Abstract
Designing and characterizing the manybody behaviors of quantum materials represents a prominent challenge for understanding strongly correlated physics and quantum information processing. We constructed artificial quantum magnets on a surface by using spin1/2 atoms in a scanning tunneling microscope (STM). These coupled spins feature strong quantum fluctuations due to antiferromagnetic exchange interactions between neighboring atoms. To characterize the resulting collective magnetic states and their energy levels, we performed electron spin resonance on individual atoms within each quantum magnet. This gives atomicscale access to properties of the exotic quantum manybody states, such as a finitesize realization of a resonating valence bond state. The tunable atomicscale magnetic field from the STM tip allows us to further characterize and engineer the quantum states. These results open a new avenue to designing and exploring quantum magnets at the atomic scale for applications in spintronics and quantum simulations.
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Introduction
Antiferromagnetism is classically described by a Néel state with alternating spin orientations on neighboring atoms. However, in lowdimensional and lowspin antiferromagnets, quantum spin fluctuations are enhanced and tend to suppress the classical Néel order. This gives rise to exotic ground states lacking longrange magnetic order, such as quantum spin liquids^{1}. A class of spin liquids is characterized by the resonating valence bond (RVB) state^{2}, in which the quantum spins continuously alter their singlet partners. RVB states have been a central topic in quantum magnetism^{3,4,5} because they are believed to describe spin1/2 Mott insulators such as the undoped copper oxides in highT_{c} superconductors^{6}. The RVBtype spin liquid has been observed by probing the spinon excitations using neutron scattering^{7,8,9} and nuclear magnetic resonance^{10}. The lowenergy excitations from the RVB states are fractional quasiparticles such as spinons and holons that enable spincharge separation^{11,12} and result in lowloss spin transmission through insulators^{11}.
RVB states in finitesized spin arrays exhibit several important spin liquid properties such as a singlet ground state lacking conventional magnetic order, strong entanglement, and a stabilizing energy benefit from coherent RVB fluctuations^{4,5}. Experimentally, these finitesize RVB states have been created in ensembles of ultracold atoms in optical lattices^{5}, and in entangled photons^{4}, providing new insights into the quantum correlations. However, RVB states with singlespin addressability have not yet been realized in a solidstate environment.
Atomicscale magnetic structures such as spin chains and arrays have been constructed on surfaces using largespin atoms in a scanning tunneling microscope (STM)^{13,14,15,16}. Recently, the high energy resolution of magnetic resonance was combined with STM to coherently probe the states of individual spins^{17} and spin pairs on a surface^{18,19,20,21,22}. Here, we used STM to perform electron spin resonance (ESR) of designed quantum magnets, in which we use spin1/2 atoms to obtain strong quantum fluctuations, and we measured their manybody states with atomic resolution (Fig. 1a). This allows for the exploration of magnetic behaviors such as RVB states, which exhibit different degrees of quantum fluctuation and different excitation properties depending on their geometry. These quantum magnets provide a versatile quantum matter toolkit with atomselective spin resonance spectroscopy that allows highly entangled states to be generated and probed in detail.
Results
Spin Hamiltonian of quantum magnets
Each quantum magnet was made by positioning Ti atomic spins on an insulating MgO film. Each Ti atom is hydrogenated, making it part of a TiH molecule^{20,23} and for simplicity, we refer to it simply as a Ti atom. Each Ti atom was adsorbed either on top of an oxygen atom (Ti_{O}) or at a bridge site (Ti_{B}) between two oxygen atoms^{20,21}. At both sites it has spin S = 1/2^{20,21}, giving it negligible magnetocrystalline anisotropy^{24} that could suppress spin fluctuations. Due to the antiferromagnetic exchange interaction at close distance^{20,21}, a pair of Ti spins forms a singlet ground state \(\circ \hskip 4pt\hskip 4pt \circ = {\kern 1pt} \downarrow \uparrow  \uparrow \downarrow\) (normalization factors omitted throughout) (Fig. 1b), known as a valence bond^{5,6}. Here ↑ and ↓ represent spinup and spindown states, respectively. In structures containing more than two spins, quantum fluctuations between many such valence bonds can give rise to the RVB state, in which the spins continuously alter their singlet partners and rearrange the pairings.
