In this thesis we consider quantum fluctuations in the ground state energy of antiferromagnetic spin configurations, and phase transitions between these configurations. We use a Holstein-Primakoff transformation up to second order to transform our spin operators, and a Bogoliubov transformation to diagonalize the corresponding Hamiltonian. First we consider a Hamiltonian with Heisenberg-exchange interaction, a magnetic field and anisotropy, for which we investigated both the the antiferromagnetic phase and the spin-flop phase. For the antiferromagnetic phase we find that quantum fluctuations lower the ground state energy. For the spin-flop phase we find that quantum fluctuations lower the ground state energy even more, such that the phase transition point between these phases is shifted to a slightly lower strength of the magnetic field. Furthermore, this shifted phase transition is caused by ground state energies of the spin-flop phase that become complex, instead of an energy crossing. This energy takes on complex values due to imaginary eigenvalues for magnons, which correspond to exponentially increasing semi-classical spin waves.\n\nNext we considered the Dzyaloshinskii-Moriya interaction (DMI) in one dimension, with the spiral phase it induces. We focused on the magnetic frustration of this phase due to anisotropy and the magnetic field. We used a superposition of two plane waves to describe the deviation due to this frustration, and minimized the ground state energy for their amplitudes numerically. By assuming only spirals of integer length and periodic boundary conditions, we constructed a semi-classical ground state phase diagram between the three mentioned phases. This phase diagram suggests a triple point between these phases when the DMI is zero, at the semi-classical phase transition point between the spin-flop phase and antiferromagnetic phase.
In this thesis we consider quantum fluctuations in the ground state energy of antiferromagnetic spin configurations, and phase transitions between these configurations. We use a Holstein-Primakoff transformation up to second order to transform our spin operators, and a Bogoliubov transformation to diagonalize the corresponding Hamiltonian. First we consider a Hamiltonian with Heisenberg-exchange interaction, a magnetic field and anisotropy, for which we investigated both the the antiferromagnetic phase and the spin-flop phase. For the antiferromagnetic phase we find that quantum fluctuations lower the ground state energy. For the spin-flop phase we find that quantum fluctuations lower the ground state energy even more, such that the phase transition point between these phases is shifted to a slightly lower strength of the magnetic field. Furthermore, this shifted phase transition is caused by ground state energies of the spin-flop phase that become complex, instead of an energy crossing. This energy takes on complex values due to imaginary eigenvalues for magnons, which correspond to exponentially increasing semi-classical spin waves.\n\nNext we considered the Dzyaloshinskii-Moriya interaction (DMI) in one dimension, with the spiral phase it induces. We focused on the magnetic frustration of this phase due to anisotropy and the magnetic field. We used a superposition of two plane waves to describe the deviation due to this frustration, and minimized the ground state energy for their amplitudes numerically. By assuming only spirals of integer length and periodic boundary conditions, we constructed a semi-classical ground state phase diagram between the three mentioned phases. This phase diagram suggests a triple point between these phases when the DMI is zero, at the semi-classical phase transition point between the spin-flop phase and antiferromagnetic phase.
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest (J1)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state (T=0) and the finite-T phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.
In this thesis we study spin systems in low dimension at zero temperature, analyzing their quantum phase transitions. In particular, we used a variational method and a linear spin wave theory for obtaining the approximate ground state. Firstly, we studied the order-disorder transitions of the quasi-one-dimensional isotropic antiferromagneticHeisenberg model of spin-1=2, dened on a square lattice with exchange interactions between nearest neighbors, J and J0. The latter interaction is responsible for onedimensional or two-dimensional behavior of the model namely, when it is zero and when it has a nonzero value, respectively. We obtained the staggered magnetizations of the antiferromagnetic and collinear antiferromagnetic phases. In the absence of exchange interactions between second neighbors, the collinear antiferromagnetic phase is obtained for J0 < 0. This model in the one-dimensional limit has a critical value for the ratio J0=J, where we have a phase transition of the order-disorder type. We founda critical value J0=J = 0, which corresponds to the value physically expected, and compare our results with the linear spin wave theory, that predicts a nonzero critical value. The variational method is also used in the study of the Heisenberg anisotropic model with frustration induced by the inclusion of exchange interactions between second neighbors J2. We renamed the exchange interactions of the rst neighbors J1 e J0 1 , so this model is known as J1 J0 1 J2. Furthermore, we include in this modelthe exchange anisotropy which follows the x and y components of the spin operators. Thus, the more general Hamiltonian presents the isotropic Heisenberg and Ising model limits for certain values of the anisotropy. With the goal to analyze the real role played by this anisotropy, we distinguish between rst and seconds neighbors by the labels 1and 2, respectively. We analyzed the staggered magnetization and the energies for the antiferromagnetic, collinear antiferromagnetic and quantum paramagnetic phase. From these analyzes, we observed quantum phase transitions for certain values of the exchange anisotropy and frustration parameter, the latter one given by the ratio J2=J1. The global phase diagram was also obtained for the cases where the rst and second neighbor exchange anisotropies are equal and where they dier from each other. In the rst case, we can observe the ordered phases, Néel and collinear antiferromagnetic, and between them a magnetically disordered phase. We checked the existence of a critical endpoint separating the three phases. In the second case, we note that the exchange anisotropy inuences the region of the disordered phase, aecting even the existence of the critical endpoint. Besides presenting, at certain intervals of the anisotropy, a behavior that was not expected a priori. We also analyze the quasi-one-dimensional behavior J01 ! 0, where the shape of the phase diagram is unchanged. We explore the degree of dimer formation via the dimerization parameter for some values of exchange anisotropy and obtain a higher degree of dimerization for the disordered phase in comparison with the ordered phases, as expected. We end this work with a generalization of the rst study without frustration in function only of the exchange anisotropy. As a challenge to the variational method, we study the more general J1 J0 1 J2 modelfor integer spin-1 without the exchange anisotropy and avoiding the unidimensional case. Dierent from what happens in the semi-integer spins case, we did not nd the disordered phase for integer spin S= 1. We note only a rst-order phase transition between the ordered phases. Finally, we return to the frustrated model with exchange anisotropies 1 and 2 to study the model from a perspective of linear spin wavetheory, in order to compare the results with those obtained via the variational method. A detailed analysis was performed based on the comparison between the two methods, which agreed qualitatively in all explored boundaries and even those behaviors that were not expected from the variational method were conrmed by the spin wave theory.This consistency of the results allowed us to get a dierent view of the role played by the exchange anisotropy when dierentiated for rst and second neighbors. If the anisotropies are considered equal they work only as a parameter that carries the model from the isotopic Heisenberg limit to the Ising model. However, when dierent, they promote disorder in the system.
