For coupled-dimer magnets with quenched disorder, we introduce a generalization of the bond-operator method, appropriate to describe both singlet and magnetically ordered phases. This allows for a numerical calculation of the magnetic excitations at all energies across the phase diagram, including the strongly inhomogeneous Griffiths regime near quantum criticality. We apply the method to the bilayer Heisenberg model with bond randomness and characterize both the broadening of excitations and the transfer of spectral weight induced by disorder. Inside the antiferromagnetic phase this model features the remarkable combination of sharp magnetic Bragg peaks and broad magnons, the latter arising from the tendency to localization of low-energy excitations.
We examine the effects of disorder on dimerized quantum antiferromagnets in a magnetic field, using the mapping to a lattice gas of hard-core bosons with finite-range interactions. Combining a strong-coupling expansion, the replica method, and a one-loop renormalization-group analysis, we investigate the nature of the glass phases formed. We find that away from the tips of the Mott lobes, the transition is from a Mott insulator to a compressible Bose glass, however the compressibility at the tips is strongly suppressed. We identify this finding with the presence of a rare Mott glass phase and demonstrate that the inclusion of replica symmetry breaking is vital to correctly describe the glassy phases. This result suggests that the formation of Bose and Mott glass phases is not simply a weak localization phenomenon but is indicative of much richer physics. We discuss our results in the context of both ultracold atomic gases and spin-dimer materials.
We employ the stochastic series expansion quantum Monte Carlo method to investigate the
S=1/2 antiferromagnetic Heisenberg model on a bilayer square lattice with diluted interlayer couplings. Both regular and random dilution patterns are considered. In systems with regular dilution, tuning the interlayer interaction drives a quantum phase transition from a N\'eel-ordered phase to a quantum disordered phase, consistent with the
O(3) universality class. In contrast, random dilution gives rise to a two-step transition: from the N\'eel phase to an intermediate Mott glass (MG) phase, followed by a transition to the quantum disordered phase. Within the MG phase, the uniform magnetic susceptibility exhibits a stretched-exponential temperature dependence ${\ensuremath{\chi}}_{u}\ensuremath{\sim}exp(\ensuremath{-}b/{T}^{\ensuremath{\alpha}}), 0<\ensuremath{\alpha}<1$. At the N\'eel-to-glass transition, the quenched disorder modifies the critical exponents in a manner consistent with the Harris criterion. These findings provide new insights into disorder-driven quantum phase transitions and the emergence of glassy phases in diluted bilayer quantum magnets.
We use quantum Monte Carlo simulations to study a disordered
S=1/2 Heisenberg quantum spin model with three different nearest-neighbor interactions, ${J}_{1}\ensuremath{\le}{J}_{2}\ensuremath{\le}{J}_{3}$, on the square lattice. We consider the regime in which
J1 represents weak bonds, and
J2 and
J3 correspond to two kinds of stronger bonds (dimers) which are randomly distributed on columns forming coupled two-leg ladders. When increasing the average intradimer coupling
(J2+J3)/2, the system undergoes a N\'eel to quantum glass transition of the ground state and later a second transition into a quantum paramagnet. The quantum glass phase is of the gapless Mott glass type (i.e., in boson language it is incompressible at temperature
T=0), and we find that the temperature dependence of the uniform magnetic susceptibility follows the stretched exponential form $\ensuremath{\chi}\ensuremath{\sim}exp(\ensuremath{-}b/{T}^{\ensuremath{\alpha}})$, with $0<\ensuremath{\alpha}<1$. At the N\'eel-glass transition, we observe the standard O(3) critical exponents, which implies that the Harris criterion for the relevance of the disorder is violated in this system.