To design a quantum magnet exhibiting RVB states, we first characterized the pairwise magnetic interaction J for different Ti pairs by measuring the splitting of ESR peaks as a function of distance^{20}, as shown in Fig. 1c. We used the Ti spins coupled dominantly by antiferromagnetic exchange (J_{ij} > 0) to build quantum magnets, including odd and evenlength spin chains, spin triangles, and spin plaquettes (Fig. 1d and Supplementary Fig. 1). The quantum states of these quantum magnets under external applied magnetic field B_{ext} (inplane) are described by the Hamiltonian^{20,25}:
where S_{i} is the spin operator for site i. The gfactor g ≈ 1.8 is obtained by ESR of isolated Ti atoms^{20,25}, and μ_{B} is the Bohr magneton. The antiferromagnetic interactions compete with the Zeeman term, which tends to align the spins along B_{ext}. Here, we used an isotropic spin1/2 model as an approximation which agrees well with most of the experimental data and thus captures the main physics. Consideration of gfactor anisotropy^{23} could further improve the accuracy of the spin Hamiltonian. Note that B_{ext} is the only source of anisotropy in the Hamiltonian, so changing its direction (here it is applied inplane) should not affect the form of the eigenstates or energies. The atomicscale tip magnetic field B_{tip} (Supplementary Fig. 2) is used both to drive ESR transitions and to tune the spin states of the quantum magnets by exerting an exchange bias only on the spin S_{n} under the tip^{25}. Here, the ESR transitions between two coupledspin states \(\left i \right\rangle\) and \(\left j \right\rangle\) are allowed if there is a nonzero matrix element \(\left\langle i \right{\Delta} {\mathbf{B}}_{{\mathrm{tip}}} \cdot {\mathbf{S}}_n\left j \right\rangle\), where ΔB_{tip} gives the field gradient^{20}. We are thus able to drive ESR transitions between different spin multiplets, which are forbidden in traditional spin resonance^{26}, offering direct access to the energy differences between multiplets.
RVB states in an odd number chain
We first built a threespin chain by alternating Ti_{O} and Ti_{B} atoms to obtain the nearestneighbor coupling of J ≈ 30 GHz (Fig. 2). This coupling strength results in a lowspin ground state, while allowing transitions between different multiplets to be visible in our ESR range of ~10–30 GHz. Each spin multiplet with a total spin S_{T} fans out into its 2S_{T} + 1 components, each having a different quantum number M, when B_{ext} is increased (Fig. 2a and Supplementary Fig. 3). The threespin chain has a ground state \(\left 1 \right\rangle = 2\left( { \downarrow \uparrow \downarrow } \right)  \left( { \downarrow \downarrow \uparrow + \uparrow \downarrow \downarrow } \right) = \circ \hskip 4pt\hskip 4pt \circ \downarrow  \downarrow \circ \hskip 4pt\hskip 4pt \circ\). This coherent superposition of valence bond states is a resonance between \(\circ \hskip 4pt\hskip 4pt \circ \downarrow\) and \(\downarrow \circ \hskip 4pt\hskip 4pt \circ\), giving an energy of −J, which is lower than the energy expectation for states having only one valence bond (\(\circ \hskip 4pt\hskip 4pt \circ \downarrow\) or \(\downarrow \circ \hskip 4pt\hskip 4pt \circ\)) or for a Néel state (↓↑↓). The case of a single valencebond state would arise in the limiting case where \(J_{12} \gg J_{23}\). The extreme case is the third spin being isolated from the other two spins, and evolution of the ESR spectra measured on the middle spin would be the same as those measured on a spin dimer^{21}. In contrast, Néel states should appear only for spins having large enough magnetic anisotropy^{27} or in arrays having sufficient disorder to mix the RVB state with higherlying states. This 3spin chain represents the simplest example of an RVB state, illustrating how a coherent superposition of valence bond states gives an energy benefit.