Magnetic frustrations are the fundamental cause of many interesting phenomena in materials that have been intensely studied since the middle of the 20th century. Magnetic systems with competing interactions, ranging from insulating antiferromagnets to arrays of superconducting qubits, exhibit nontrivial quantum e ects that arise from the need to satisfy a jagged free energy landscape. The search for and characterization of quantum spin liquids, nontrivial topological phases, and quantum phase transitions is conducted through investigations of such frustrated models. Within the large umbrella of frustrated magnetism I have studied some speci c models with the purpose of answering the following questions: (1) What are the interesting phases and less-interesting phases, and how do they fall in the phase diagram relative to one another? (2) What properties do the phases have, and how can we characterize what happens when one phase transitions to another? Armed with the answers to these questions, researchers can further explore these interesting phenomena and apply them towards new materials and devices. In this thesis, I rst develop a computational methodology for studying both classical and quantum spin models using various forms of Monte Carlo simulation. I use classical Monte Carlo to study highly-frustrated models of Ising spins both at nite temperatures and in the T = 0 limit, to map the phase diagram and determine the universality class of phase transitions. Using these methods, I show that in a dissipative chain of qubits, bath-induced qubit-qubit interactions lead to collective decoherence above a critical dissipation strength. This causes the qubits to lose all stored information and become fully correlated with one another, an e ect that one would need to mitigate in order to perform quantum computation. In another study, I show that in antiferromagnetic insulators with anisotropic interactions, frustration can cause an intermediate paramagnetic phase with oblong \droplets" of correlated spins. This phenomenon is quite general and the presence of these droplets can help explain experimental signatures that occur at unexpected temperatures in such materials. In addition to the classical simulations, I develop a variational Monte Carlo (VMC) technique to search for ground states in Heisenberg-type models, speci cally for spins S = 1. The case of S = 1 is more complicated than the case for S = 1=2 due to an additional local degree of freedom, which has interesting implications for the possible states that can be realized. Using semiclassical energy minimization as well as VMC, I show that spin S = 1 diamond-lattice antiferromagnets experience higher-order spinspin interactions that destroy previously theorized liquid-like spiral spin orderings. However, the presence of these higher-order interactions can also potentially allow other phases such as quantum spin liquids and topological paramagnets. Such results can provide an avenue for explaining anomalous phenomena like low-temperature paramagnetism in future experimental materials.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.
In this thesis, we investigate strongly correlated phenomena in quantum spin systems. In the first part of this work, we study geometrically frustrated antiferromagnets (AFMs). Generalizing the SU(2) Heisenberg Hamiltonian to Sp(N) symmetry, we obtain, in the large-N limit, the mean-field phase diagrams for the planar pyrochlore and cubic AFMs. We then use gauge theories to consider fluctuation effects about their respective mean-field configurations. We find, in addition to conventional Neel states, a plethora of novel magnetically disordered phases: two kinds of spin liquids, Z2 in 2+1D and U(1)in 3+1D, and several valence bond solids such as two and three-dimensional plaquette and columnar singlet states. We use the same approach to study the diamond lattice AFM which possesses extended classical ground state degeneracy. We demonstrate that quantum and entropic fluctuations lift this degeneracy in different ways. In the second part of the thesis, we study ultracold spinor atoms confined in optical lattices. We first demonstrate the feasibility of experimental realization of rotor models using ultracold spin-one Bose atoms in a spin-dependent and disordered optical lattice. We show that the ground state of such disordered rotor models with quadrupolar interactions can exhibit biaxial nematic ordering in the disorder-averaged sense, and suggest an imaging experiment to detect the biaxial nematicity in such systems. Finally, using variational wavefunction methods, we study the Mott phases and superfluid-insulator transition of spin-three bosons in an optical lattice with an anisotropic two dimensional optical trap. We chart out the phase diagrams for Mott states with n = 1 and n = 2 atoms per lattice site. We show that the long-range dipolar interaction stabilizes a state characterized by antiferromagnetic chains made of ferromagnetically aligned spins. We also obtain the mean-field phase boundary for the superfluid-insulator transition, and show that inside the superfluid phase and near the superfluid-insulator phase boundary, the system undergoes a first order antiferromagnetic-ferromagnetic spin ordering transition.
We study the ground-state (GS) properties of the frustrated spin-21J1--J2--J3 Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (J2>0) and next-next-nearest-neighbor (J3>0) exchanges, for the case J3=J2. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at J2cl=71 ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region 101<J2<51. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with J1=+1). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter J2. However, there are considerable differences in the behavior of the order parameters for the two models for ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with J1=+1) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. Unlike in the antiferromagnetic model (with J1=+1), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for J2c1<J2<J2c2, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range 101<J2<51.