Simulating low-temperature properties of three-dimensional frustrated quantum magnets is challenging due to the sign problem and the system sizes required to mitigate substantial finite-size effects. However, there are many experimental examples of three-dimensional crystals that could host exotic low-temperature states of matter, such as quantum spin liquids. We calculate the ground-state phase diagrams of frustrated quantum spin models on the body-centered cubic lattice using neural quantum states. First, we study the antiferromagnetic
J1−J2 model where we find a direct first-order phase transition between Néel and collinear long-range-ordered phases at
(J2/J1)c=0.705, consistent with previous studies. Then, in a tetragonally-distorted variant, proposed as a minimal model of NaCa
2Cu
2(VO
4)
3, we find no evidence of a quantum paramagnetic ground state, with a first-order phase transition between Néel and chain phases at
(J2ab/J1)c=1.0375. Therefore, the ground state of the tetragonally-distorted model does not reproduce the low-temperature magnetic properties of NaCa
2Cu
2(VO
4)
3, and the inclusion of other effects is necessary to rationalize experimental observations.
In the present work low energy excitations in various antiferromagnetic systems are investigated.The first part is devoted to the antiferromagnetic Heisenberg chain with two different alternating magnetic ions having spins S A and S B .The system can be characterised as a ferrimagnet with a long range ordered ground state.Linear spin wave theory (LSWT) leads to a qualitatively correct description of two excitation branches.The low energy spectrum contains ferromagnetic magnons which are discussed in the introductory chapter.The LSWT results are compared with the numerical data from exact diagonalisation for the case S A =1 and S B =1/2.In the second part of the thesis, the quantum phase transitions from the Néel phase to the Heisenberg phase and from the Néel phase to the Haldane phase for half-odd-integer spins S =1 /2, 3/2 and integer spins S =1 , 2, respectively, are studied in context of the Haldane conjecture.The models are analysed by means of high-order series expansions for the ground state properties and excitation energies.The expansion parameter is the anisotropy λ where the unperturbed case λ =0corresponds to the Ising model.For the integer-S systems the following scenario is obtained: The phase transitions from the Néel phase to the Haldane phase are determined by the condensation of the so-called spin zero defects (SZD) at finite anisotropy.These domain wall type excitations can be characterised as S z =0states embedded in the antiferromagnetic order.The critical exponents β of the order parameter and η of the correlation function are those of the two-dimensional Ising model.In contrast to the above scenario, the half-odd-integer S =1/2 chain undergoes the Kosterlitz-Thouless phase transition at the isotropic point.In the case of the S =3/2 chain the critical behaviour of the static structure factor S (zz) (π) is consistent with the exponentially divergent correlation length at the isotropic point.Finally, kink excitations of the S =1 /2 Heisenberg antiferromagnet chain with nearest and next nearest neighbour interactions (the zigzag chain) are investigated.Here a kink is defined as a free spin S =1 /2 mediating two dimerised ground states.The phase diagram of the zigzag chain contains the Majumdar-Ghosh model as well as the S =1/2 spin ladder and the S =1Heisenberg antiferromagnet.Starting from the Majumdar-Ghosh model, a two-kink variational wave function with S tot =1is constructed.The variational parameter is the characteristic distance between the two kinks.The variational wave function describes the low energy triplet excitation of the gapped zigzag chain in a qualitatively correct way.Within this approach, the elementary excitation of the spin ladder as well as the Haldane triplet of the S =1chain appears as a bound state of two kinks.The phase diagram of the zigzag chain contains the incommensurate phase where the wave vector of the lowest mode k 0 runs from k 0 =0to k 0 = π.At the qualitative level, this behaviour is correctly described by the variational approach.