To probe the energy spectrum and eigenstates of this threespin chain, we performed ESR on the middle spin and studied the evolution of the ESR spectra as a function of B_{tip} (Fig. 2b, c). Since different ESR transitions respond differently with increasing B_{tip}, this allows us to easily identify the corresponding initial and final states, in accordance with the quantitative simulations (Fig. 2d).
For example, the lowestfrequency ESR peak (I) reveals the transition from the ground state \(\left 1 \right\rangle = \circ \hskip 4pt\hskip 4pt \circ \downarrow  \downarrow \circ \hskip 4pt\hskip 4pt \circ\) (S_{T} = 1/2, M = −1/2), to the ferromagnetic state \(\left 2 \right\rangle = \downarrow \downarrow \downarrow\) (S_{T} = 3/2, M = −3/2). The average spin polarization \(\langle S_2^z\rangle\) (where z is the direction of B_{ext}) of the middle spin is zero in the state \(\left 1 \right\rangle\), while in state \(\left 2 \right\rangle\) it is fully polarized. This difference in local spin polarization is directly visualized as the frequency shift of peak I with increasing B_{tip}, which favors the spin polarization of the middle spin in state \(\left 2 \right\rangle\). For the higherenergy states \(\left 4 \right\rangle = {\,}\) and \(\left 6 \right\rangle = {\,}\), the two end spins form a spinsinglet. These two states are also eigenstates of the spin Hamiltonian having zero energies at zero magnetic field since the direct coupling between the two end spins is negligible. The polarization of the center spin is opposite for these two states, so the corresponding ESR transition (IV) shifts to higher frequency rapidly as we increase B_{tip}.
The observation of the ESR transitions I and III allows us to determine the energy of the RVB ground state with respect to the ferromagnetic state, which is a conventional measure of RVB energy^{28} that gives the quantitative energy benefit for forming this coherent superposition of singlets. Here, it is given by \( \left( {f_{\mathrm{I}} + f_{{\mathrm{III}}}} \right)/3 = \left( {  0.497 \pm 0.010} \right){J}\) per spin at zero magnetic field (Supplementary Fig. 5), in agreement with the calculated value of −0.5 J.
RVB states in an even number chain
Quantum magnets having an even number of spins^{14} can exhibit a nonmagnetic RVB ground state because all the spins can simultaneously pair into valence bonds. Resonance among different ways to pair the spins results in intriguing spin entanglement between pairs that are not directly coupled^{4,29,30}. To explore these properties, we built a fourspin chain with a strong pairwise coupling of ~65 GHz (Fig. 3a), which exceeds the Zeeman energy of individual spins at 0.9 T so that the RVB state, which is nonmagnetic (S_{T} = 0), becomes the ground state (Fig. 3b and Supplementary Fig. 8). The RVB state is the superposition of two pairing configurations: \(\left( { \downarrow _1 \uparrow _2  \uparrow _1 \downarrow _2} \right)\left( { \downarrow _3 \uparrow _4  \uparrow _3 \downarrow _4} \right)\) and \(\left( { \downarrow _1 \uparrow _4  \uparrow _1 \downarrow _4} \right)\left( { \downarrow _2 \uparrow _3  \uparrow _2 \downarrow _3} \right)\) (Fig. 3a, bottom). Even though there is negligible direct pairwise interaction between the end spins S_{1} and S_{4}, the singlet bonding configuration for these spins still appears in the ground state. Thus, the RVB state can be envisaged as a fluctuating pattern of valence bonds having different lengths^{2}. We measured the ESR spectra on different sites as a function of B_{tip} (Fig. 3c). This allowed us to extract information about the spin wavefunction, including spin polarization on different sites and spin pairings. Conventional tunneling spectra (dI/dV) are thermally limited so they resolve only a single broadened excitation for this chain (Supplementary Fig. 6).
