Quantum phase transition and composite excitations of antiferromagnetic spin trimer chains in a magnetic field

Understanding the profound physical nature of the strongly correlated many-body systems is a challenging and fascinating task in modern condensed-matter physics. Among the various physical properties, magnetic excitation plays a crucial role in understanding the magnetic structures of quantum materials and can be studied both theoretically and experimentally^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22}. In particular, the strong quantum fluctuations in low-dimensional systems give rise to a variety of exotic ground states and excitations, such as the Luttinger liquid and spinon excitation, which have attracted significant interest^{1,2,3,4,5,6,7,8,9,10,11,12}. Quasi one-dimensional (1D) magnetic materials can be effectively described by the Heisenberg spin chain, and its various extensions have been extensively investigated. For example, the gapless two-spinon continuum² has been observed through inelastic neutron scattering in quasi 1D material KCuF₃^3,4 and the frustrated ferromagnetic spin-1/2 chain compound LiCuVO₄⁵. Multi-spin excitations can be detected using the resonant inelastic X-ray scattering (RIXS) technique in the material Sr₂CuO₃^6,7. Furthermore, the high-energy string excitations have been proposed as the dominant excitations in the isotropic Heisenberg antiferromagnet based on the Bethe ansatz⁸, and have recently been observed in an antiferromagnetic Heisenberg-Ising chain compounds SrCo₂V₂O₈ and BaCo₂V₂O₈ under strong longitudinal magnetic fields using the high-resolution terahertz spectroscopy^9,10.

Besides the uniform spin chains, quantum materials often exhibit structures that consist of more than one spin per unit cell, resulting in more rich magnetic properties. Among that, ladder systems are well-studied examples of quasi-1D systems with more spins in a unit cell, where the gapless or gapped excitation spectrum depends on whether the rungs contain an odd or even number of S = 1/2 spins, respectively²³. This behavior is analogous to the Haldane’s conjecture regarding spin chains with half-odd-integer or integer spins²⁴. Among the experimental realizations, the two-leg ladder compound (C₇H₁₀N)₂CuBr₄ is noteworthy due to the excellent agreement between its inelastic neutron scattering spectrum and the dynamic spin structure factor derived from the model calculations²⁵. Besides the ladder spin system, spatial inhomogeneous is another way to enlarge the unit cell with more spins. For instance, the model featuring two-spin unit cells with alternating couplings J₁ − J₂, called alternating or dimerized Heisenberg chain, has been well studied. A gap appears in the spectrum if J₁ ≠ J₂, as this modulation causes the spinons of the uniform Heisenberg antiferromagnetic chain (J₁ = J₂) to be confined into triplons that can be considered as weakly bound of spinons when J₂ ≈ J₁²⁶. Extended to trimerized system with three-spin unit cells and repeated couplings J₁ − J₁ − J₂, the so-called trimerized Heisenberg chain is less studied and would exhibit very different magnetic excitations due to odd number of spins in a unit cell¹¹. Moreover, this trimerized structure has been observed in real materials like A₃Cu₃(PO₄)₄ (A = Ca, Sr, Pb)^{27,28,29,30,31,32}, (C₅H₁₁NO₂)₂ · 3CuCl₂ · 2H₂O³³, and Ba₄Ir₃O₁₀^34,35. In the iridate Ba₄Ir₃O₁₀, where three-spin unit cells form layered trimers, fractional spinon excitation has have been observed in the RIXS experiments^34,35.

In our previous work¹¹, we have investigated the spin dynamics of trimer chain characterized by repeated couplings J₁ − J₁ − J₂, whereas the intratrimer J₁ is larger than intertrimer J₂. Our findings show that the low-energy excitation corresponds to the two-spinon continuum can be well described by the uniform Heisenberg model of effective trimer block spins¹¹. Most interestingly, some new composite excitations of the novel quasi-particles, known as doublons and quartons, have been predicted in the intermediate-energy and high-energy spectra, respectively, and have been subsequently confirmed in the inelastic neutron scattering measurements on Na₂Cu₃Ge₄O₁₂¹². As J₂/J₁ → 1, the doublons and quartons lose their identities and fractionalize into the conventional two-spinon continuum. We have also extended the doublon and quarton excitations to 2D trimer systems²². Even though we have a clear understanding these exotic excitations, how to control these excitations using magnetic field is still an interesting topic.

