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arXiv:1407.7870 · Physical Review B

Nonlinear bond-operator theory and 1/d expansion for coupled-dimer magnets. I. Paramagnetic phase

Darshan G. Joshi, K. Coester, Kai Phillip Schmidt, Matthias Vojta C

Published Mar 4, 2015• 28 citations
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Abstract

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.

DOI
10.1103/physrevb.91.094404

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For coupled-dimer Heisenberg magnets, a paradigm of magnetic quantum phase transitions, we develop a systematic expansion in 1/d1/d1/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→\ensuremath∞d\ensuremath{\rightarrow}\ensuremath{\infty}d\ensuremath→\ensuremath∞ limit, with corrections vanishing as 1/d1/d1/d. We demonstrate the approach for a model of dimers on a hypercubic lattice, which generalizes the square-lattice bilayer Heisenberg model to arbitrary ddd. In this paper, we use the 1/d1/d1/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=2d=2d=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/d1/d1/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 epsilonepsilonepsilon-expansion are some of the few successful cornerstones. In this doctoral thesis work, we have developed a novel large-ddd expansion method, ddd 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-ddd 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/d1/d1/d. It turns out that, due to momentum summation properties of the interaction structure factor, all diagrams are suppressed in powers of 1/d1/d1/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=1/21/21/2, we have calculated the 1/d1/d1/d expansion of static and dynamic observables for coupled dimers on a hypercubic lattice at zero temperature. Analyticity of the 1/d1/d1/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-1/21/21/2 in arbitrary dimensions. We also discuss the Brueckner approach, which relies on small quasiparticle density, and derive the same 1/d1/d1/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=2d=2d=2, the results from our calculations are in qualitative agreement with the quantum Monte Carlo study of the square-lattice bilayer Heisenberg AF spin-1/21/21/2 model. In particular, our results are useful to identify the amplitude (Higgs) mode in the QMC data. The ideas of large-ddd 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 extend to magnetically ordered phases a recently developed expansion in 1/d1/d1/d for coupled-dimer Heisenberg magnets, where ddd is the number of space dimensions. This extension utilizes generalized bond operators describing spin excitations on top of a reference state involving triplet condensates. We explicitly consider a model of dimers on a hypercubic lattice which displays, in addition to the paramagnetic singlet phase, a collinear antiferromagnetic phase for which we calculate static and dynamic observables at zero temperature. In particular, we show that the 1/d1/d1/d expansion smoothly connects the paramagnetic and antiferromagnetic phases and produces sensible results at and near the quantum phase transition point. Among others, we determine the dispersion and spectral-weight distribution of the amplitude (i.e., Higgs) mode of the ordered phase. In the limit of vanishing intradimer coupling, we connect our approach to spin-wave theory.
For arbitrary space dimension ddd, we investigate the quantum phase transitions of two paradigmatic spin models defined on a hypercubic lattice, the coupled-dimer Heisenberg model and the transverse-field Ising model. To this end, high-order linked-cluster expansions for the ground-state energy and the one-particle gap are performed up to order 9 about the decoupled-dimer and high-field limits, respectively. Extrapolations of the high-order series yield the location of the quantum phase transition and the correlation-length exponent $\ensuremath{\nu}$ as a function of space dimension ddd. The results are complemented by 1/d1/d1/d expansions to next-to-leading order of observables across the phase diagrams. Remarkably, our analysis of the extrapolated linked-cluster expansion allows to extract the coefficients of the full 1/d1/d1/d expansion for the phase-boundary location in both models exactly in leading order and quantitatively for subleading corrections.
Large-NNN SU(4) Schwinger boson theory for coupled-dimer antiferromagnets
We develop a systematic large-NNN expansion based on the Schwinger boson representation of SU(4) coherent states of dimers for the paradigmatic spin-1/21/21/2 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′/Jg = J'/Jg=J′/J, which is defined as the ratio between the inter-dimer J′J'J′ and intra-dimer JJJ 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≈0.42g_c \approx 0.42gc​≈0.42 and the dynamic spin structure factor reproduce quantum Monte Carlo results with high precision. Notably, the 1/N1/N1/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-NNN SU(N)SU(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.
We present a bond-operator theory (BOT) suitable for the description of both magnetically ordered phases and paramagnetic phases with singlet ground states in spin-1/2 magnets. This technique allows one to trace the evolution of quasiparticles across the transition between the phases. Some elementary excitations described in the theory by separate bosons appear in conventional approaches as bound states of well-known quasiparticles (magnons or triplons). The proposed BOT provides a regular expansion of physical quantities in powers of 1/n1/n1/n, where nnn is the maximum number of bosons that can occupy a unit cell (physical results correspond to n=1n=1n=1). Two variants of BOT are suggested: for two and for four spins in the unit cell (two-spin and four-spin BOTs, respectively). We consider spin-1/2 Heisenberg antiferromagnets (HAF) on a simple square lattice bilayer by the two-spin BOT. The ground-state energy E\mathcal{E}E, the staggered magnetization MMM, and quasiparticle spectra found within the first order in 1/n1/n1/n are in good quantitative agreement with previous results both in paramagnetic and in ordered phases not very close to the quantum critical point between the phases. By doubling the unit cell in two directions, we discuss spin-1/2 HAF on a square lattice using the suggested four-spin BOT. We identify the magnon and the amplitude (Higgs) modes among fifteen spin-2, spin-1, and spin-0 quasiparticles arising in the theory. The magnon spectrum, E\mathcal{E}E, and MMM found in the first order in 1/n1/n1/n are in good quantitative agreement with previous numerical and experimental results. We observe a special moderately damped spin-0 quasiparticle (``singlon'' for short) whose energy is smaller than the energy of the Higgs mode in the most part of the Brillouin zone. By considering HAF with Ising-type anisotropy, we find that both Higgs and singlon modes stem from two-magnon bound states, which merge with two-magnon continuum not far from the isotropic limit. We demonstrate that singlons appear explicitly in ``scalar'' correlators one of which describes the Raman intensity in B1g{B}_{1g}B1g​ symmetry. The latter is expressed in the leading order in 1/n1/n1/n via the singlon Green's function at zero momentum, which shows an asymmetric peak. The position of this peak given by the singlon energy coincides with the position of the so-called ``two-magnon'' peak observed experimentally in, e.g., layered cuprates.
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.