The ESR transition from the RVB ground state \(\left 1 \right\rangle\) to the first excited state \(\left 2 \right\rangle\) (S_{T} = 1, M = −1) evolves differently depending on where \(B_{{\mathrm{tip}}}\) is applied. Since all spins have zero polarization in state \(\left 1 \right\rangle\), this evolution gives us information about the spin distributions of the state \(\left 2 \right\rangle\) with atomic resolution. Examination of the quantitative spin states shows that for state \(\left 2 \right\rangle\) the zaxis spin polarization of S_{1} (~85%) is larger than S_{2} (~15%) (Supplementary Fig. 7). Accordingly, the frequency of the ESR transition from the state \(\left 1 \right\rangle\) to \(\left 2 \right\rangle\) shifts faster when \(B_{{\mathrm{tip}}}\) is applied to S_{1} than the case with B_{tip} applied on S_{2} (Fig. 3c). At higher frequency, the transition from \(\left 2 \right\rangle\) to \(\left 3 \right\rangle\) is visible when measuring the spin at either end of the chain.
This atomicscale ESR offers a direct measurement of the energy differences between spin multiplets, due to its distinct selection rule based on atomically local excitation, while these relative energies are not accessible in conventional bulk ESR^{26}. For example, the sum of the frequencies of the two detected ESR transitions gives the energy difference between the two lowest spin multiplets (Fig. 3d), which is \((  0.652 \pm 0.010)\,{J}\) at zero field, in agreement with the calculated value of −0.659 J.
The atomicscale tip field also influences the pairwise entanglement in the RVB ground state through its competition with the exchange coupling and the external field. To qualitatively understand this competition, we plot the concurrence C_{ij}, a measure of entanglement^{4,29,30}, between two spin sites i and j computed for the ground state (Fig. 3e and Supplementary Note 4). The concurrences varied differently depending on whether B_{tip} was applied on S_{1} or S_{2}. This sensitivity directly reflects the spinpairing information. The tip field only has to compete with \(J{\mathbf{S}}_1 \cdot {\mathbf{S}}_2\) to polarize spin 1, while it has to compete with both \(J{\mathbf{S}}_1 \cdot {\mathbf{S}}_2\) and \(J{\mathbf{S}}_2 \cdot {\mathbf{S}}_3\) when B_{tip} is applied on spin 2. Note that the decrease of C_{12} is accompanied by the increase of C_{23}, showing the “monogamy” of entanglement^{4,29}.
RVB states in a spin plaquette
While the fourspin chain shows an unequal superposition of the valence bond basis states, we can engineer an equal superposition by building closedchain structures, with additional translational symmetry. A closed chain of four spins is obtained by assembling a 2by2 plaquette^{31} (Fig. 4a), the smallest configuration for simulating a quantum spin liquid on a twodimensional square lattice^{3,4,5}. In such a geometry, there are two valencebond basis states (Supplementary Figs. 10, 11): \( {{\mathrm{{\Phi} }}_ = } \rangle = \left( { \downarrow _1 \uparrow _2  \uparrow _1 \downarrow _2} \right)\left( { \downarrow _4 \uparrow _3  \uparrow _4 \downarrow _3} \right)\) and \( {{\mathrm{{\Phi} }}_\parallel } \rangle = \left( { \downarrow _1 \uparrow _4  \uparrow _1 \downarrow _4} \right)\left( { \downarrow _2 \uparrow _3  \uparrow _2 \downarrow _3} \right)\). The RVB state has swave symmetry and consists of the coherent superposition \( {{\mathrm{{\Phi} }}_ = } \rangle +  {{\mathrm{{\Phi} }}_\parallel } \rangle\), resulting in the entanglement of each spin with its two neighbors (Fig. 4a, right). The dwave superposition \( {{\mathrm{{\Phi} }}_ \times } \rangle =  {{\mathrm{{\Phi} }}_ = } \rangle   {{\mathrm{{\Phi} }}_\parallel } \rangle\), in contrast, is not an RVB state and lies at higher energy (Supplementary Fig. 10).
