Milestones of low-D quantum magnetism

There is a long time gap between the formulation of the basic theory of low-dimensional (low-D) magnetism as advanced by Ising, Heisenberg and Bethe and its experimental verification. The latter started not long before the discovery of high-TC superconductivity in cuprates and has been boosted by this discovery result in an impressive succession of newly observed physical phenomena. Milestones on this road were the compounds which reached their quantum ground states upon lowering the temperature either gradually or through different instabilities. The gapless and gapped ground states for spin excitations in these compounds are inherent for isolated half-integer spin and integer spin chains, respectively. The same is true for the compounds hosting odd and even leg spin ladders. Some complex oxides of transition metals reach gapped ground state by means of spin-Peierls transition, charge ordering or orbital ordering mechanisms. However, the overwhelming majority of low-dimensional systems arrive to a long-range ordered magnetic state, albeit quite exotic realizations. Under a magnetic field some frustrated magnets stabilize multipolar order, e.g., showing a spin-nematic state in the simplest quadropolar case. Finally, numerous square, triangular, kagome and honeycomb layered lattices, along with Shastry–Sutherland and Nersesyan–Tsvelik patterns constitute the playground to check the basic concepts of two-dimensional magnetism, including resonating valence bond state, Berezinskii–Kosterlitz–Thouless transition and Kitaev model.


INTRODUCTION
Milestones in the field of low-dimensional magnetism, similar to posts along the road, lead to a very attractive destination, i.e., the formulation of a coherent and unified picture of quantum cooperative phenomena in solids. The story begins in 1925 when Ising following the advice of his tutor Lenz considered infinite chain of magnetic moments with nearest neighbor interaction only. 1 The Hamiltonian considered in this case is valid for the preferred component of the spin S H I sin g ¼ J X n ðS z n S z nþ1 Þ (1) No spontaneous magnetization at any finite temperature was found within frames of this model. Opposite is the isotropic Heisenberg case 2 H Heisenberg ¼ J X n ðS x n S x nþ1 þ S y n S y nþ1 þ S z n S z nþ1 Þ The ground states of uniform S = 1/2 chains are different in these two models. While the chain becomes ordered at zerotemperature in the Ising limit, it remains disordered even at zero-temperature in the Heisenberg limit. In 1931, Bethe introduced his famous "ansatz" method to find the exact quantum ground state of the antiferromagnetic Heisenberg model in one dimension. 3 The extension of classical Ising model to two dimensions was provided by Onsager in 1944. 4 Such a system orders magnetically at finite temperature comparable to the value of exchange interaction parameter J. Twodimensional Heisenberg system remains disordered at finite temperatures, but its ground state is ordered. The basic role in low dimensional magnetism belongs to Mermin-Wagner theorem formulated in 1966. 5 It states that no onedimensional or two-dimensional isotropic Heisenberg spin system can order either ferromagnetically or antiferromagnetically at any non-zero temperature.
In the case when the moments lie perpendicular to the chosen axis the model Hamiltonian is Two dimensional antiferromagnets of XY type form magnetic vortices and antivortices within the plane. The diameters of these objects grow upon cooling. The vortices contact each other at Berezinskii-Kosterlitz-Thouless (BKT) temperature resulting in a unique form of long-range order without spontaneous magnetization. [6][7][8] An important difference between integer and half-integer spin chains was admitted by Haldane in 1983. 9 The uniform spin-1/2 chain is gapless, it has fractionalized excitations-domain walls carrying spin S = 1/2. These excitations are confined when chains are coupled into ladders or when there is an alternation of exchange interaction. The uniform spin-1 chain is gapped and the excitations are triplets.
Overall, the properties of magnetic systems depend on their symmetry and dimensionality D. Discrete symmetry (Ising model) can be broken at T = 0 in D = 1 and cannot be broken at finite T. Continuous Abelian symmetry (XY model) cannot be broken in D = 1, but at T = 0 the correlations decay as power law. In D = 2 correlations decay as power law at finite T and become exponential above BKT transition. Continuous non-Abelian symmetry SU(2) (Heisenberg model) cannot be broken in D = 1 even at T = 0 and can be broken spontaneously only at T = 0 in D = 2.
The exotic phenomena mentioned in this review are realized in quantum magnets being localized low-spin systems, either spin-1/ 2 or spin-1, not large classical moments. An attractive feature of some low-dimensional magnets is a spin-liquid state defying a long-range order. The liquid may be either gapped or gapless, dependent on the type of quantum statistics. Adjacent to lowdimensionality is the field of spin frustration, these two phenomena being frequently coexisting in real magnets. The spin-liquid ground state may survive in the presence of strong interaction between magnetic entities, albeit long-range magnetic order establishes itself most frequently in solids upon lowering the temperature due to residual interactions in three dimensions. Numerous attempts were undertaken to find higher-dimensional analogs of the one-dimensional spin−1/2 Heisenberg antiferromagnet (AFM) in a state that breaks neither translational nor spin rotational symmetry. Apart from the isolated ions, the dimers, i.e., two spins coupled by positive (antiferromagnetic) exchange interaction J, seem to be the simplest objects of lowdimensional magnetism. The singlet rotationally invariant ground state of isolated dimers is separated in energy from the triplet excited state by the excitation gap Δ = J. The interaction between moments belonging to different dimers, may it be of spin or spinlattice type, leads to remarkable cooperative phenomena.
