetic field and compared to ESR experiments. Zero-temperature calculations for momentum- and frequency-resolved dynamical quantities give direct access to the intensity of the elementary excitations and go beyond the low-energy description by the quantum sine-Gordon model. Thus, a deviation from the Lorentz invariant dispersion for the single-soliton resonance is found. The presence of the strongest boundary bound state previously predicted from a boundary sine-Gordon field theory is confirmed, while composite boundary-bulk excitations have too low intensities to be found in the numerical results. At finite temperature, there is a temperature-induced crossover of the soliton. Moreover, additional temperature effects such as interbreather transitions emerge, which is confirmed by accompanying ESR experiments on Cu-PM over a wide range of the applied field strength. The second system studied in this thesis is the compound BaCu $_2$ V $_2$ O $_8$ . It is shown that the magnetic properties of this quasi-1D material can be described by a strongly alternating antiferromagnetic-ferromagnetic spin-1/2 Heisenberg chain. As found for other dimerized systems, the strong correlations in BaCu $_2$ V $_2$ O $_8$ persist even at elevated temperatures. Moreover, these correlations lead to an asymmetric lineshape broadening of the magnetic excitations at finite temperature. Upon raising the temperature, an increasingly asymmetric lineshape is observed in high-resolution INS experiments for BaCu $_2$ V $_2$ O $_8$ . In this thesis, the lineshape is calculated by the MPS finite-temperature method developed as a part of this work. Comparing these results to the INS data and the lineshape obtained by a diagrammatic approach, excellent agreement is found over a broad temperature range. This demonstrates that coherent quantum behavior persists at elevated temperatures in BaCu $_2$ V $_2$ O $_8$ and that it can be predicted quantitatively.This thesis is concerned with the numerical study of one-dimensional (1D) spin-1/2 quantum magnets and related method development. Its focus is on the calculation of dynamical spin correlation functions both at zero and finite temperature. This is motivated by the accessibility of dynamical quantities in experiments such as inelastic neutron scattering (INS) and electron spin resonance (ESR). The numerical methods used in this thesis are based on extensions of the density-matrix renormalization group (DMRG) and are formulated in the framework of matrix product states (MPS). While zero-temperature dynamical correlation functions are computed with existing MPS frequency-domain methods, an MPS frequency-domain approach for their calculation at finite temperature is developed in this thesis. The new method combines the Liouville-space formulation of the dynamics with a moment expansion of the dynamical correlation function. The majority of the results are obtained via MPS-based Chebyshev expansions. These numerical techniques are applied to two different model systems describing real materials. 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Moreover, additional temperature effects such as interbreather transitions emerge, which is confirmed by accompanying ESR experiments on Cu-PM over a wide range of the applied field strength. The second system studied in this thesis is the compound BaCu$_2$V$_2$O$_8$. It is shown that the magnetic properties of this quasi-1D material can be described by a strongly alternating antiferromagnetic-ferromagnetic spin-1/2 Heisenberg chain. As found for other dimerized systems, the strong correlations in BaCu$_2$V$_2$O$_8$ persist even at elevated temperatures. Moreover, these correlations lead to an asymmetric lineshape broadening of the magnetic excitations at finite temperature. Upon raising the temperature, an increasingly asymmetric lineshape is observed in high-resolution INS experiments for BaCu$_2$V$_2$O$_8$. In this thesis, the lineshape is calculated by the MPS finite-temperature method developed as a part of this work. Comparing these results to the INS data and the lineshape obtained by a diagrammatic approach, excellent agreement is found over a broad temperature range. This demonstrates that coherent quantum behavior persists at elevated temperatures in BaCu$_2$V$_2$O$_8$ and that it can be predicted quantitatively.

