The physics of one-dimensional interacting bosonic systems is reviewed. Beginning with results from exactly solvable models and computational approaches, the concept of bosonic Tomonaga-Luttinger liquids relevant for one-dimensional Bose fluids is introduced, and compared with Bose-Einstein condensates existing in dimensions higher than one. The effects of various perturbations on the Tomonaga-Luttinger liquid state are discussed as well as extensions to multicomponent and out of equilibrium situations. Finally, the experimental systems that can be described in terms of models of interacting bosons in one dimension are discussed.

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One dimensional bosons: From condensed matter systems to ultracold gases

The Bose–Einstein condensate (BEC) is a fascinating state of matter predicted to occur for particles obeying Bose statistics. Although the BEC has been observed with bosonic atoms in liquid helium and cold gases, the concept is much more general. We here review analogous states, where excitations in magnetic insulators form the BEC. In antiferromagnets, elementary excitations are magnons, quasiparticles with integer spin and Bose statistics. In certain experiments their density can be controlled by an applied magnetic field leading to the formation of a BEC. Furthermore, interactions between the excitations and the interplay with the crystalline lattice produce very rich physics compared with the canonical BEC. Studies of magnon condensation in a growing number of magnetic materials thus provide a unique window into an exciting world of quantum phase transitions and exotic quantum states, with striking parallels to phenomena studied in ultracold atomic gases in optical lattices. A collection of bosonic particles, such as liquid helium or ultracold gases, can condense into a ground state in which the atoms flow as a ‘superfluid’ without scattering. Magnetic materials further illustrate the generality of the effect, as described in this review.

Bose–Einstein condensation in magnetic insulators

The current efforts to find the materials hosting Kitaev model physics have been focused on Mott insulators of ${d}^{5}$ pseudospin-1/2 ions ${\mathrm{Ir}}^{4+}$ and ${\mathrm{Ru}}^{3+}$ with ${t}_{2g}^{5}(S=1/2,L=1)$ electronic configuration. Here we propose that the Kitaev model can be realized in materials based on ${d}^{7}$ ions with ${t}_{2g}^{5}{e}_{g}^{2}(S=3/2,L=1)$ configuration such as ${\mathrm{Co}}^{2+}$, which also host the pseudospin-1/2 magnetism. Considering possible exchange processes, we have derived the ${d}^{7}$ pseudospin-1/2 interactions in ${90}^{\ensuremath{\circ}}$ bonding geometry. The obtained Hamiltonian comprises the bond-directional Kitaev $K$ and isotropic Heisenberg $J$ interactions as in the case of ${d}^{5}$ ions. However, we find that the presence of additional spin-active ${e}_{g}$ electrons radically changes the balance between Kitaev and Heisenberg couplings. Most remarkably, we show that the exchange processes involving ${e}_{g}$ spins are highly sensitive to whether the system is in Mott ($Ul\mathrm{\ensuremath{\Delta}}$) or charge-transfer ($Ug\mathrm{\ensuremath{\Delta}}$) insulating regime. In the latter case, to which many cobalt compounds do actually belong, the antiferromagnetic Heisenberg coupling $J$ is strongly suppressed and spin-liquid phase can be stabilized. The results suggest cobalt-based materials as promising candidates for the realization of the Kitaev model.

Pseudospin exchange interactions in d 7 cobalt compounds: Possible realization of the Kitaev model

Thallium copper chloride is a quantum spin liquid of $S=1/2$ ${\mathrm{Cu}}^{2+}$ dimers. Interdimer superexchange interactions give a three-dimensional magnon dispersion and a spin gap significantly smaller than the dimer coupling. This gap is closed by an applied hydrostatic pressure of approximately 2 kbar or by a magnetic field of 5.6 T, offering a unique opportunity to explore both types of quantum phase transition and their associated critical phenomena. We use a bond-operator formulation to obtain a continuous description of all disordered and ordered phases, and thus of the transitions separating these. Both pressure- and field-induced transitions may be considered as the Bose--Einstein condensation of triplet magnon excitations, and the respective phases of staggered magnetic order as linear combinations of dimer-singlet and dimer-triplet modes. We focus on the evolution with applied pressure and field of the magnetic excitations in each phase, and in particular on the gapless (Goldstone) modes in the ordered regimes which correspond to phase fluctuations of the ordered moment. The bond-operator description yields a good account of the magnetization curves and of magnon dispersion relations observed by inelastic neutron scattering under applied fields, and a variety of experimental predictions for pressure-dependent measurements.