In this paper, we want to study the effect of magnetic field on the doublons and quartons in the trimer chain illustrated in Fig. 1(a). To achieve this, we employ various techniques, including exact diagonalization (ED), density matrix renormalization group (DMRG)^36,37,38, time-dependent variational principle (TDVP)^39,40, and cluster perturbation theory (CPT)^41,42,43,44 to investigate the excitation spectra of trimer chain under the magnetic field. By mapping the entanglement entropy onto the parameter space, we identify the XY-I, the 1/3 magnetization plateau, the XY-II and the ferromagnetic phases. In the gapless XY-I and XY-II phases, both central charges c ≃ 1 indicate that these two phases are well described by the conformal field theory. More importantly, we investigate the intermediate-energy and high-energy excitations for small g in the XY-I and 1/3 magnetization plateau phases. Our analysis demonstrates that the intermediate-energy and high-energy modes are primarily governed by the internal trimer excitations, referred to as the doublons and quartons, respectively. Furthermore, these features of the excitation spectra can also be observed in the spin chain with a trimer structure that is closely associated with the quantum magnet Na₂Cu₃Ge₄O₁₂¹². The magnetic field drives the lower quarton toward zero energy, suggesting the potential for observing the magnetic-field-induced quarton Bose-Einstein condensation (BEC) in the quantum magnet Na₂Cu₃Ge₄O₁₂. Our results may facilitate further exploration of the high-energy spin excitation mechanisms in other systems containing clusters with odd spins.

a Schematic representation of a trimer spin chain subjected to a longitudinal magnetic field. The analysis focuses on systems characterized by the condition J₁ ≥ J₂ > 0, with the letters a, b, c denoting the three spins within a unit cell. b The phase diagram is obtained by employing the DMRG method to map the entanglement entropy onto the parameter space (g, H_z) for a system with L = 180 spins. c Magnetization curves are illustrated as a function of H_z for different g in a system with L = 180. Inset shows the width of 1/3 magnetization plateau as a function of g. d Entanglement entropy S(L_A) as a function of the subsystem size L_A under open boundary conditions. Solid lines in the inset represent the best fits to the CFT scaling form, with the optimal values for the central charges provided. e Ground state energy of per spin e₀ = E₀/L and its first derivative de₀/dH_z as a function of H_z for the system with L = 210. f The second derivative ({d}^{2}{e}_{0}/d{H}_{z}^{2}) as a function of H_z for the system with L = 210. g Magnetization of each spin obtained by DMRG for four distinct phases where g = 0.6 and H_z = 0.3, 1.0, 1.5, 2.0.

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Model

The Hamiltonian of the spin-1/2 antiferromagnetic trimer chain subjected to a longitudinal magnetic field reads

where S_i,γ is the spin-1/2 operator at the γ-th sublattice site of the i-th trimer, the intratrimer labels γ ∈ {a, b, c} are explained in Fig. 1a. H_z represents the strength of external magnetic field, which breaks the SU(2) symmetry. The system comprises a total of N trimers, resulting in a system length of L = 3N. The tuning parameter g is defined as g = J₂/J₁. For simplicity, we set the intratrimer interaction J₁ = 1 as the energy unit, so that intertrimer interaction J₂ = g. Our interest is in the range of coupling ratios g ∈ [0, 1], where the system evolves between the isolated trimers and the isotropic Heisenberg antiferromagnetic chain.

Quantum phase transition

In the absence of a magnetic field, the antiferromagnetic quantum spin trimer chain exhibits a gapless low-energy excitation known as the two-spinon continuum¹¹. When a magnetic field is applied, the SU(2) symmetry is broken, leading to the emergence of a quantum phase transition driven by the competition between the interaction and magnetic field. In this subsection, we aim to investigate the detailed phase diagram using the DMRG method.