The outline of the review is as follows. The 0D objects of quantum low-dimensional magnetism, i.e., spin-1/2 dimers, exhibit a cooperative behavior when coupled. Multiple plateaus appear in magnetization of Shastry-Sutherland dimers network due to the formation of regular patterns of triplet excitations. The plateaus in magnetization can be considered as intermediate Mott insulator phases separating the domes of Bose-Einstein condensation of magnons. Giamarchi, Ruegg and Tchernyshyov underpin the basic concepts of this phenomenon including a detailed correspondence between a Bose gas and a quantum antiferromagnet. 10 Bose-Einstein condensation of magnons is not restricted to spin-1/2 dimers, the representative examples are spin-1 dimers and spin-1 ions with strong anisotropy also.
The chains, either gapped or gapless, represent an evident move from 0D to 1D. The description of uniform and alternating half-integer spin chains is complimented by short detour to integer spin chains. Various phase transitions may bring uniform spin-1/2 chain into ground state with a gap in magnetic excitation spectrum. Among them, spin-Peierls transition, charge ordering and orbital ordering effects can be distinguished. The routine scenario for quasi-one-dimensional system is an eventual 3D order, albeit quite exotic sometimes. Numerous features of this order-chirality, incommensurability, nematicity-make this field of study attractive.
Spin-ladders, to be considered intermediate between 1D and 2D systems, may show either gapped or gapless behavior dependent on leg number. The ladder system can be extended to Nersesyan-Tsvelik network, whose properties are described in some details. As the dimensionality of the system increases, the richer becomes the spectrum of observed magnetic phenomena. It is out of question to list all of them in a short review, but some issues related to 2D square, triangular, kagome and honeycomb lattices are mentioned. The conclusion is supplemented by list of selected spin gap compounds of authors' choice.

DIMERS, SHASTRY-SUTHERLAND NETWORK
A set of orthogonal dimers coupled by frustrated interdimer interaction constitutes the network described by the Shastry-Sutherland model 11,12 It is assumed that both intradimer J and interdimer J′ exchange interactions are positive. Dependent on α = J′/J ratio, the ground state of this model is either spin singlet (α < α C ) or Neel order (α > α C ), where α C~0 .7.
A good realization of the Shastry-Sutherland model is the SrCu 2 (BO 3 ) 2 . 13 The temperature dependence of the magnetic susceptibility χ in SrCu 2 (BO 3 ) 2 is shown in Fig. 1a. At around T = 20 K, a steep drop evidences the presence of a spin gap and the singlet/non-magnetic ground state. Evidently, neither the isolated spin dimer model (solid line) nor its mean field modification (dashed line) describes the experimental data. The numerical analysis 12 has shown that the peculiar shape of the χ(T) curve is due to the fact that the ratio of J = 100 K and J′ = 68 K in SrCu 2 (BO 3 ) 2 is quite close to the transition point α C . The almost dispersionless spin gap Δ = 34 K was evaluated in inelastic neutron scattering. 14 Another attractive feature of SrCu 2 (BO 3 ) 2 is the sequence of plateaus in magnetization, as shown in the inset to Fig. 1a. Plateaus are due to the strong localization of the triplet excitations within the set of orthogonal dimers. At the fractions of magnetization, where the triplets create a superstructure, the energy is at a local minimum. At present, the magnetization in SrCu 2 (BO 3 ) 2 investigated in static magnetic fields up to 34 T has revealed the plateaus at 1/8, 2/15, 1/6 and 1/4 of the saturation, 15 while the measurements in pulsed magnetic fields identified additionally 1/3 and 1/2 plateaus. 16 The plateau regions correspond to the spin-gapped states with the stripe order of triplets. It has been argued that the sequence of field-induced phases observed in SrCu 2 (BO 3 ) 2 represents the first example of an incomplete devil's staircase concerning magnetization of the quantum AFM 15 where the lower plateaus should be described by a superlattice of triplets of the four-Cu spins, instead of dimer triplets. 17 The plaquette phase intermediate between the dimerized spinsinglet state (α = 0) and Neel order (α → ∞) has been predicted in zero magnetic field for the Shastry-Sutherland network at 0.68 < α < 0.86. 18 This phase has been identified recently in inelastic neutron scattering performed on SrCu 2 (BO 3 ) 2 single crystal under pressure up to 60 kbar. 19 As shown in Fig. 1b (left panel), the dimer phase exists up to P = 16 kbar, where both the gap and the energy of bound triplets decrease. In the range P = 21.5 ÷ 40 kbar, the new plaquette phase with a spin gap Δ~23 K is identified, as shown in Fig. 1b (middle panel). Eventually, the gap closes under pressure and Neel ordering takes place at P = 40 kbar. The transition between dimer and plaquette phases is of the first order in the range P = 16 ÷ 21.5 kbar, while the transition between the plaquette singlet and Neel phase is of the second order. As shown in Fig. 1b (right panel), a further increase in pressure results in a tetragonal monoclinic structural phase transition, where the Cu 2+ spin dimers are no more orthogonal.