https://arxiv.org/pdf/cond-mat/0309440.pdf

Field- and pressure-induced magnetic quantum phase transitions in TlCuCl 3

The temperature and field variations of the magnetization have been measured for which has a singlet ground state with an excitation gap. It is found that undergoes three-dimensional ordering in magnetic fields. The phase boundaries between the paramagnetic and ordered states are determined for and . The phase boundaries for these field directions coincide when normalized by the g-factor. The excitation gap at zero temperature is re-evaluated as .

https://iopscience.iop.org/article/10.1088/0953-8984/11/1/021/pdf

Field-induced three-dimensional magnetic ordering in the spin-gap system

A new type of spin ordering in a spin-pair system is theoretically investigated. In the system, the intra-pair exchange interaction is antiferromagnetic and is much stronger than the inter-pair exchange interaction. In an external magnetic field, the excited triplet of the pair splits, and the lowest component of the triplet crosses the ground singlet at a certain magnitude of the field. In the vicinity of the point of level crossing, even the weak inter-pair exchange interaction causes a considerable amount of mixing between the singlet and the lowest component of the triplet. Due to the mixing, an ordering of the spin component perpendicular to the external field occurs. This ordering explains an anomaly which has been found by Haseda et al. in their experiment of cooling by adiabatic magnetization of Cu(NO 3 ) 2 2.5H 2 O.

Spin Ordering in a Spin-Pair System

It has recently been shown that in spin-pair systems a new type of spin ordering occurs under magnetic fields. In the present paper, the spin ordering in general cases of the inter-pair exchange interaction and the thermodynamical properties of these systems are investigated on the basis of molecular field theory. Various types of spin ordering appear depending on the strength of the inter-pair exchange interaction and the external magnetic field. The specific heat originating from the short range order of spins in a one-dimensional lattice is also calculated for a purpose of comparison with experiments in Cu(N03)2 2. 5 HzO.

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Spin Ordering and Thermodynamical Properties in Spin-Pair Systems under Magnetic Fields

It has recently been shown, by using a molecular field theory, that in these systems a new type of spin ordering occurs under external magnetic fields. In the present paper, the physical properties of these systems at low temperatures are investigated on the basis of spin wave theory. In the ordered state, the spin wave frequency is proportional to the wave number when it is small, and the proportionality coefficient is strongly dependent on the intensity of external field. The field dependence of spin temperature in adiabatic magnetization process is modified from that obtained by the molecular field theory.

Spin Waves in a Spin-Pair System and in a System of Magnetic Ions with Large Anisotropy Energy

In Cu(NO 3 ) 2 2.5H 2 O, the Cu 2+ ions are coupled in almost isolated pairs by antiferromagnetic exchange interaction, and the pairs are considered to be linked in one-dimensional way by weak inter-pair exchange interaction. In an external magnetic field, the ground singlet of the pair crosses the lowest component of the excited triplet at about H =36 kOe. In the vicinity of the point of level crossing, a large amount of the short range order of the spin component perpendicular to the field appears at sufficiently low temperatures. An anomaly found by Haseda et al . in their experiment of cooling by adiabatic magnetization of the crystal can be explained as the occurrence of this short range order.

Takemi Yamada

Papers

Spin Ordering in a Spin-Pair System

Spin Ordering in a Spin-Pair System

Spin Ordering and Thermodynamical Properties in Spin-Pair Systems under Magnetic Fields

Spin Waves in a Spin-Pair System and in a System of Magnetic Ions with Large Anisotropy Energy

Short Range Order of Spins in Cu(NO 3 ) 2 2.5H 2 O under Magnetic Field