Quantum dimer models are believed to capture the essential physics of antiferromagnetic phases dominated by short-ranged valence bond configurations. We show that these models arise as particular limits of Ising
(Z2) gauge theories, but that in these limits the system develops a larger local U(1) invariance that has different consequences on different lattices. Conversely, we note that the standard
Z2 gauge theory is a generalized quantum dimer model, in which the particular relaxation of the hardcore constraint for the dimers breaks the U(1) down to
Z2. These mappings indicate that at least one realization of the Senthil--Fisher proposal for fractionalization is exactly the short ranged resonating valence bond (RVB) scenario of Anderson and of Kivelson, Rokhsar and Sethna. They also suggest that other realizations will require the identification of a local low energy, Ising link variable and a natural constraint. We also discuss the notion of topological order in
Z2 gauge theories and its connection to earlier ideas in RVB theory. We note that this notion is not central to the experiment proposed by Senthil and Fisher to detect vortices in the conjectured
Z2 gauge field.
Recent work [Kundu, Damle, Phys. Rev. X 15, 011018 (2025)] has demonstrated that a class of
S=1 kagome lattice antiferromagnets with competing anisotropies can exhibit unusual low-temperature behavior that is best described in terms of the statistical mechanics of an interesting interpolation between the fully-packed dimer model and the fully-packed loop model on the bipartite honeycomb lattice. On this and other bipartite lattices in two and three dimension, such fully-packed dimer-loop models generically have two distinct power-law correlated Coulomb phases, both of which admit a low-energy description in terms of a fluctuating divergence-free polarization vector field [Phys. Rev. X 15, 011018 (2025); arXiv:2510.11607]. The transition between these phases corresponds to a confinement transition for the
Z2 part of the corresponding
U(1) flux [Phys. Rev. X 15, 011018 (2025); arXiv:2510.11607]. Here we demonstrate that a variant of such a topological confinement transition can also exist in dimer-loop models on non-bipartite lattices such as the kagome and triangular lattices although the low-energy description does not feature any conserved
U(1) flux. In these non-bipartite dimer-loop models, both phases are short-range correlated liquids, and the topological distinction between them is in terms of a
Z2-valued winding number instead of the flux of a divergence-free polarization field. In spite of this key difference, this transition appears to be in the same Ising universality class as the analogous transition of the bipartite dimer-loop models. These kagome and triangular lattice dimer-loop models correspond respectively to the low-energy limit of a class of
S=1 antiferromagnets with competing anisotropies on the ruby lattice comprising a corner-sharing arrangement of fully-frustrated squares and a lattice of corner-sharing fully-frustrated hexagons.
We introduce quantum dimer models on lattices made of corner-sharing triangles. These lattices include the kagome lattice and can be defined in arbitrary geometry. They realize fully disordered and gapped dimer-liquid phase with topological degeneracy and deconfined fractional excitations, as well as solid phases. Using geometrical properties of the lattice, several results are obtained exactly, including the full spectrum of a dimer liquid. These models offer a very natural-and maybe the simplest possible-framework to illustrate general concepts such as fractionalization, topological order, and relation to Z2 gauge theories.
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 investigate the nonequilibrium relaxation dynamics of an interacting monomer-dimer model with nearest neighbor repulsion on a square lattice, which possesses two symmetric absorbing states. The model is known to exhibit two nearby continuous transitions: the Z 2 symmetry-breaking order-disorder transition and the absorbing transition with directed percolation criticality. We performed a more detailed analysis of our extensive simulations on bigger lattice systems which reaffirms that the symmetry-breaking transition exhibits a non-Ising critical behavior with β ≃ 0.149(2) and η ≃ 0.30(1) that are distinct from those values of a pure two dimensional Ising model. Finite size scaling of dimer density near the symmetry breaking transition gives logarithmic scaling ( α = 0.0) which is consistent with the hyperscaling relation but the corresponding exponent of ν B ≃ 1.37(2) exhibits a conspicuous deviation from the pure Ising value of 1. The value of dynamic critical exponent z , however, is found to be close to that of the kinetic Ising model as 1/ z ≃ 0.466(5) from the relaxation of staggered magnetization (and also similar but slightly smaller values from coarsening).
We study a one-dimensional (1D) system that shows many analogies to proposed two-dimensional (2D) deconfined quantum critical points (DQCP). Our system is a translationally invariant spin-
21 chain with onsite ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry and time-reversal symmetry. It undergoes a direct continuous transition from a ferromagnet (FM), where one of the
Z2 symmetries and the time reversal are broken, to a valence bond solid (VBS), where all onsite symmetries are restored while the translation symmetry is broken. The other
Z2 symmetry remains unbroken throughout, but its presence is crucial for both the direct transition (via specific Berry phase effect on topological defects, also related to a Lieb-Schultz-Mattis--type theorem) and the precise characterization of the VBS phase (which has crystalline-symmetry-protected-topological--like property). The transition has a description in terms of either two domain-wall species that ``fractionalize'' the VBS order parameter or in terms of two partons that ``fractionalize'' the FM order parameter, with each picture having its own
Z2 gauge theory structure. The two descriptions are dual to each other and, at long wavelengths, take the form of a self-dual gauged Ashkin-Teller model, reminiscent of the self-dual easy-plane noncompact
CP1 model that arises in the description of the 2D easy-plane DQCP. We also find an exact reformulation of the transition that leads to a simple field-theory description that explicitly unifies the FM and VBS order parameters; this reformulation can be interpreted as a new parton approach that does not attempt to fractionalize either of the FM and VBS order parameters but instead encodes them in instanton operators. Aside from providing explicit realizations of many ideas proposed in the context of the 2D DQCP, here in the simpler and fully tractable 1D setting with continuous transition, our study also suggests a possible line of approach to the 2D DQCP.