Quantum entanglement provides a distinctive framework for unveiling the ground-state properties of many-body systems and has been extensively used to study the quantum phase transitions^{45,46,47,48,49,50}. Entanglement entropy, a crucial metric for assessing bipartite quantum entanglement, can be easily derived from DMRG calculations. Its definition is given by

where the reduced density matrix ρ_A is the partial trace of the density matrix of the whole system ρ, ({rho }_{{rm{A}}}=T{r}_{{rm{B}}}left[rho right]). If A and B are entangled, the reduced density matrix must be a mixed state, and the entanglement entropy quantifies the degree of this mixing. By effectively analyzing the entanglement entropy, the characteristics of ground states in various quantum phases can be extracted. Therefore, the entanglement entropy serves as a viable and useful tool for investigating the quantum phase transitions. As illustrated in Fig. 1b, the entanglement entropy reveals four distinct phases within the (g, H_z) parameter space. When an external magnetic field is applied, a transition from the Néel phase to an incommensurate phase occurs, propelling the system into the XY-I phase. However, the magnetic field is insufficient to open a gap, resulting in a ground state that remains gapless with a nonzero entanglement entropy. In Fig. 1c, the magnetization increases with the magnetic field in the XY-I phase until a fractional magnetization plateau is reached.

The fractional magnetization plateau observed in the magnetization curves can be understood through the Oshikawa-Yamanaka-Affleck criterion⁵¹:

where n is the number of spins in a unit cell, m is the magnetization per site and s denotes the magnitude of spin. For the trimer chain, it has n = 3, s = 1/2, so when n(s − m) = 0 it results in m = 1/2, which corresponds to the full polarized state. Conversely, when n(s − m) = 1 it yields m = 1/6 corresponding to the 1/3 magnetization plateau. Fig. 1c clearly illustrates the existence of these two plateaus. In the 1/3 magnetization plateau phase, the external magnetic field is insufficient to decouple the singlets. As g increases, the width of magnetization plateau decreases, ultimately vanishing at g = 1 where the trimer chain transitions into the uniform Heisenberg chain. From the magnetization of each spin, as shown in Fig. 1g, we can observe that the magnetization exhibits a periodic pattern corresponding to the trimerized structure. Specifically, the magnetization of the central spin in each trimer increases as H_z rises. Importantly, the magnetization of each spin remains fixed even as the magnetic field increases in the 1/3 magnetization plateau phase. The ground state of an isolated trimer in the presence of a magnetic field is described by,

which is also the antiferromagnetic trimer state of the Haldane plateau observed in one-dimensional (S, s) = (1, 1/2) mixed spin chain⁵². In the 1/3 magnetization plateau phase, as shown in Fig. 1g, the expectation values of the z components of three spins, labeled a, b, c, are 0.322, −0.144, and 0.322, respectively. These values are approximately coincide with the ideal state, which has expectation values of 1/3, −1/6, and 1/3. This indicates that the 1/3 magnetization plateau state exhibits Néel order along the direction of magnetic field, with each trimer effectively possessing a spin of 1/2, thereby creating the appearance of polarization for each trimer.

As the magnetic field increases, the singlets are disrupted, resulting in to the emergence of the XY-II phase. The system remains gapless, exhibiting a nonzero entanglement entropy in the ground state. Additionally, the average magnetization is greater than that in the XY-I phase and increases with the magnetic field. As long as the magnetic field is sufficiently strong, all spins become polarized, leading to the formation of an additional magnetization plateau and the emergence of a ferromagnetic phase. We also utilize the properties of entanglement to suggest a potential conformal field theory description of the gapless phases. The Rényi entanglement entropy of subsystem A is defined as follows:

In the limit ν → 1, the above expression simplifies to the von Neumann entanglement entropy,

The Rényi entanglement entropy of subsystem A follows the scaling form^53,54,55:

where

and

Here, η = 1, 2 corresponds to periodic and open boundary conditions, respectively. The central charge c, the Fermi momentum k_F, and the scaling dimension Δ₁ are universal parameters. F_ν(L_A/L) is a universal scaling function and ({tilde{c}}_{nu }) is a nonuniversal constant. By fitting the DMRG data with these functions for ν = 1, we extract the central charges of the two XY phases, which serve as indicators of their universality classes. As shown in Fig. 1d, the both XY phases are described by the conformal field theory with central charges c ≃ 1.