DIMERS, BOSE-EINSTEIN CONDENSATION
The ground state energy of a system consisting of integer spin particles, bosons, can be minimized via spontaneous Bose-Einstein condensation (BEC), without any interaction. This phenomenon predicted initially for the photons is considered responsible for superfluidity and superconductivity, refer to condensation of trapped atomic gases and can also be applied to quasiparticles in a solid. 20 Note that the concept of BEC with regard to the spin systems is only an approximation: exchange anisotropy and single-ion anisotropy always break rotational symmetry. Magnons are bosons irrespective of the ions magnetic moment. At low temperatures, the BEC was observed in Heisenberg or axially symmetric low-dimensional magnets with spin-singlet ground state, e.g., spin-1/2 dimers 21-25 and composite integer spin chains 26 or in the systems with strong single-ion anisotropy. 27 The BEC occurs under action of magnetic field which splits the triplet S = 1 and lowers the energy of S z = 1 level. At the first critical field H C1 , the S = 0 and S z = 1 levels cross starting the process of BEC. At low temperatures, the formation of new state is manifested by sharp anomalies in magnetization and specific heat, whose magnitude increases with magnetic field. In the range H C1 < H < H C2 , the canted AFM state exists with net magnetization proportional to the magnetic field. At the second critical field H C2 , the magnetization saturates. In the vicinity of H C1 , the phase boundary for a three-dimensional system should follow the power law H C1 (T) -H C1 (0)~T 3/2 .
At present, BEC phenomena have been reported at finite temperatures for a number of quantum magnets. 28 Three representative cases are the systems of spin-1/2 dimers, spin-1 dimers and isolated S = 1 ions. The ancient Han purple pigment, BaCuSi 2 O 6 , is a spin gap compound with a square lattice of dimers forming the bilayer structure. 23 Along the c axis, the Cu 2+ ions, S = 1/2, are coupled by J = Δ = 52 K. In addition, these ions are coupled in the ab plane by J′ = 7 K. The BEC critical fields in BaCuSi 2 O 6 are μ 0 H C1 = 23.5 T and μ 0 H C2 = 49 T with T max = 3.8 K. The dome shaped phase boundary of the ordered phase in BaCuSi 2 O 6 is shown in Fig. 2a. This boundary is marked by sharp anomalies in specific heat and magnetocaloric effect.
The singlet−triplet and triplet−quintuplet BEC was studied in the spin-1 dimer compound, Ba 3 Mn 2 O 8 . 25 Its magnetic subsystem is comprised of pairs of Mn 5+ ions arranged on triangular lattice and coupled along the c axis by J = 19 K. The multiple interactions between dimers amount to J′~1 K, while the single-ion uniaxial anisotropy is D~0.3 K. The ground state of S = 1 dimers is the singlet; the first excited state is the triplet S = 1 (Δ 1 = J); and the second one is the quintuplet S = 2 (Δ 2 = 3 J). Under magnetic field  Fig. 2b. The regions I and II correspond to different phases of the triplet condensate; the region III corresponds to the quintuplet condensate. 29 A qualitatively different case of BEC is represented by NiCl 2 -4SC (NH 2 ) 2 (DTN), where the gap between singlet S = 0 and doublet S z = ±1 levels is due to a single-ion anisotropy D = Δ = 8 K of isolated S = 1 Ni 2+ ions. 27 In this compound, the nickel ions are surrounded by four polar molecules of thiourea, making the DTN a molecular magnet. Similar to dimers, an external magnetic field shifts down the S z = 1 level. The critical fields in DTN are μ 0 H C1 = 2 T and μ 0 H C2 = 12.5 T with T max = 1.2 K (Fig. 2c). In variance with the dimer systems where the gap is basically isotropic, the properties of DTN strongly depend upon the direction of the magnetic field with respect to the easy magnetization axis.

CHAINS, SPIN LIQUIDS
The uniform half-integer spin chain does not present a gap in the triplet excitation spectrum. The chain is disordered in an isotropic case but the anisotropy of exchange interaction results in a longrange order at T = 0 K. 30 The system with small exchange anisotropy can be described by the pure Heisenberg form JS i S j while the Ising form JS i z S j z should be applied to the highly anisotropic case. 31 The χ(T) curve of Heisenberg AFM spin-1/2 chain demonstrates a broad maximum at T max~0 .64 J. Below this temperature, it is reduced by~15%. At the same time, the Ising chain demonstrates a broad maximum in χ(T) curve at T max = 0.5 J and its reduction to zero at T = 0 K. There are many good examples of uniform Heisenberg spin-1/2 chains, among them AE 2 Cu(PO 4 ) 2 (AE = Sr, Ba). 32 The broad maxima in χ(T) curves are seen at T max = 92 K (J = 143 K) for the Sr-and T max = 82 K (J = 132 K) for the Ba-compounds (Fig. 3a). Although the Sr 2 Cu(PO 4 ) 2 seems to be the best realization of a Heisenberg spin-1/2 chain, 33 the measurements of ac-susceptibility identified the long-range order in the Sr-compound at T N = 0.085 K. 34 An alternation of the exchange interaction along the spin-1/2 chain, J 1 -J 2 , leads to the appearance of a spin gap in the excitation spectrum. The confined excitations carry spin 0 and 1, the gap for triplet excitations is located at q = π. 35 Depending on the alternation parameter 0 ≤ α = J 2 /J 1 ≤ 1 two limiting cases can be considered. For α = 0, the chain transforms into a set of isolated dimers; for α = 1, it is the uniform chain. In the range 0 ≤ α ≤ 0.9, the spin gap is defined as 36 In the case of ferromagnetic (FM) exchange interaction J 2 alternating with AFM J 1 , the energy spectrum is gapped also. The gap is located at q = π/2 similar to the AFM Heisenberg S = 1 chain. 