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 anisotropy²⁴ 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 $$ \circ -\circ =\downarrow \uparrow -\uparrow \downarrow $$ (normalization factors omitted throughout) (Fig. 1b), known as a valence bond^5,6. Here ↑ and ↓ represent spin-up and spin-down 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 distance²⁰, 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 even-length 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 (in-plane) are described by the Hamiltonian^20,25:

RVB states in an odd number chain

We first built a three-spin chain by alternating Ti_O and Ti_B atoms to obtain the nearest-neighbor coupling of J ≈ 30 GHz (Fig. 2). This coupling strength results in a low-spin 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 three-spin chain has a ground state $$ ∣1⟩=2\left(\downarrow \uparrow \downarrow \right)-\left(\downarrow \downarrow \uparrow +\uparrow \downarrow \downarrow \right)=\circ -\circ \downarrow -\downarrow \circ -\circ $$. This coherent superposition of valence bond states is a resonance between $$ \circ -\circ \downarrow $$ and $$ \downarrow \circ -\circ $$, giving an energy of −J, which is lower than the energy expectation for states having only one valence bond ($$ \circ -\circ \downarrow $$ or $$ \downarrow \circ -\circ $$) or for a Néel state (↓↑↓). The case of a single valence-bond 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 dimer²¹. In contrast, Néel states should appear only for spins having large enough magnetic anisotropy²⁷ or in arrays having sufficient disorder to mix the RVB state with higher-lying states. This 3-spin chain represents the simplest example of an RVB state, illustrating how a coherent superposition of valence bond states gives an energy benefit.

Fig. 2

Magnetic states of a three-spin chain.

a Energy level diagram as a function of B_ext (up to 0.9 T), and B_tip with J ≈ 30 GHz, calculated using Hamiltonian (1). The scale shows the sum of B_ext and B_tip. States are labeled by their wavefunctions at B_tip = 0. Blue and orange arrows represent spin-down and spin-up states, respectively (same below). Dashed arrows indicate detected ESR transitions. b, c ESR spectra measured on the middle (Ti_O) spin at different B_tip as labeled, shown as a graph (b) and lightness scale (c). (V_DC = 60 mV, I_DC = 5–40 pA, V_RF = 7–22 mV, B_ext = 0.9 T). Signals near 20.6 and 25.9 GHz are experimental artifacts due to the frequency-dependent RF transmission. d, Simulated ESR spectra. Weak anisotropy⁴⁸ (−0.02 J) is added to the exchange Hamiltonian to reproduce the observed slightly broken degeneracy, yielding distinct peaks II and III (Supplementary Note 5).

To probe the energy spectrum and eigenstates of this three-spin 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 lowest-frequency ESR peak (I) reveals the transition from the ground state $$ ∣1⟩=\circ -\circ \downarrow -\downarrow \circ -\circ $$ (S_T = 1/2, M = −1/2), to the ferromagnetic state $$ ∣2⟩=\downarrow \downarrow \downarrow $$ (S_T = 3/2, M = −3/2). The average spin polarization $$ ⟨S_2^z⟩$$ (where z is the direction of B_ext) of the middle spin is zero in the state $$ ∣1⟩$$, while in state $$ ∣2⟩$$ 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 $$ ∣2⟩$$. For the higher-energy states $$ ∣4⟩=$$ and $$ ∣6⟩=$$, the two end spins form a spin-singlet. 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 energy²⁸ 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 spins¹⁴ can exhibit a non-magnetic 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 four-spin 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 non-magnetic (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 S₁ and S₄, 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 lengths². 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).

Fig. 3

Spin pairings in an even-length chain.

a Experimental setup showing a four-spin chain of Ti_B atoms (J ≈ 65 GHz). The double arrow indicates the superposition of two spin-pairing configurations of the ground state $$ ∣1⟩$$. b Energy level diagram as a function of B_ext and B_tip, applied to S₁ (solid lines) or S₂ (dashed lines). Dashed arrows indicate detected ESR transitions. c ESR spectra measured on each of the 4 spins as a function of B_tip (V_DC = 60 mV, I_DC = 10–160 pA, V_RF = 24–38 mV, B_ext = 0.9 T). Vertical stripes are due to the frequency-dependent RF transmission. d Energy difference between states $$ ∣1⟩$$ and $$ ∣3⟩$$ at different B_tip, with a linear fit (solid line). Dashed arrow indicates the theoretical value at B_tip = 0. Error bars are from the fitting uncertainties of the ESR frequencies with a 95% confidence. e Ground-state concurrence C_ij between the three different pairs of neighboring spins, as a function of B_tip. Solid (dotted) lines show concurrence when the B_tip is applied to S₁ (S₂) as illustrated.

The ESR transition from the RVB ground state $$ ∣1⟩$$ to the first excited state $$ ∣2⟩$$ (S_T = 1, M = −1) evolves differently depending on where $$ B_{\mathrm{tip}}$$ is applied. Since all spins have zero polarization in state $$ ∣1⟩$$, this evolution gives us information about the spin distributions of the state $$ ∣2⟩$$ with atomic resolution. Examination of the quantitative spin states shows that for state $$ ∣2⟩$$ the z-axis spin polarization of S₁ (~85%) is larger than S₂ (~15%) (Supplementary Fig. 7). Accordingly, the frequency of the ESR transition from the state $$ ∣1⟩$$ to $$ ∣2⟩$$ shifts faster when $$ B_{\mathrm{tip}}$$ is applied to S₁ than the case with B_tip applied on S₂ (Fig. 3c). At higher frequency, the transition from $$ ∣2⟩$$ to $$ ∣3⟩$$ is visible when measuring the spin at either end of the chain.

