For coupled-dimer Heisenberg magnets, a paradigm of magnetic quantum phase transitions, we develop a systematic expansion in
1/d, the inverse number of space dimensions. The expansion employs a formulation of the bond-operator technique and is based on the observation that a suitably chosen product-state wave function yields exact zero-temperature expectation values of local observables in the $d\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, with corrections vanishing as
1/d. We demonstrate the approach for a model of dimers on a hypercubic lattice, which generalizes the square-lattice bilayer Heisenberg model to arbitrary
d. In this paper, we use the
1/d expansion to calculate static and dynamic observables at zero temperature in the paramagnetic singlet phase, up to the quantum phase transition, and compare the results with numerical data available for
d=2. Contact is also made with previously proposed refinements of bond-operator theory as well as with a perturbative expansion in the interdimer coupling. In a companion paper, the present
1/d expansion will be extended to the ordered phase, where it is shown to consistently describe the entire phase diagram including the quantum critical point.
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.
We develop a systematic large-
N expansion based on the Schwinger boson representation of SU(4) coherent states of dimers for the paradigmatic spin-
1/2 bilayer square lattice Heisenberg antiferromagnet. This system exhibits a quantum phase transition between a quantum paramagnetic state and a Néel order state, driven by the coupling constant
g=J′/J, which is defined as the ratio between the inter-dimer
J′ and intra-dimer
J exchange interactions. We demonstrate that this approach accurately describes static and dynamic properties on both sides of the quantum phase transition. The critical coupling constant
gc≈0.42 and the dynamic spin structure factor reproduce quantum Monte Carlo results with high precision. Notably, the
1/N corrections reveal the longitudinal mode of the magnetically ordered phase along with the overdamping caused by its decay into the two-magnon continuum. The present large-
N SU(N) Schwinger boson theory can be extended to more general cases of quantum paramagnets that undergo a quantum phase transition into magnetically ordered states.
Quantum phase transition from the N\'eel to the dimer states in an antiferromagnetic (AF) Heisenberg model on square lattice is studied. We introduce a control parameter $\ensuremath{\alpha}$ for the exchange coupling that connects the N\'eel $(\ensuremath{\alpha}=0)$ and the dimer $(\ensuremath{\alpha}=1)$ states. We employ the
CP1 representation of the
s=21 spin operator and integrate out the half of the
CP1 variables at odd sites to obtain a
CP1 nonlinear $\ensuremath{\sigma}$ model. The effective coupling constant is a function of $\ensuremath{\alpha}$, and at $\ensuremath{\alpha}=0$, the
CP1 model is in the ordered phase that corresponds to the N\'eel state of the AF Heisenberg model. A phase transition to the dimer state occurs at a certain critical value of ${\ensuremath{\alpha}}_{C}$ as $\ensuremath{\alpha}$ increases. In the N\'eel state, the dynamical composite U(1) gauge field in the
CP1 model is in a Higgs phase, and low-energy excitations are gapless spin waves. In the dimer phase, a confinement phase of the gauge theory with
s=1 excitations is realized. For the critical point, we argue that a deconfinement phase, which is similar to the Coulomb phase in three spatial dimensions, is realized and
s=21 spinons appear as low-energy excitations.
We study the spin-1/2 Heisenberg chain with alternating nearest neighbor interactions ${J}_{1}(1+\ensuremath{\delta})$ and ${J}_{1}(1\ensuremath{-}\ensuremath{\delta})$ and a uniform second neighbor interaction ${J}_{2}=y(1\ensuremath{-}\ensuremath{\delta})$ by series expansions around the limit of decoupled dimers $(\ensuremath{\delta}=1).$ By extrapolating to $\ensuremath{\delta}=0$ and tuning y, we study the critical point separating the power-law and spontaneously dimerized phases of the spin-1/2 antiferromagnet. We then focus on the disorder line
y=0.5, $0<~\ensuremath{\delta}<~1,$ where the ground states are known exactly. We calculate the triplet excitation spectrum, their spectral weights, and wave vector dependent static susceptibility along this line. It is well known that as $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\delta}}0,$ the spin gap is still nonzero but the triplets are replaced by spinons as the elementary excitations. We study this dynamical transition by analyzing the series for the spectral weight and the static susceptibility. In particular, we show that the spectral weight for the triplets vanishes and the static spin susceptibility changes from a simple pole at imaginary wave vectors to a branch cut at the transition.