37 The χ(T) curves show the correlation maximum, which shifts from T max~0 .64J 1 at α = |J 2 |/J 1 = 0 to lower temperatures with α increasing. 38 An alternating spin-1/2 chain compound with a large spin gap is BaCu 2 V 2 O 8 . 39 The χ(T) curve demonstrates a broad maximum at T max ≈ 280 K (Fig. 3b) which enables estimation of a leading exchange interaction as J 1 = 460 K. The spin gap found in 51 V NMR measurements amounts to 380 K. 40 While first principles calculations identified BaCu 2 V 2 O 8 as an AFM-AFM chain compound with α equal to either 0.16 (Ref. 41 ) or 0.05 (Ref. 42 ), recent highresolution inelastic neutron scattering data unveiled an AFM-FM alternating chain with J 1 = 475 K and J 2 = −140 K. 43 Integer spin chains with sufficiently weak anisotropy are characterized by a nonmagnetic singlet ground state and a nonzero excitation-energy gap. 44 Hamiltonian of spin-1 chain is where λ is the exchange interaction anisotropy and μ is the crystal field splitting of the single ion levels. The gapped phase exists in an extended range of exchange anisotropy 0 ≤ λ ≤ 1.18 for μ = 0. 45 Moreover, for λ ≈ 1, the gap decreases, goes through a minimum, estimated to be zero, and then increases with positive μ. Monte Carlo calculations performed for spin-1 AFM Heisenberg chain estimate the gap Δ = 0.41 J. 46 PbNi 2 V 2 O 8 is considered to be an example of the Haldane chain conjecture. The χ(T) curve evidences the presence of the gap in the energy spectrum (Fig.  3c). The position of the broad maximum at T max = 120 K allows estimating J = 95 K and Δ = 39 K. 47 However, the determination of the spin gap from magnetization curves gives a lower value Δ = (2Δ ⊥ + Δ || )/3 = 22 K. This is ascribed to the presence of both interchain interactions J ⊥ and negative single ion anisotropy D. In inelastic neutron scattering the main magnetic parameters in PbNi 2 V 2 O 8 were estimated as J = 110 K, J ⊥ = 1 K and D = −2.7 K. The values of transverse and longitudinal gaps constituted Δ ⊥ = 48 K and Δ || = 43 K. 48 The D/J and J ⊥ /J ratios put Pb 2 Ni 2 V 2 O 8 system in the Haldane phase near the border with the ordered Ising-like phase in the D-J ⊥ phase diagram. 46

CHAINS, PHASE TRANSITIONS, SPIN GAP
The uniform spin-1/2 chains are unstable with respect to various effects leading to a spin gap formation. The interactions of spin, charge and orbital degrees of freedom with the lattice lead to the spin-Peierls transition, charge and orbital order driven transitions. All of them include structural distortion and in every case a loss in elastic energy is compensated by a gain in magnetic energy. The spin-Peierls transition, being the most unusual kind of magnetoelastic transition, relates to the particular quantum mechanical nature of quasi-one-dimensional AFM. Similar to the Peierls transition in quasi-one-dimensional conductors, the spin-Peierls transition integrates spin gap formation and dimerization of the underlying crystal lattice. This phenomenon, found initially in tetrathiafulvalene-CuS 4 C 4 (CF 3 ) 4 at T SP = 12 K, 49 was observed later in CuGeO 3 . 50 In contrast to the AFM transition, the reduction of magnetic susceptibility χ at the spin-Peierls transition is isotropic. This is illustrated by χ(T) curves measured along three principal axes in CuGeO 3 (Fig. 4a). The broad correlation maximum is reached at T max = 56 K, which defines the intrachain exchange interaction along the c axis J = 88 K. The spin-Peierls transition is manifested by a sharp drop in the χ(T) curve at T SP = 14 K. Under magnetic field, the transition shifts to lower temperatures~αH 2 with α = 0.46. 51 At T < T SP , two alternating J's form, i.e., J 1,2 (T) = J(1 ±δ(T)). The spin gap Δ = 24 K is proportional to the alternation δ = 0.17. The values of interchain exchange interactions along the b axis J b = 0.1 J and c axis J c = −0.01 J were provided by inelastic neutron scattering. 52 All spin-Peierls compounds obey universal magnetic phase diagram comprised of uniform, dimerized and intermediate phases. 53 The last one is considered to be a commensurate, discommensurate (a magnetic soliton), or incommensurate phase. The critical field of the transition between dimerized and intermediate phases in CuGeO 3 is 12-13 T. 54 Full saturation of magnetization in CuGeO 3 was achieved at μ 0 H = 253 T in pulsed magnetic field measurements. 55 A charge-ordering-driven phase transition into the spin gap state was observed in the NaV 2 O 5 at T C = 35 K (Fig. 4b). 56 At elevated temperatures, the average oxidation state of vanadium ions is V 4.5+ . Below T C , two distinctly different oxidation states were evidenced in V 51 NMR measurements which also identified a spin gap value Δ = 108 K. 57 At low temperatures, the monoclinic A112 structure of NaV 2 O 5 is constituted by enlarged unit cell (ab) × 2b × 4c, where a, b and c are the crystal lattice parameters of the high-temperature orthorhombic phase. 