In the study of strongly interacting condensed-matter systems controlled microscopic theories hold a key position. Spin-wave theory, large-N expansion, and
epsilon-expansion are some of the few successful cornerstones. In this doctoral thesis work, we have developed a novel large-
d expansion method,
d being the spatial dimension, to study model Hamiltonians hosting a quantum phase transition between a paramagnet and a magnetically ordered phase. A highlight of this technique is that it can consistently describe the entire phase diagram of the above mentioned models, including the quantum critical point. Note that most analytical techniques either efficiently describe only one of the phases or suffer from divergences near the critical point. The idea of large-
d formalism is that in this limit, non-local fluctuations become unimportant and that a suitable product state delivers exact expectation values for local observables, with corrections being suppressed in powers of
1/d. It turns out that, due to momentum summation properties of the interaction structure factor, all diagrams are suppressed in powers of
1/d leading to an analytic expansion. We have demonstrated this method in two important systems namely, the coupled-dimer magnets and the transverse-field Ising model. Coupled-dimer magnets are Heisenberg spin systems with two spins, coupled by intra-dimer antiferromagnetic interaction, per crystallographic unit cell (dimer). In turn, spins from neighboring dimers interact via some inter-dimer interaction. A quantum paramagnet is realized for a dominant intra-dimer interaction, while a magnetically ordered phase exists for a dominant (or of the same order as intra-dimer interaction) inter-dimer interaction. These two phases are connected by a quantum phase transition, which is in the Heisenberg O(3) universality class. Microscopic analytical theories to study such systems have been restricted to either only one of the phases or involve uncontrolled approximations. Using a non-linear bond-operator theory for spins with S=
1/2, we have calculated the
1/d expansion of static and dynamic observables for coupled dimers on a hypercubic lattice at zero temperature. Analyticity of the
1/d expansion, even at the critical point, is ensured by correctly identifying suitable observables using the mean-field critical exponents. This method yields gapless excitation modes in the continuous symmetry broken phase, as required by Goldstone\'s theorem. In appropriate limits, our results match with perturbation expansion in small ratio of inter-dimer and intra-dimer coupling, performed using continuous unitary transformations, as well as the spin-wave theory for spin-
1/2 in arbitrary dimensions. We also discuss the Brueckner approach, which relies on small quasiparticle density, and derive the same
1/d expansion for the dispersion relation in the disordered phase. Another success of our work is in describing the amplitude (Higgs) mode in coupled-dimer magnets. Our novel method establishes the popular bond-operator theory as a controlled approach. In
d=2, the results from our calculations are in qualitative agreement with the quantum Monte Carlo study of the square-lattice bilayer Heisenberg AF spin-
1/2 model. In particular, our results are useful to identify the amplitude (Higgs) mode in the QMC data. The ideas of large-
d are also successfully applied to the transverse-field Ising model on a hypercubic lattice. Similar to bond operators, we have introduced auxiliary Bosonsic operators to set up our method in this case. We have also discussed briefly the bilayer Kitaev model, constructed by antiferromagnetically coupling two layers of the Kitaev model on a honeycomb lattice. In this case, we investigate the dimer quantum paramagnetic phase, realized in the strong inter-layer coupling limit. Using bond-operator theory, we calculate the mode dispersion in this phase, within the harmonic approximation. We also conjecture a zero-temperature phase diagram for this model.
In this work we study the effects of quantum fluctuations in a disordered magnetic system. We consider a cluster-spin Ising model with antiferromagnetic short range interactions (J0) and disordered long range interactions (J) between clusters, where the system is under the influence of a transverse magnetic field Γ. The analytic treatment of the model is done by the replica method and Trotter formalism, in which a cluster mean field, replica symmetric and static approximations are used. This allows us to obtain an effective model of a single cluster that is exactly diagonalized. Thermodynamic quantities like order parameter and free energy are obtained then, the phase diagrams are analyzed. They are shown in temperature versus J0=J phase diagrams for some intensities of Γ. With Γ = 0, when we decrease the temperature (for lower intensities of J0=J) a continuous phase transition between the paramagnetic and cluster spin glass state (CSG) is found. When we increase the ratio J0=J, the system presents the antiferromagnetic phase (AF). The CSG state and the AF phase are separated by first order transitions. In the intermediate region of disorder, there exist a small interval in the AF-PM transition that is first order. When Γ is increased, the transitions PM-AF and PM-VSC decrease toward critical points in the lower and higher disorder, respectively. It is worth to note, in the intermediate range of disorder, that the transition CSG-AF disappears, letting a PM phase between them, even for zero temperature. Besides that, the first-order region, in the PM-AF transition, turns into a second order phase transition, changing its criticality in the low temperature regime. Our results show that quantum fluctuations affects the phase transitions of the model being capable to suppress and/or change his criticality. The present work suggests a model and a mechanism of the change in the criticality away from the quantum critical point, helping the investigation of this interesting and recent topic.