We constructed a plaquette with an interatomic distance d = 8.7Å (Fig. 4a), yielding a nearestneighbor coupling \(J \approx 6\,{\mathrm{GHz}}\), and negligible secondnearestneighbor coupling (∼0.1 GHz). At B_{ext} = 0.9 T, the Zeeman energy dominates, and therefore the ferromagnetic state \(\downarrow _1 \downarrow _2 \downarrow _3 \downarrow _4\) is the ground state (Fig. 4b). Many transitions are accessible to ESR in such a plaquette because of the thermal occupation of several lowerlying states (Fig. 4c, d). These transitions are in excellent agreement to the energies, amplitudes, and the dependence on B_{tip} given by diagonalization of the model Hamiltonian (quantum states detailed in Supplementary Fig. 11).
We obtain direct access to the RVB state by driving an ESR transition (peak III in Fig. 4) into the RVB state. We can thus measure the relative energy between the RVB state and the ferromagnetic state (\(\downarrow _1 \downarrow _2 \downarrow _3 \downarrow _4\)), giving \((  0.738 \pm 0.030)\,{J}\) per spin (Fig. 4b, upper inset). This value agrees with the expected value of −0.75 J for an ideal spin1/2 plaquette, and is much lower than the −0.5 J of a Néel state (\(\downarrow _1 \uparrow _2 \downarrow _3 \uparrow _4\)), thus confirming the RVB nature of this spin plaquette. Theoretical studies show that the energy per spin decreases in magnitude with an increasing number of spins, and reaches −0.693 J in the thermodynamic limit of an infinite chain^{28}, which is remarkably similar to the fourspin value. We further brought four Ti atoms closer to form a smaller plaquette \((d = 7.4\,{\mathrm{{\AA}}})\), which increased the coupling J to 25 GHz, making the RVB state lowest in energy even in the presence of the magnetic field (Supplementary Note 8).
Discussion
The formation of the RVB state studied here is due to the competition between different configurations of spinsinglet pairings. This competition exists even though the Heisenberg antiferromagnet on a square lattice does not have classical magnetic frustration, which requires oddlength cycles^{2}. We also generated such classical frustration, coexisting with RVB behavior, by assembling Ti atoms into a spin triangle (Supplementary Note 9), and observe that the frustration brings the two lowspin doublets closer in energy than in the 3spin chain, leading towards the highdegeneracy ground state characteristic of frustrated systems. Geometrically frustrated lattices with competing interactions are highly challenging for analytical and numerical studies. Employing the atomicscale ESR technique developed here for probing coupledspin states, and using a different decoupling layer such as hexagonal boron nitride, one could measure the emergent nontrivial phases of the spin1/2 triangular lattice.
Customdesigned quantum magnets assembled on a surface combined with singleatom ESR provide a flexible platform to explore the quantum states of finitesize spin systems. This technique can introduce precisely characterized disorder by placing point defects, vacancies, and adjusted couplings by repositioning the atoms. These artificial nanomagnets could aid in the design and complement the use of chemically synthesized molecular nanomagnets^{32,33,34}, which have emerged as promising vehicles for spintronics^{35}, quantum computing^{34,36,37}, and quantum simulations^{38}. The precisely engineered finitesize quantum manybody systems demonstrated here may serve as versatile analog quantum simulators^{31,34,38,39} because they can be assembled, modified, and probed in situ with singlespin selectivity. The spin plaquette constructed here is the fundamental building block of the square lattice spin liquids^{40}. A unique opportunity provided by the STM is to explore the realspace response of the spin liquid to point defects such as individual pinned magnetic moments, which can better reveal the character of the quantum spin liquids^{41,42}. In addition, studies of larger spin arrays or using different atom species with larger singleion anisotropy should allow exploration of the quantumclassicaltransition, and of competition between quantum fluctuations and Néel order^{8,43}. Another natural extension of the current work is to use atomic spins on MgO to realize simulations of the magnetic phases of the Mott insulating states in copper oxide highT_{c} superconductors, and this could also provide a realization of the deconfined quantum critical point by tuning the quantum magnets^{44}.