58 At T ≤ T C , the temperature-dependent charge disproportionation V 4.5±δc/2 was observed with continuous variation of δ c . 59 The fully charged zigzag-type pattern differs distinctly from the chain-type considered a prerequisite to the spin-Peierls state. At present, the alternation of exchange interaction within zigzag chain is considered to be responsible for a spin gap. The low temperature crystal structure in NaV 2 O 5 is fixed by both lattice distortion and Coulomb repulsion. These two factors are responsible also for the "devil's staircase" phase transitions between commensurate phases with 2a×2b×zc type superstructures found in NaV 2 O 5 . 60 A spin-Peierls-like phase transition 61 driven by spin-orbital fluctuations 62 was observed in NaTiSi 2 O 6 . The transition takes place at T C = 210 K which is higher than the temperature of correlation maximum in χ(T) curve. 31 In this case, the short-range order within the chains is not fully developed and solely magnetic fluctuations cannot be considered to be the driving force. NaTiSi 2 O 6 hosts the skew-edge-sharing chain of slightly distorted TiO 6 octahedra in monoclinic C2/c structure. 63 At elevated temperatures, the fluctuations of the orbital degrees of freedom allow NaTiSi 2 O 6 to be considered as a dynamic Jahn−Teller phase. 64 At T C = 210 K, NaTiSi 2 O 6 transforms to triclinic P1modification 63 which is accompanied by a gradual decrease in magnetic susceptibility (Fig. 4c). The exchange interaction J along the chain is provided by the overlap of nearly degenerated xy j i and yz j i orbitals ( xz j i orbitals are non-bonding). Taking into consideration the orbital degree of freedom, the Hamiltonian of this system can be written as 64 where the orbital operator T z i = 1/2 corresponds to an occupied xy j i orbital and T z i = −1/2 to an occupied yz j i orbital. The ground state of this Hamitonian is a dimerized orbital-ordered one hosting the spin singlet on each bond. The states with either xy j i or yz j i occupied are degenerated. The condensation of the system in either one of these states explains the appearance of a large singlet-triplet spin gap. The value of gap Δ = 620 K was estimated in time-of-flight neutron spectroscopy 65 in good correspondence with the first principles calculations. 66

CHAINS, PHASE TRANSITIONS, LONG-RANGE ORDER
The long-range order is the final destination for numerous quasione-dimensional magnets not protected by a spin gap. This is because the weak interchain exchange interactions J′ inevitably come into play upon lowering the temperature. The ground state of these systems also depends on both signs and values of nearest neighbor J nn and next nearest neighbor couplings J nnn within the chains. 67 The chain Hamiltonian in a magnetic field H is In the case of J nnn being antiferromagnetic (J nnn > 0), the chain is frustrated independent of the sign of J nn . If both J nn and J nnn are positive, a spin gap opens at J nnn /J nn = α > α C = 0.241. 68 At α = 0.5, the Majumdar-Ghosh ground state is represented by a superposition of spin singlets. Tentatively, copper chromate, CuCrO 4 , is the best realization of this model with J nn = 54 K and J nnn = 27 K. 69 It should be noted, however, that no spin gap was observed experimentally in the compounds which satisfy the criterion α > α C . If J nn is FM (J nn < 0), FM order within the chain is established in the range −0.25 < α ≤ 0. At α = −0.25, the system undergoes quantum phase transition to an incommensurate spin helix state. 70 Among recently found species of this type there are several chain cuprates, e.g., LiCu 2 O 2 , 71 Li 2 CuZrO 4 , 72 LiCuSbO 4 , 73 etc., which adopt non-collinear magnetic structure. Of special interest is LiCuVO 4 which exhibits ferroelectricity at low temperatures and nematicity at high magnetic fields.
In LiCuVO 4 , the Cu 2+ ions form isolated spin-1/2 chains along the orthorhombic b axis. 74 The signs of the exchange interactions within the chains differ, i.e., J nn = −19 K while J nnn = 44 K. The long-range helix order at T N = 2.4 K is triggered by the interchain interaction J′ = −4.6 K. 75 The ordered moments of Cu 2+ ions form a spiral spin ground state in the ab plane with incommensurate propagation vector Q = (0;0.532;0). LiCuVO 4 is an improper ferroelectric with the long-range polar order induced at the onset of a spiral spin order. Measurements of magnetic-field-dependent dielectric constant ε and electrical polarization P allow the construction of a magnetoelectric phase diagram (Fig. 5a). 76 At T < T N and H < H 1~2 .5 T, the normal vector e to (a,b) helix in LiCuVO 4 is parallel to the c axis. At a critical field H 1~2 .5 T, the vector e is turned into the direction of the external field. According to the symmetry rule of spiral magnets, ferroelectric order is established with the polarization P ∝ e × Q along the a axis. In the range H 1 < H < H 2~7 .5 T, the normal vector e reorients along the external magnetic field and, thus, the electrical polarization depends on the direction of the magnetic field. Finally, for external magnetic fields above H 2 , the helical spin structure is destroyed and the system is paraelectric for all field directions.