Antiferromagnetic Heisenberg integer-spin chains are characterized by a spin-liquid ground state with no long-range order, due to the relevance of quantum fluctuations. Spin anisotropy, however, freezes quantum fluctuations, and the system is magnetized in the presence of a sufficiently large easy-axis anisotropy. We numerically investigate the case S=1, by means of the density-matrix renormalization group, and find that the freezing of the spin liquid into a Néel spin solid, with increasing easy-axis anisotropy, is a continuous quantum phase transition. Numerical evidence indicates that the transition is not in the two-dimensional Ising universality class.
The ground-state phase diagram of
S=2 antiferromagnetic Heisenberg chains with coexisting uniform and alternating single-site anisotropies is investigated by the numerical exact diagonalization and density matrix renormalization group methods. We find the Haldane, large-
D, Néel, period-doubled Néel, gapless spin fluid, quantized and partial ferrimagnetic phases. The Haldane phase is limited to the close neighborhood of the isotropic point. Within numerical accuracy, the transition from the gapless spin-fluid phase to the period-doubled Néel phase is a direct transition. Nevertheless, the presence of a narrow spin-gap phase between these two phases is suggested on the basis of the low-energy effective theory. The ferrimagnetic ground state is present in a wide parameter range. This suggests the realization of magnetized single-chain magnets with a uniform spin magnitude by controlling the environment of each magnetic ion without introducing ferromagnetic interactions.
The ground-state properties of the one-dimensional spin-1/2 alternating Heisenberg-Ising antiferromagnet are studied both analytically and numerically. The phase diagram is qualitatively predicted by the use of the phase Hamiltonian: The ground state is either the Néel or the effective singlet state according to the values of the alternation and anisotropy parameters. The phase transition between these states is of the first order. The ground state energy, its derivative with respect to the alternation parameter, excitation gap, and the various long-range order parameters are exactly calculated by the numerical method for finite systems ( N ≦20). The numerical calculations support the analytically predicted phase diagram. The correlation length is also numerically calculated at finite temperatures by the use of the quantum transfer-matrix method.
The quantum phases of one-dimensional spin
s=1/2 chains are discussed for models with two parameters, frustrating exchange
g=J2>0 between second neighbors and normalized nonfrustrating power-law exchange with exponent $\ensuremath{\alpha}$ and distance dependence ${r}^{\ensuremath{-}\ensuremath{\alpha}}$. The ground state (GS) at
g=0 has a long-range order (LRO) for $\ensuremath{\alpha}<2$ and long-range spin fluctuations for $\ensuremath{\alpha}>2$. The models conserve total spin
S=SA+SB, have singlet GS for any
g, $\ensuremath{\alpha}\ensuremath{\ge}0$ and decouple at
1/g=0 to linear Heisenberg antiferromagnets on sublattices
A and
B of odd- and even-numbered sites. Exact diagonalization of finite chains gives the sublattice spin $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$, the magnetic gap
Em to the lowest triplet state, and the excitation ${E}_{\ensuremath{\sigma}}$ to the lowest singlet with opposite inversion symmetry to the GS. An analytical model that conserves sublattice spin has a first-order quantum transition at
gc=1/4ln2 from a GS with perfect LRO to a decoupled phase with
SA=SB=0 for $g\ensuremath{\ge}4/{\ensuremath{\pi}}^{2}$ and no correlation between spins in different sublattices. The model with $\ensuremath{\alpha}=1$ has a first-order transition to a decoupled phase that closely resembles the analytical model. The bond order wave (BOW) phase and continuous quantum phase transitions of finite models with $\ensuremath{\alpha}\ensuremath{\ge}2$ are discussed in terms of GS degeneracy where ${E}_{\ensuremath{\sigma}}(g)=0$, excited state degeneracy where ${E}_{\ensuremath{\sigma}}(g)={E}_{m}(g)$, and $\ensuremath{\langle}{S}_{A}^{2}\ensuremath{\rangle}$. The decoupled phase at large frustration has nondegenerate GS for any exponent $\ensuremath{\alpha}$ and excited states related to sublattice excitations.