Furthermore, to identify the types of quantum phase transitions present, we have conducted an analysis involving the computation of the first and second derivatives of the ground state energy with respect to the magnetic field H_z, see Fig. 1e, f. According to the Hellmann-Feynman theorem, the magnetization curves (see Fig. 1c) and the first derivative of the ground state energy de₀/dH_z exhibit the similar behaviors, characterized by continuity but a lack of differentiability near the critical points. The second derivative ({d}^{2}{e}_{0}/d{H}_{z}^{2}) displays nonanalytic behavior in the vicinity of these critical points. Collectively, these results suggest that the quantum phase transitions between the XY-I, 1/3 magnetization plateau, XY-II, and ferromagnetic phases are second-order quantum phase transitions. In Supplementary note 1, we also provide the real-space spin-spin correlation function for the four phases. The correlation functions in the XY-I and XY-II phases decay according to power laws, with the critical exponents converging towards 1, which is analogous to the S = 1/2 isotropic Heisenberg chain⁵⁶.

Excitation spectra

In this section, we present the spin excitation spectra of the trimer chain in an external magnetic field, utilizing the dynamical structure factor (DSF):

where α, β refer to the spin components x, y, and z. We calculated the DSF using the CPT and DMRG-TDVP methods to study the spin dynamics under the modification of control parameters, g and H_z. Detailed calculations can be found in “Methods”. In our previous study, we have utilized the quantum Monte Carlo methods with subsequent numerical analytic continuation to investigate the spin dynamics of trimer chain in the absence of magnetic field, revealing the doublons and quartons in the intermediate-energy and high-energy regimes, respectively¹¹. For comparison, the spectral characteristics are also assessed using the DMRG-TDVP and CPT calculations, with results available in Supplementary note 2. Furthermore, we provide the spin excitation spectra obtained from ED calculation in Supplementary note 5.

In Fig. 2, the transverse excitation spectra ({{mathcal{S}}}^{xx}(q,omega )) for four distinct phases are presented. The excitations are gapless in two XY phases and gapped in the 1/3 magnetic plateau and ferromagnetic phases. To understand the spin dynamics of the XY-I and XY-II phases, we will examine the zero-energy excitations. The incommensurability observed in the spin dynamics of an AF spin-1/2 chain subjected to a longitudinal magnetic field can be interpreted using the language of spinless fermions⁵⁷. The longitudinal magnetic field acts as a chemical potential, which alters the band filling and breaks the degeneracy of the electron-hole bands. The intraband and interband zero-energy excitations correspond to the longitudinal and transverse fluctuations, respectively. For the trimer chain subjected to a longitudinal magnetic field, incommensurability arises from the splitting of the bands. Fig. 2a, e display the transverse excitations S^xx(q, ω) as the number of particles is varied, leading to fluctuations that reach zero energy at incommensurate wave numbers q = m_zπ and q = (2 − m_z)π in addition to q = π. The spectral weight is concentrated at the commensurate positions corresponding to each reciprocal lattice points at q = π in the XY-I and XY-II phases. In the high-energy regime, a continuum is observed in both the 1/3 magnetization plateau and the XY-II phases. At the ferromagnetic phase, see Fig. 2d, h, all spins are polarized, and the spin excitation continues to propagate as magnons. Energy gaps are observed at the edges of the Brillouin zone, specifically at q = π/3, 2π/3, 4π/3, 5π/3 where spin waves are diffracted due to the periodic potential of the trimerized interaction. Consequently, the magnons at the edges of the Brillouin zone exhibit two distinct energy levels for the same wave vector. In Supplementary note 3, we also present the longitudinal excitation spectrum and discuss the zero-energy excitations that correspond to the longitudinal fluctuations.