Above H 2 , an incommensurate, collinear spin density wave of bound magnon pairs is stabilized in medium magnetic fields by a FM J nn . In high fields just below the saturation of magnetization, these pairs experience a Bose-Einstein condensation into quantum multipolar states. One of these states expected just below the saturation H S is a quadrupolar state of magnon pairs called a spin nematic state, analogous to a nematic liquid crystal. In a spin nematic state, an energy gap develops in the transverse spin-excitation spectrum making the energy of the two-magnon bound state lower than the energy of the single-magnon state. 77 The microscopic experimental evidence for the formation of a homogeneous, field-dependent, longitudinal spin state without transverse dipolar order was obtained in nuclear magnetic resonance (NMR) measurements on LiCuVO 4 single crystals up to 56 T for both H∥c and H∥b orientations. 78 Observed was the field-dependent NMR line position without change of its width with respect to the saturated phase, as predicted for a spin nematic phase. Figure 5b shows the field dependencies of the 51 V spectra for H∥b taken at T = 1.3 K. The internal local field H int generated on 51 V by the transferred hyperfine coupling from the neighboring Cu 2+ moments directly measures the local magnetization M, and is thus extracted using H int = ν( 51 V)/ 51 γ − μ 0 H where ν is the frequency and γ is the gyromagnetic ratio.
Three different regions can be identified in these NMR spectra. At H > 50.55 T (43.55 T for H∥c), the spectra are field independent and consist of narrow and symmetric lines which is characteristic for a saturated homogeneous magnetic phase. At H < 48.95 T (42.41 T for H∥c) there appears a strong line broadening; both linewidth and line position are field dependent, which is consistent with the previously identified spin density wave state. This phase is characterized by a modulated spin polarization, where the moments are collinear with the external field. In the field ranges 48.95÷50.55 T for H∥b (42.41÷43.55 T for H∥c), the line positions change with H as in the spin density waves phase, but their widths remain unchanged relative to those of the saturated phase. This behavior corresponds to the formation of a homogeneous magnetic state as expected for a spin-nematic state.
In variance with LiCuVO 4 , its newly synthesized counterpart LiCuSbO 4 does not exhibit long-range order down to 0.1 K, signifying the weakness or frustration of interchain exchange interactions. Indications on the presence of a field-induced spinnematic state were obtained in measurements of temperature dependencies of 7 Li nuclear spin lattice relaxation rate, T 1 −1 , at various fields. 79 Below threshold field μ 0 H c1 = 13 T, T 1 −1 diverges at lowering temperature pointing to approach of magnetically ordered phase. Surprisingly, above this field T 1 −1 shows drastic suppression of relaxation rate at lowering temperature evidencing a gap in magnetic excitation spectrum. Excluding well established mechanisms for the spin gap formation, i.e., Zeeman effect at saturation magnetization and Dzyaloshinskii-Moriya interaction, it was concluded that an external magnetic field induces a multicomponent spin liquid in LiCuSbO 4 . According to phase diagram, shown in Fig. 5c, a collinear incommensurate spin density wave phase precedes spin-nematic phase, both being gapped. The range of possible multipolar nematic phase was narrowed to 12.5-13 T, recently. 80 Thus, the study of LiCuSbO 4 gives further support to the concept that the fragile multipolar phases may survive in low-dimensional magnets due to enhancement of quantum fluctuations in the presence of competing and strongly anisotropic exchange interactions.

LADDERS, NERSESYAN-TSVELIK NETWORK
Isolated magnetic entities consisting of exchange-coupled chains constitute the multitude of spin ladders. 81 The magnetic excitation spectra are gapped for half-integer even leg ladders and gapless for odd leg ladders. In the former case the fractionalized spin -1/2 excitations are confined so that the excitations carry spin 0 and 1.
Depending on the ratio of the rung J r and the leg J l exchange interactions, various ground states could be formed in these objects. In the case of J r ≫J l , the even leg ladder can be considered as the collection of weakly interacting dimers. In the opposite case of J r ≪J l , independent on the number of legs, the pattern is that of weakly interacting gapless chains. In the case of the spin-1 ladder the ground state is gapped for any ratio of J r and J l . Of special interest is the Nersesyan-Tsvelik network, which is an extension of the spin-ladder pattern to the layer where both rung J r and plaquette-diagonal J d exchange interaction are taken into account. 82 In the coupled chains model the spin liquid state of the twodimensional counterpart of one-dimensional spin-1/2 Heisenberg AFM can be realized when the exchange is frustrated in the direction perpendicular to the chains and can be fine-tuned. In the case for which the interchain couplings satisfy the relation J r = 2J d , the interaction between staggered magnetizations is eliminated completely. Both frustration and spatial anisotropy of exchange interactions are essential ingredients of the Nersesyan-Tsvelik model. The Hamiltonian in this case is H ¼ X j;n J l S j;n Á S jþ1;n þ X μ¼ ± 1 J r S j;n þ J d S jþ1;n þ S jÀ1;n À Á S j;nþμ Â Ã ( ) where S j,n are spin-1/2 operators, J l ,J r ,J d > 0 and J l ≫J r ,J d . The spatially anisotropic square lattice quantum AFM was analyzed by Starykh and Balents who showed that to realize the Nersesyan-Tsvelik model just the reduction of the coupling of staggered magnetization of different chains is needed, not full elimination. 83 An attempt to verify this model involves the (NO)Cu (NO 3 ) 3 . The layered crystal structure of this compound is organized by weakly coupled chains running along the b axis, as shown in Fig. 6a. 84 The intrachain exchange interaction, J l , passes through NO 3 − group which bounds neighboring Cu 2+ (S = 1/2) ions. Within the bc plane these ions are coupled by rung exchange interaction, J r , involving two NO + groups and diagonal exchange interaction, J d , which passes through one NO + group. It allows presuming that J r = 2J d . The interplane exchange interaction along the a axis is considered to be small.