Jan 1, 20161 citation

Zero-temperature dynamics of the one-dimensionalXXZandt-Jmodels: A weak-coupling continued-fraction analysis

V. S. Viswanath, Shu Zhang, Gerhard H. Muller +1 more

We use the recursion method to study the spectral and dynamical properties of the one-dimensional (1D) s=1/2 XXZ method with planar anisotropy at T=0. Distinct methods of continued-fraction analysis have been developed for the weak-coupling and strong-coupling regimes of the corresponding lattice fermion system. The weak-coupling analysis presented here yields detailed information on the spectral-weight distribution in dynamic structure factors and spin autocorrelation functions, notably on the infrared singularities produced by critical fluctuations, and on the bound states for the case of attractive fermion interaction. The same method is then applied to the charge dynamics of the 1D t-J model for strongly correlated electrons. There it yields similar yet distinct results in the regime of weak exchange coupling. The results for renormalized bandwidths of particle-hole excitations are consistent with available results for charge velocities, and the results for the infrared exponent in the charge dynamic structure factor agree with existing results for the exponent of the equal-time charge correlation function.We use the recursion method to study the spectral and dynamical properties of the one-dimensional (1D) s=1/2 XXZ method with planar anisotropy at T=0. Distinct methods of continued-fraction analysis have been developed for the weak-coupling and strong-coupling regimes of the corresponding lattice fermion system. The weak-coupling analysis presented here yields detailed information on the spectral-weight distribution in dynamic structure factors and spin autocorrelation functions, notably on the infrared singularities produced by critical fluctuations, and on the bound states for the case of attractive fermion interaction. The same method is then applied to the charge dynamics of the 1D t-J model for strongly correlated electrons. There it yields similar yet distinct results in the regime of weak exchange coupling. The results for renormalized bandwidths of particle-hole excitations are consistent with available results for charge velocities, and the results for the infrared exponent in the charge dynamic structure factor agree with existing results for the exponent of the equal-time charge correlation function.

Physical review. B, Condensed matterJan 1, 19959 citations

Quantum Monte Carlo methods and strongly correlated electrons on honeycomb structures

Thomas C. Lang

In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.

Jan 1, 20102 citations

Frustration-induced Quantum Spin Liquid Behavior in the s = 1/2 Random-bond Heisenberg Antiferromagnet on the Zigzag Chain

Kazuki Uematsu, Toshiya Hikihara, Hikaru Kawamura

Recent studies have revealed that the randomness-induced quantum spin liquid (QSL)-like state is stabilized in certain frustrated quantum magnets in two and three dimensions. In order to clarify the nature of this gapless QSL-like state, we investigate both zero- and finite-temperature properties of the random-bond one-dimensional (1D) $s=12$ Heisenberg model with the competing nearest-neighbor and next-nearest-neighbor antiferromagnetic interactions, $J_1$ and $J_2$ , by means of the exact diagonalization, density-matrix renormalization-group and Hams--de Raedt methods. We find that, on increasing the frustration $J_2$ , the gapless nonmagnetic state stabilized in the unfrustrated model with $J_2=0$ , the {\it unfrustrated\} random-singlet (RS) state, exhibits a phase transition into different gapless nonmagnetic state, the {\it frustrated\} RS state. This frustrated RS state in 1D has properties quite similar to the randomness-induced QSL-like state recently identified in 2D and 3D frustrated magnets exhibiting the $T$ -linear low-temperature ( $T$ ) specific heat, while the unfrustrated RS state is more or less specific to the unfrustrated 1D system exhibiting the $\sim 1(\log T)^3$ low- $T$ specific heat. Universal features and the robustness against perturbations of the frustrated RS state are emphasized.Recent studies have revealed that the randomness-induced quantum spin liquid (QSL)-like state is stabilized in certain frustrated quantum magnets in two and three dimensions. In order to clarify the nature of this gapless QSL-like state, we investigate both zero- and finite-temperature properties of the random-bond one-dimensional (1D) $s=12$ Heisenberg model with the competing nearest-neighbor and next-nearest-neighbor antiferromagnetic interactions, $J_1$ and $J_2$, by means of the exact diagonalization, density-matrix renormalization-group and Hams--de Raedt methods. We find that, on increasing the frustration $J_2$, the gapless nonmagnetic state stabilized in the unfrustrated model with $J_2=0$, the {\it unfrustrated\} random-singlet (RS) state, exhibits a phase transition into different gapless nonmagnetic state, the {\it frustrated\} RS state. This frustrated RS state in 1D has properties quite similar to the randomness-induced QSL-like state recently identified in 2D and 3D frustrated magnets exhibiting the $T$-linear low-temperature ($T$) specific heat, while the unfrustrated RS state is more or less specific to the unfrustrated 1D system exhibiting the $\sim 1(\log T)^3$ low-$T$ specific heat. Universal features and the robustness against perturbations of the frustrated RS state are emphasized.

arXiv:2009.08630Journal of the Physical Society of JapanNov 9, 20218 citations

J\\sb1

) and next-nearest neighbor (

J\\sb2

) interactions, the 1D t-J model, and the 1D t-

J\\sb{z}

model.