In addition to exhibiting stationary magnetic orderings, quantum spin arrays can also carry spinon spin current based on quantum fluctuations^{11}. The combination of pump–probe electronic pulses^{45} with pulsed ESR^{22} could allow further exploration of the quantum dynamics of quasiparticles in artificial spin structures with atomic resolution, including, for example, the dynamical evolution of the spin transmission in spin arrays on surfaces, or the operation of nanoscale devices based on spin currents in insulators^{11,46}. Studying the time evolution and confinement of these elementary excitations in 1D and 2D could help to reveal how information propagates in manybody systems, complementing numerical simulations and analytical studies^{47}.
Methods
Sample preparation
Measurements were performed in a homebuilt ultrahighvacuum (<10^{−9} Torr) STM operating at 1.2 K. MgO is two monolayers (ML) thick (referred to as bilayer MgO) and was grown on an atomically clean Ag(001) single crystal by thermally evaporating Mg in an ~10^{–6} Torr O_{2} environment^{17}. Ti and Fe atoms were deposited in situ from pure metal rods by ebeam evaporation onto the sample held at ~10 K. An external magnetic field (0.44 T, 0.5 T or 0.90 T as indicated in the figure captions) was applied at ~8° off the surface, with the inplane component aligned along the [100] direction of the MgO lattice. STM images were acquired in constantcurrent mode and all voltages refer to the sample voltage with respect to the tip.
Spinpolarized tip
The iridium STM tip was coated with silver by indentations into the Ag sample until the tip gave a good lateral resolution in the STM image. To prepare a spinpolarized tip, ~1–5 Fe atoms were each transferred from the MgO onto the tip by applying a bias voltage (~0.55 V) while withdrawing the tip from near point contact with the Fe atom. The degree of spin polarization was verified by the asymmetry in dI/dV spectra of Ti with respect to voltage polarity^{20}.
RF measurement
The continuous wave electron spin resonance spectra were acquired by sweeping the frequency of an RF voltage V_{RF} generated by the RF generator (Agilent E8257D) across the tunneling junction and monitoring changes in the tunneling current. The current signal was modulated at 95 Hz by chopping V_{RF}, which allowed the readout of the current by a lockin technique^{17}. The RF and DC voltages were combined at room temperature using an RF diplexer, and guided to the STM tip through semirigid coaxial cables with a loss of ~30 dB at 20 GHz^{17}.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
We thank Bruce Melior for expert technical assistance; J.L. Lado and J. FernándezRossier for helpful discussions. We gratefully acknowledge financial support from the Office of Naval Research. S.H.P., Y.B., T.E., P.W. and A.J.H. acknowledge support from Institute for Basic Science (IBSR027D1). A.A. acknowledges support from the Engineering and Physical Sciences Research Council (EP/L011972/1 and EP/P000479/1), the QuantERA European Project SUMO, and the European Union’s Horizon 2020 research and innovation program under grant agreement No 863098 (SPRING).
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K.Y. and C.P.L. designed the experiment. K.Y., S.H.P., Y.B., T.E. and P.W. carried out the STM measurements. K.Y. and C.P.L. performed the analysis and wrote the manuscript with help from all authors. All authors discussed the results and edited the paper.
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Yang, K., Phark, SH., Bae, Y. et al. Probing resonating valence bond states in artificial quantum magnets. Nat Commun 12, 993 (2021). https://doi.org/10.1038/s41467021212745
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DOI: https://doi.org/10.1038/s41467021212745
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