The temperature dependencies of both magnetic susceptibility χ and electron spin resonance intensity χ ESR in (NO)Cu(NO 3 ) 3 have been described by the formalism appropriate for isolated halfinteger spin chains with J l = 170 K. However, the value of χ at low temperatures was found to be significantly smaller than expected for an isolated spin-1/2 Heisenberg chain. Another probe of the spin liquid state was Raman spectroscopy which evidenced a gapless continuum of magnetic origin (Fig. 6b). 85 The position of the maximum in this continuum defines the major exchange coupling along the chains as J l = 150 K. That same spinon continuum was observed in inelastic neutron scattering (Fig. 6c). 86 The condition J r = 2J d requires a subtle fine tuning of the couplings. The deviation from this ratio may lead to formation of the Neel state at low temperatures. In the band structure calculations, it was admitted that J r = 2J d ratio may not be  Fig. 6c may contribute to J r but may not contribute to J d . 87 Indeed, the long-range magnetic order occurs at the highly reduced Neel temperature T N = 0.58(5)K. 86 The large ratio J l /T N~2 .5 × 10 2 marks the strong suppression of magnetic order. Furthermore, the specific heat C p and muon spectroscopy (μSR) imply a small ordered moment m while the neutron diffraction gives an upper limit of m~0.01 μ B . Evidence that the interchain interactions are competing comes from μSR, which shows that the magnetic order is an incommensurate spin density wave. Since the inelastic neutron scattering reveals commensurate magnetism along the chains, the order must be incommensurate perpendicular to the chains. Hence, the (NO)Cu (NO 3 ) 3 can be considered as a highly one-dimensional chain compound with frustrated interchain interactions. Tentatively, it corresponds to the Nersesyan−Tsvelik model with finite and competing values of J r and J d , although the ratio of these interactions and the proximity of the system to the special point J r = 2J d is still unknown.

LAYERS, TRIANGULAR, KAGOME AND HONEYCOMB LATTICES
The interest to layered magnets has been triggered by discovery of superconductivity in La 2 -x Ba x CuO 4 which possesses a square layered magnetic lattice. 88 The issue of quantum ground state in such a lattice belongs to the most complicated ones since the competition of intralayer exchange interactions along with interlayer interactions and anisotropy may significantly influence the long-range ordering processes. The introduction of holes into the copper layers leads to frustration of magnetic interactions and formation of resonating dimer singlets, i.e., mobile Cooper pairs. This allowed Anderson advance a concept of high-T C superconductivity in cuprates based on idea of resonating valence bond (RVB) state. 89 Such a quantum spin liquid state was suggested initially to describe valence bond interactions in geometrically frustrated 2D system of Mott insulator. 90 To realize the spin liquid state in two dimensions the most obvious candidates are triangular and kagome lattices.
Among triangular 2D spin-1/2 Heisenberg antiferromagnets, Cs 2 CuCl 4 is considered to be a closest realization of a quantum spin liquid. 91 Within planes of this compound the copper spins form an anisotropic frustrated network with linear chain coupling J along the b axis and zigzag inter-chain coupling J′~J/3 along the c axis, as shown in the inset to Fig. 7b. Present are also order of magnitude smaller inter-plane coupling J′′ and in-plane Dzyaloshinskii-Moriya term D responsible for incommensurate spiral order at T N = 0.62(1)K. 92 A distinctive feature of Cs 2 CuCl 4 revealed by inelastic neutron scattering is the presence of highly dispersive excitation continuum indicative of fractionalization of S = 1 spin waves into pairs of deconfined S = 1/2 spinons, as shown in Fig. 7a. Below T N , the sharp excitations appear at low energies, but the dominant continuum at higher energies remains basically unchanged. It was argued by Kohno, Starykh and Balents, 93 that the sharp excitations represent the spinon bound states, i.e., triplons, rather than magnons which are modes of a long-range ordered magnet. The data obtained suggest that Cs 2 CuCl 4 could be placed into close proximity to quantum critical point separating fractional resonating-valence-bond (RVB) spin liquid and a magnetically ordered state, as shown in Fig. 7b. In a magnetic field, the phase diagram of Cs 2 CuCl 4 has been found to be quite sensitive to smallest interactions. 94 These interactions may induce entirely new phases and are responsible for commensurate-incommensurate transition. A cascade of energy scales pertinent to Cs 2 CuCl 4 in a magnetic field oriented along the b axis is represented by Fig. 7c.  6 Cl 2 , the last one being a subject of strictest scrutiny. The main copper-copper antiferromagnetic exchange parameter within network of cornersharing triangles in ZnCu 3 (OH) 6 Cl 2 was estimated as J~200 K from the slope of χ(T) curve at T > 200 K, but no evidence on long-range order was obtained down to 50 mK. 98 A peculiar feature of herbertsmithite masking its ground properties is an inevitable partial substitution of inter-plane Zn 2+ by Cu 2+ . These defects largely define the low temperature magnetic susceptibility χ of ZnCu 3 (OH) 6 Cl 2 . To reveal the intrinsic properties of a kagome layer much better is the local probe, i.e., 17 O NMR lineshift. 99 Temperature dependence of this property is markedly different from that of a bulk probe. While the χ(T) dependence resembles the Curie law, 17 O NMR lineshift passes through broad maximum at elevated temperatures and becomes temperature-independent at lowest temperatures. The fractional spin excitations in ZnCu 3 (OH) 6 Cl 2 form flat continuum evidenced in neutron scattering measurements. This is a signature of a quantum spin liquid.