({{mathcal{S}}}^{xx}(q,omega )) in (a, e) XY-I phase, (b, f) 1/3 magnetization plateau phase, (c, g) XY-II phase, and (d, h) Ferromagnetic phase. All results are derived from the case where g = 0.8, and the DMRG-TDVP calculations are conducted for a system with length L = 120. The color coding of ({{mathcal{S}}}^{xx}(q,omega )) uses a piecewise function with the boundary value U₀ = 2. Below this boundary, the low-intensity portion is characterized by a linear mapping of the spectral function to the color bar, while above the boundary a logarithmic scale is used, (U={U}_{0}+{log }_{10}[{{mathcal{S}}}^{zz}(q,omega )]-{log }_{10}({U}_{0})).

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In our previous study¹¹, we found that a smaller g induces rich intermediate-energy and high-energy excitations beyond the spin wave. Therefore, it is of great interest to investigate the evolution of intermediate-energy and high-energy quasiparticles, referred to as the doublons and quartons, under the influence of a magnetic field. In this study, we focus on the weak intertrimer coupling g = 0.3 to examine their dynamical evolutions. In this case, the system behaves as isolated trimers, allowing for straightforward analysis. From Fig. 3a, e, we observe that the low-energy excitation resembles the excitation spectrum of conventional spinons in a magnetic field. A splitting of the dispersion relation occurs, characterized by the emergent fermions of the Heisenberg chain in the presence of magnetic field⁵⁸. The intermediate-energy spectrum shows minimal separation near q = π/3 and q = 5π/3, with a continuum emerges, possibly due to the propagation of doublons dressed by spinons¹¹. The high-energy spectrum is distinctly split into two branches by the magnetic field.

({{mathcal{S}}}^{xx}(q,omega )) in (a, e) XY-I phase, (b, f) 1/3 magnetization plateau phase, (c, g) XY-II phase, and (d, h) Ferromagnetic phase. All results are derived from the case where g = 0.3, and the DMRG-TDVP calculations are conducted for a system with length L = 120. The color coding of ({{mathcal{S}}}^{xx}(q,omega )) uses a piecewise function with the boundary value U₀ = 2.

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As H_z increases, see Fig. 3b, f, the excitation gap opens, leading the system into a 1/3 magnetization plateau phase. Consequently, the lower spectrum of high-energy excitations with ΔM = 1 transitions to the low-energy regime. When H_z = 1.5, the system evolves into the gapless XY-II phase (see Fig. 3c, g), where the low-energy, intermediate-energy and high-energy spectra exhibit clear differentiation. In the ferromagnetic phase H_z = 2.0, shown in Fig. 3d, h, the spin excitation are predominantly characterized by spin waves due to complete spin polarization. Additionally, some energy gaps are observed at the Brillouin zone edges q = π/3, 2π/3, 4π/3, 5π/3, which is consistent with the results of Fig. 2d, h.

The preceding discourse primarily focuses on the impact of a magnetic field on the spin excitation spectra of a trimer chain with a constant intertrimer interaction strength. To enhance our understanding of the spin dynamics within such a trimer chain under the influence of a magnetic field, we present additional results obtained through DMRG-TDVP calculations for varying values of the g parameter. Further details are provided in Supplementary Note 4 for clarification.

Excitations mechanisms

To gain a deeper understanding of the intermediate-energy and high-energy spin dynamics, it is instructive to analyze the complete level spectrum and the corresponding eigenvectors of a single trimer. As depicted in Fig. 4, the application of a magnetic field causes the splitting of three energy levels into eight distinct levels. Notably, the eigenvectors, spin quantum numbers, and magnetic quantum numbers remain invariant. When H_z ≤ 1.5, the ground state of the trimer is denoted as (leftvert 0rightrangle) with an energy E₀ = − J₁ − H_z/2. Considering the excitations with (leftvert Delta Mrightvert =1) from (leftvert 0rightrangle), only four cases satisfy this condition, as indicated in the last column of Fig. 4. For small g, the coupling between trimers can be treated as a perturbation of the product state of isolated trimers. This approach has been validated in our previous study of a trimer chain without an external magnetic field¹¹. Here, the perturbative analysis remains an effective tool to handle with the spin excitations of trimer chain under a magnetic field, particularly in the XY-I and 1/3 magnetization plateau phases. In the XY-I phase with a small g, a weak magnetic field induces an incommensurate ground state with slight magnetization. By utilizing the ground state (leftvert 0rightrangle) and the first excited state (leftvert 1rightrangle) of a single trimer, we can construct an approximate ground state with antiferromagnetic order, such as ({leftvert psi rightrangle }_{g}=leftvert 0101ldots 01rightrangle). Consequently, we are able to calculate the dispersion relations corresponding to the intermediate-energy and high-energy excitations with (leftvert Delta Mrightvert =1) by employing only N = 4 trimers, details can be found in Supplementary note 6. Regarding the intermediate-energy excitations, four dispersion relations emerge at ω ∝ J₁, as shown in Fig. 3e, which describe the localized excitations from (leftvert 0rightrangle) to (leftvert 3rightrangle),