This atomic-scale 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 ESR²⁶. 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 $$ \left(-0.652\pm 0.010\right)J$$ at zero field, in agreement with the calculated value of −0.659 J.

The atomic-scale 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 S₁ or S₂. This sensitivity directly reflects the spin-pairing 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 C₁₂ is accompanied by the increase of C₂₃, showing the “monogamy” of entanglement^4,29.

RVB states in a spin plaquette

While the four-spin chain shows an unequal superposition of the valence bond basis states, we can engineer an equal superposition by building closed-chain structures, with additional translational symmetry. A closed chain of four spins is obtained by assembling a 2-by-2 plaquette³¹ (Fig. 4a), the smallest configuration for simulating a quantum spin liquid on a two-dimensional square lattice^3–5. In such a geometry, there are two valence-bond basis states (Supplementary Figs. 10, 11): $$ \mid {\mathrm{\Phi }}_{=}⟩=\left({\downarrow }_1{\uparrow }_2-{\uparrow }_1{\downarrow }_2\right)\left({\downarrow }_4{\uparrow }_3-{\uparrow }_4{\downarrow }_3\right)$$ and $$ \mid {\mathrm{\Phi }}_{\parallel }⟩=\left({\downarrow }_1{\uparrow }_4-{\uparrow }_1{\downarrow }_4\right)\left({\downarrow }_2{\uparrow }_3-{\uparrow }_2{\downarrow }_3\right)$$. The RVB state has s-wave symmetry and consists of the coherent superposition $$ \mid {\mathrm{\Phi }}_{=}⟩+\mid {\mathrm{\Phi }}_{\parallel }⟩$$, resulting in the entanglement of each spin with its two neighbors (Fig. 4a, right). The d-wave superposition $$ \mid {\mathrm{\Phi }}_{\times }⟩=\mid {\mathrm{\Phi }}_{=}⟩-\mid {\mathrm{\Phi }}_{\parallel }⟩$$, in contrast, is not an RVB state and lies at higher energy (Supplementary Fig. 10).

Fig. 4

RVB state in a spin plaquette.

a STM image of a spin plaquette of Ti_B atoms (J ≈ 6 GHz). Right: Schematic of the RVB state (red) consisting of the superposition of two spin pairings (green). b Energy level diagram as a function of B_ext and B_tip. Dashed arrows indicate detected ESR transitions. Red line is the RVB state energy. Upper inset: energy per spin of the RVB state relative to the ferromagnetic state $$ {\downarrow }_1{\downarrow }_2{\downarrow }_3{\downarrow }_4$$ at B_ext = 0, determined in two ways, by measuring $$ \left(f_{\mathrm{I}}+f_{\mathrm{III}}-2g{\mu }_{\mathrm{B}}{B}_{\mathrm{ext}}\right)/4J$$ (red circles) or $$ \left(f_{\mathrm{II}}+f_{\mathrm{III}}+f_{\mathrm{IV}}-f_{\mathrm{VI}}-2g{\mu }_{\mathrm{B}}{B}_{\mathrm{ext}}\right)/4J$$ (blue circles) and extrapolating to $$ B_{\mathrm{tip}}=0$$. Solid lines are linear fits. Dashed arrows indicate the theoretical values for N = 4 spins and N = ∞²⁸ at $$ B_{\mathrm{tip}}=0$$. Error bars are from the fitting uncertainties of the ESR frequencies with a 95% confidence. c ESR spectra taken on S₁ (V_DC = 50 mV, I_DC = 200 pA, V_RF = 8 mV, B_ext = 0.9 T). Red curve is a fit to a sum of asymmetric Lorentzian peaks. d ESR spectra as a function of B_tip of the spin plaquette (V_DC = 50 mV, I_DC = 50–350 pA, V_RF = 3–13 mV, B_ext = 0.9 T). Lower panel: Simulated ESR spectra. Blue and orange arrows represent spin-down and spin-up states, respectively.

We constructed a plaquette with an interatomic distance d = 8.7Å (Fig. 4a), yielding a nearest-neighbor coupling $$ J\approx 6\mathrm{GHz}$$, and negligible second-nearest-neighbor 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 lower-lying 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 $$ \left(-0.738\pm 0.030\right)J$$ per spin (Fig. 4b, upper inset). This value agrees with the expected value of −0.75 J for an ideal spin-1/2 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 chain²⁸, which is remarkably similar to the four-spin value. We further brought four Ti atoms closer to form a smaller plaquette $$ \left(d=7.4\mathrm{\AA }\right)$$, 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).

Sample preparation

Measurements were performed in a home-built ultrahigh-vacuum (<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 O₂ environment¹⁷. Ti and Fe atoms were deposited in situ from pure metal rods by e-beam 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 in-plane component aligned along the [100] direction of the MgO lattice. STM images were acquired in constant-current mode and all voltages refer to the sample voltage with respect to the tip.

Spin-polarized 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 spin-polarized 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 polarity²⁰.

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 lock-in technique¹⁷. The RF and DC voltages were combined at room temperature using an RF diplexer, and guided to the STM tip through semi-rigid coaxial cables with a loss of ~30 dB at 20 GHz¹⁷.