The key issue in this respect is the presence (or absence) of a spin gap. While it was not established unambiguously, the neutron scattering data set an upper limit for the spin gap value of about 0.1 J, if any. 100 Evidence for a gapped spin-liquid ground state was obtained from the 17 O NMR lineshift measurements on herbertsmithite single crystal. It was demonstrated that the intrinsic local susceptibility of kagome lattice tends to zero at T < 0.03 J. 101 These experimental data are crucial to distinguish between various theories on quantum ground state of spin-1/2 Heisenberg AFM on a kagome lattice, including valence-bond solid, gapped and gapless spin liquids.
Among quantum theoretical models of two-dimensional magnets an important role belongs to the Kitaev model where an exact solution for a spin-1/2 honeycomb lattice with anisotropic bond-dependent interactions exists. 102 The ground state in the pure Kitaev model is a quantum spin liquid, either gapped or gapless depending on the exchange interaction parameters (Fig. 8a). Beyond the pure Kitaev limit, four other types of the ground state can be realized in honeycomb lattice dependent on anisotropy and frustration triggered by competition of exchange interactions: FM, Neel's AFM, AFM zigzag and AFM stripe order (Fig. 8b).
The Kitaev model has generated a new trend in the study of quantum spin liquids due to the topological nature of its solution: in contrast to conventionally ordered magnets, which possess bosonic elementary excitations (magnons), in such a state spins S = 1/2 are predicted to fractionalize into itinerant Majorana fermions and localized Z 2 fluxes. [103][104][105] The quantum liquid state preserves all the symmetries of the high-temperature paramagnet even at T = 0 K and evade a description by conventional local order parameters, because the fractionalization affects both thermal and dynamic properties of these topological phases. The signatures of the Kitaev spin liquid are (i) two peaks at T l and T h in specific heat curves, C p (T), caused by the fractionalization of spins; (ii) a plateau at 1/2Rln2 in C p (T) curves in between these peaks; (iii) incoherent spectra of dynamical spin structure factors S (q,ω), iv) small ratio T l /T h ≤ 0.03. 106 The most popular candidates for the experimental verification of the Kitaev model have been limited up to now to spin-1/ 2 systems with 5d and 6d elements, and most importantly A 2 IrO 3 (A = Li,Na) and α−RuCl 3 . Strong spin-orbit coupling was found to play a key role in the formation of anisotropic bond-dependent interactions on the honeycomb lattice in this case. Despite expectations, however, all these compounds do not have a true spin-liquid ground state because they demonstrate long-range AFM order at low temperatures, preceded by a wide maximum on the temperature dependence of the magnetic susceptibility. This cannot be related to Kitaev interactions, originating from the direct exchange between the transition metal ions. At the same time, the properties of these compounds at elevated temperatures reflect the proximity to Kitaev model and remain to be of great interest.
For instance, two peaks in the C p (T) caused by the fractionalization of spin to two types of Majorana fermions and plateau/ shoulder pinned at 1/2Rln2 in S mag (T) have been observed recently for Na 2 IrO 3 , as shown in Fig. 8c. 107 Fractionalized elementary excitations, reflecting the peculiarity of quantum spin liquid, have been identified in inelastic neutron scattering where they constitute a continuum, sharply distinct from the magnon modes inherent for ordered magnets. Such incoherent spectra were observed in inelastic neutron scattering 108,109 (Fig. 8d) and Raman experiments (Fig. 8e) in α−RuCl 3 . 110,111 The BKT paradigm formulated initially for the frustrated square lattice can be extended to triangular, kagome and honeycomb systems also. This concept presumes a phase transition from unbound vortex and antivortex state of two-dimensional magnet to the coupled vortex-antivortex phase at low temperatures. Below critical temperature of this transition, the formation of topological defects (vortex-antivortex pairs) leads to the appearance of additional degree of freedom, i.e., chirality.

CONCLUSION
The versatile phenomena seen in low-D quantum magnets are just mentioned here in an introductory manner. Each of these phenomena deserves a separate review papers, interested readers are respectfully referred to them. The choice of milestones in the field of low-dimensional magnetism is highly debatable. There cannot be unambiguous criteria for importance, timeliness or impact on the scientific community. Several advanced models and concepts of low-dimensional magnetism, for example the BKT transition or the Kitaev model, are still waiting for a rigorous experimental verification. Quite recently, a new member of honeycomb iridates family, Cu 2 IrO 3 , becomes available. Its C2/c structure with bond angles close to 120°fits almost perfectly the Kitaev model. Although Cu 2 IrO 3 experiences weak magnetic order at 2.7 K, its high frustration ratio of about 40 and sensitivity of the transition to magnetic field evidences its proximity to quantum spin liquid state. 112 Similarly, a new candidate for the realization of quantum spin liquid state on a kagome lattice has appeared recently. It is kapellasite-type cuprate YCu 3 (OH) 6 Cl 3 where no mixing of Y 3+ and Cu 2+ suggests even better realization of perfect kagome than herbertsmithite. 113 Despite high Curie-Weiss temperature of about 100 K this compound exhibit no longrange order down to 2 K. The list of chosen spin-gap compounds is given in Table 1. There are not many, and the gapless spinliquids are even scarcer. Fortunately, every new compound with an exotic ground state and non-trivial excitations brings new colors to the palette of quantum cooperative phenomena in solids and brings new inspiration to researches concentrated on this fascinating topic. Milestones of low-D quantum magnetism A Vasiliev et al.