It can be observ ed that these dispersion relations depend on the energy gap between the ground state and the first excited state of one single trimer in the absence of magnetic field, and they increase with the application of magnetic field H_z. In comparison to the case H_z = 0¹¹, only one branch of doublons remains under the influence of magnetic field; thus, the intermediate-energy excitation corresponds to the generation of doublons. These dispersion lines do not align well with the spectrum due to the approximation of the ground state. Additionally, we observe a continuum that may originate from bound spinons. The central doublet, which is dressed by these spinons, propagates through the system, resulting in various internal modes of these composite excitations. This, in turn, leads to a band of finite width in the energy of these excitations. For further details, see the propagation of doublons in Supplementary note 2.

The second column lists the wave functions in the spin-z basis, while the third column presents the spin structures using a basis of singlets (gray ovals and rounded shapes), zero-magnetization triplets (gray square shapes), and unpaired spins (arrows). The last column lists the total spin quantum number S, magnetic quantum number M, and the internal trimer excitations with ΔM = ±1.

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For the high-energy excitations, the application of a magnetic field results in the division of the spectrum into two distinct branches, as shown in Fig. 3a, e. Both branches originate from the high-energy internal trimer excitations. We designate the upper branch (excitation from (leftvert 0rightrangle) to (leftvert 6rightrangle)) as the upper quarton, and the lower one (excitation from (leftvert 0rightrangle) to (leftvert 4rightrangle)) as the lower quarton. The dispersion relations for the upper quarton are given by,

and the ones for lower quarton are given by,

These dispersion relations depend on the energy gap between the ground state and the second excited state of one single trimer in the absence of magnetic field, and they exhibit a significant concordance with the DMRG-TDVP results concerning the positioning of these excitations and their bandwidths. This alignment indicates the conceptualization of localized excitations is valid, despite the fact that the calculation relies on a rather coarse approximation of the ground state. Consequently, the high-energy quartons remain persist in the XY-I phase at low values of g.

In the 1/3 magnetization plateau phase, each trimer exhibits an effective magnetic quantum number 1/2, resembling a polarized spin as a unit cell. We can construct the ground state of the 1/3 magnetization plateau using the ground state of single trimer, ({leftvert psi rightrangle }_{g}=leftvert 000ldots 00rightrangle), to study the spin dynamics. The low-energy spin wave is generated by the flipping of one spin within the ferromagnetic state, a phenomenon effectively described by the propagation of magnons. In this scenario, we can manipulate the effective spin of one trimer; for instance, by altering one trimer from (leftvert 0rightrangle) to (leftvert 1rightrangle) in state ({leftvert psi rightrangle }_{g}), that results in the dispersion relation,

which coincides well with the low-energy excitation spectrum, irrespective of the magnitude of the magnetic field, as illustrated in Fig. 5. We designate this excitation as the reduced spin wave inspire of the conventional magnon picture. Moving on to the intermediate-energy excitations, where one trimer is excited from (leftvert 0rightrangle) to (leftvert 3rightrangle) with ΔM = 1, the associated dispersion relation is

