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Register a new userTensor network investigation of the double layer Kagome compound Ca$_{10}$Cr$_7$O$_{28}$
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Quantum spin liquids are exotic quantum phases of matter that do not order even at zero temperature. While there are several toy models and simple Hamiltonians that could host a quantum spin liquid as their ground state, it is very rare to find actual, realistic materials that exhibits their properties. At the same time, the classical simulation of such instances of strongly correlated systems is intricate and reliable methods are scarce. In this work, we investigate the quantum magnet Ca$_{10}$Cr$_7$O$_{28}$ that has recently been discovered to exhibit properties of a quantum spin liquid in inelastic neutron scattering experiments. This compound has a distorted bilayer Kagome lattice crystal structure consisting of Cr$^{5+}$ ions with spin-$1/2$ moments. Coincidentally, the lattice structure renders a tensor network algorithm in 2D applicable that can be seen as a new variant of a projected entangled simplex state algorithm in the thermodynamic limit. In this first numerical investigation of this material that takes into account genuine quantum correlations, good agreement with the experimental findings is found. We argue that this is one of the very first studies of physical materials in the laboratory with tensor network methods, contributing to uplifting tensor networks from conceptual tools to methods to describe real two-dimensional quantum materials.
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Recently, the bi-layer Kagome lattice material Ca$_{10}$Cr$_7$O$_{28}$ has been shown to be a quasi-two-dimensional quantum spin liquid (QSL) where the frustration arises from a balance between competing ferromagnetic and antiferromagnetic exchange within a bi-layer. In an attempt to understand what happens when this balance is disturbed, we present a magnetic dilution study. Specifically, we have synthesized Ca$_{10}$(Cr$_{1-x}$V$_x$)$_7$O$_{28}$ (0 $leq$ x $leq$ 0.5) where magnetic Cr$^{5+}$ ($S = 1/2$) is partially replaced by non-magnetic V$^{5+}$ ($S = 0$). We also synthesized the fully non-magnetic isostructural material Ca$_{10}$V$_7$O$_{27.5}$. We report a detailed structural, magnetic and heat capacity study on these materials. A monotonic increase in the unit cell parameters is found for the Ca$_{10}$(Cr$_{1-x}$V$_x$)$_7$O$_{28}$ materials with increasing $x$. An order of magnitude decrease in the Curie-Weiss temperature from $4$ to $0.5$~ K is found for the partial V substituted samples, which indicates a relative increase in antiferromagnetic exchange with increase in V content. However, despite this change in the relative balance in the exchange interactions and the large disorder introduced, no magnetic ordering or spin-glass state is observed down to $2$~K in the V substituted samples. The QSL state of the parent compound thus seems surprisingly robust against these large perturbations.
Ca$_{10}$Cr$_7$O$_{28}$ is a novel spin-$1/2$ magnet exhibiting spin liquid behaviour which sets it apart from any previously studied model or material. However, understanding Ca$_{10}$Cr$_7$O$_{28}$ presents a significant challenge, because the low symmetry of the crystal structure leads to very complex interactions, with up to seven inequivalent coupling parameters in the unit cell. Here we explore the origin of the spin-liquid behaviour in Ca$_{10}$Cr$_7$O$_{28}$, starting from the simplest microscopic model consistent with experiment - a Heisenberg model on a single bilayer of the breathing-kagome (BBK) lattice. We use a combination of classical Monte Carlo (MC) simulation and (semi-)classical Molecular Dynamics (MD) simulation to explore the thermodynamic and dynamic properties of this model, and compare these with experimental results for Ca$_{10}$Cr$_7$O$_{28}$. We uncover qualitatively different behaviours on different timescales, and argue that the ground state of Ca$_{10}$Cr$_7$O$_{28}$ is born out of a slowly-fluctuating spiral spin liquid, while faster fluctuations echo the U(1) spin liquid found in the kagome antiferromagnet. We also identify key differences between longitudinal and transverse spin excitations in applied magnetic field, and argue that these are a distinguishing feature of the spin liquid in the BBK model.
Distinctive effects of dopant valency is discussed using an impurity level 1% doping each of Cr$^{3+}$ and Ni$^{2+}$ in CaBaCo$_4$O$_7$. Through a comparative study of the magnetic and dielectric properties of multiferroic CaBaCo$_{3.96}$Cr$_{0.04}$O$_7$ and CaBaCo$_{3.96}$Ni$_{0.04}$O$_7$, we highlight that Cr doping does not significantly alter the properties in spite of differences in magnetic spin and ionic radius compared to Co$^{3+}$ and Co$^{2+}$ while Ni doping induces spectacular changes. Particularly, a manifold increase of electric polarization change up to 650 ${rm mu}$C/m$^2$ at 5.6 kV/cm is observed in CaBaCo$_{3.96}$Ni$_{0.04}$O$_7$ compared to CaBaCo$_{3.96}$Cr$_{0.04}$O$_7$ and a stronger magnetoelectric coupling leading to a polarization change of $sim$ 12% in 15 T magnetic field and below 40 K. Further, magnetodielectric effects hint to competing magnetic phases over the temperature range of magnetic transitions. We discuss the observed disparity in the light of a possible site selective doping of Cr$^{3+}$ and Ni$^{2+}$ in the triangular and kagome layers, respectively, of the parent compound.
Using maximally localized Wannier functions obtained from DFT calculations, we derive an effective Hubbard Hamiltonian for a bilayer of Sr$_3$Cr$_2$O$_7$, the $n=2$ member of the Ruddlesden-Popper Sr$_{n+1}$Cr$_n$O$_{3n+1}$ system. The model consists of effective $t_{2g}$ orbitals of Cr in two square lattices, one above the other. The model is further reduced at low energies and two electrons per site, to an effective Kugel-Khomskii Hamiltonian that describes interacting spins 1 and pseudospins 1/2 at each site describing spin and orbitals degrees of freedom respectively. We solve this Hamiltonian at zero temperature using pseudospin bond operators and spin waves. Our results confirm a previous experimental and theoretical study that proposes spin ordering antiferromagnetic in the planes and ferromagnetic between planes, while pseudospins form vertical singlets, although the interplane separation is larger than the nearest-neighbor distance in the plane. We explain the physics behind this rather unexpected behavior.
We present several results relating to the contraction of generic tensor networks and discuss their application to the simulation of quantum many-body systems using variational approaches based upon tensor network states. Given a closed tensor network $mathcal{T}$, we prove that if the environment of a single tensor from the network can be evaluated with computational cost $kappa$, then the environment of any other tensor from $mathcal{T}$ can be evaluated with identical cost $kappa$. Moreover, we describe how the set of all single tensor environments from $mathcal{T}$ can be simultaneously evaluated with fixed cost $3kappa$. The usefulness of these results, which are applicable to a variety of tensor network methods, is demonstrated for the optimization of a Multi-scale Entanglement Renormalization Ansatz (MERA) for the ground state of a 1D quantum system, where they are shown to substantially reduce the computation time.
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