S\\sb{\\mu\\mu}(q,\\omega),\\ \\mu = x,\\ z

are found to bear the characteristic signatures of several phase transitions between the Haldane phase, the Neel phase, the singlet phase, and two different critical phases. The second model investigated is the 1D s = 1/2 nearest-neighbor (nn) XXZ model with an additional isotropic next-nearest-neighbor (nnn) coupling. The focus here is on the dynamic structure factors

S\\sb{zz}(q,\\omega)

and

S\\sb{DD}(q,\\omega

), which describe (for q =

\\pi

) the fluctuations of the Neel and dimer order parameters, respectively. By means of a weak-coupling continued-fraction analysis, the dependence on the exchange constants of the infrared exponent, the renormalized bandwidth of spinon excitations, and the spectral-weight distributions of S

\\sb{zz}(\\pi,\\omega

) and

S\\sb{DD}(\\pi,\\omega

) in the weak-coupling regime of the spin-fluid phase are calculated. For some parameter values, a discrete branch of excitations above the spinon continuum is found. The discrete states contribute to

S\\sb{zz}(q,\\omega

) but not to

S\\sb{DD}(q,\\omega

). A strong-coupling continued-fraction analysis is employed for the study of the dynamically relevant dispersions and spectral-weight distributions of

S\\sb{zz}(q,\\omega

) and

S\\sb{DD}(q,\\omega

) in the strong-coupling regime of the spin-fluid phase and in the dimer or Neel ordered phases. Special attention is given to the case where the pure dimer ground state is realized exactly. The third model investigated is the 1D s = 1 XXX model with bilinear and biquadratic exchange. Here the study is focused on the special coupling ratio where the valence-bond-solid ground state is realized. The excitation spectra which are dynamically relevant for the dynamic spill and dimer structure factors,

S\\sb{zz}(q,\\omega

) and

S\\sb{DD}(q,\\omega

), are investigated via two different types of continued-fraction analysis. The spectral weight in

S\\sb{ZZ}(q,\\omega

) is found to be carried predominantly by a single branch of quasi-particles and a continuum of two-particle scattering states thereof, whereas the spectral weight in

S\\sb{DD}(q,\\omega

) is dominated by the continuum alone, which shrinks to a single mode at

q = \\pi

_2

_2

_8

. It is shown that the magnetic properties of this quasi-1D material can be described by a strongly alternating antiferromagnetic-ferromagnetic spin-1/2 Heisenberg chain. As found for other dimerized systems, the strong correlations in BaCu

_2

_2

_8

persist even at elevated temperatures. Moreover, these correlations lead to an asymmetric lineshape broadening of the magnetic excitations at finite temperature. Upon raising the temperature, an increasingly asymmetric lineshape is observed in high-resolution INS experiments for BaCu

_2

_2

_8

. In this thesis, the lineshape is calculated by the MPS finite-temperature method developed as a part of this work. Comparing these results to the INS data and the lineshape obtained by a diagrammatic approach, excellent agreement is found over a broad temperature range. This demonstrates that coherent quantum behavior persists at elevated temperatures in BaCu

_2

_2

_8

and that it can be predicted quantitatively.

s=12

Heisenberg model with the competing nearest-neighbor and next-nearest-neighbor antiferromagnetic interactions,

J_1

and

J_2

, by means of the exact diagonalization, density-matrix renormalization-group and Hams--de Raedt methods. We find that, on increasing the frustration

J_2

, the gapless nonmagnetic state stabilized in the unfrustrated model with

J_2=0

, the {\it unfrustrated\} random-singlet (RS) state, exhibits a phase transition into different gapless nonmagnetic state, the {\it frustrated\} RS state. This frustrated RS state in 1D has properties quite similar to the randomness-induced QSL-like state recently identified in 2D and 3D frustrated magnets exhibiting the

T

-linear low-temperature (

T

) specific heat, while the unfrustrated RS state is more or less specific to the unfrustrated 1D system exhibiting the

\sim 1(\log T)^3

low-

T

specific heat. Universal features and the robustness against perturbations of the frustrated RS state are emphasized.