Here, the intermediate-energy mode is termed as the doublon rather than the magnon, as it arises from the excitation of localized trimers and exhibits a higher energy gap compared to the low-energy magnon. For the high-energy excitations, two distinct branches of the excitation spectra emerge, which corresponds to the excitation (leftvert 0rightrangle to leftvert 6rightrangle) with ΔM = − 1 and (leftvert 0rightrangle to leftvert 4rightrangle) with ΔM = 1. The corresponding dispersion relations are given by,

which are referred to as the high-energy quartons. Notably, it can be observed that the reduced spin wave ((leftvert 0rightrangle to leftvert 1rightrangle)), doublon ((leftvert 0rightrangle to leftvert 3rightrangle)) and upper quarton ((leftvert 0rightrangle to leftvert 6rightrangle)) share the same magnetization quantum number ΔM = −1, and they collectively increase in energy as the magnetic field intensifies. Conversely, the lower quarton descends independently due to its distinct magnetization quantum number ΔM = 1, ultimately becoming the low-energy spectrum when H_z ≥ 0.9. More interestingly, as shown in Fig. 3c, g, it is noteworthy that even in the XY-II phase, the excitations characterized by ΔM = −1 remain observable in the high-energy regime.

All results are obtained by DMRG-TDVP calculations for a system with length L = 120, and the color coding of ({{mathcal{S}}}^{xx}(q,omega )) uses a piecewise function with a boundary value U₀ = 2. a–h Respectively correspond to the cases with H_z = 0.3,0.4,⋯,0.9,1.0. The dispersion lines, distinguished by colors and numerical labels, correspond to various localized excitations within a single trimer. (1)(2)(4) are the excitations from (leftvert 0rightrangle to leftvert 1rightrangle), (leftvert 0rightrangle to leftvert 3rightrangle) and (leftvert 0rightrangle to leftvert 6rightrangle) with ΔM =− 1, respectively. (3) is the excitations from (leftvert 0rightrangle to leftvert 4rightrangle) with ΔM = 1.

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Quantum magnets

It has been discovered that Na₂Cu₃Ge₄O₁₂ is an ideal realization of the spin-1/2 antiferromagnetic trimer chain, wherein Cu₃O₈ comprises the trimers formed by three edge-sharing CuO₄ square planes arranged linearly. The magnetic Cu²⁺ ions within the CuO₄ square planes demonstrate quantum spin-1/2 characteristics^12,59. Figure 6a presents a more realistic spin model that incorporates an additional next-nearest neighbor intratrimer exchange coupling J₃. The Hamiltonian for this system is given by

where the experimental measurements have established the coupling strengths as J₁ = 235 K and J₂ = J₃ = 0.18J₁. When H_z = 0, the eight energy levels (see Fig. 6c) reduce to three ones, ({E}_{0}^{{prime} }=-0.955{J}_{1}), ({E}_{1}^{{prime} }=-0.135{J}_{1}) and ({E}_{2}^{{prime} }=0.545{J}_{1}). The doublon and quarton manifest at (omega sim {E}_{1}^{{prime} }-{E}_{0}^{{prime} }=0.82{J}_{1}) and (omega sim {E}_{2}^{{prime} }-{E}_{0}^{{prime} }=1.5{J}_{1}), respectively, as shown in Fig. 6e. In the inelastic neutron scattering measurements conducted on Na₂Cu₃Ge₄O₁₂¹², three excitation modes have been identified. The two-spinon modes are observed below 5 meV, while, the doublon and quarton states appear in the intermediate (17–22 meV) and high (32–37 meV) energy ranges, respectively. These findings are corroborated by the intermediate-energy (at ω ~ 0.82J₁) and high-energy (at ω ~ 1.5J₁) excitations observed in our numerical simulations.

a Schematic representation of the trimer model with next-nearest neighbor intratrimer exchange couplings J₃, and J₂ = J₃ = 0.18J₁. The spins labeled as a, b and c are the three Cu²⁺ spins within a trimer unit. b Energy levels as functions of the magnetic field H_z. c Eigenevergies, wave functions, and quantum numbers of an isolated trimer unit in the material under the magnetic field H_z. d The magnetization curves obtained by experimental measurements and DMRG calculation, with the results normalized to the maximum of magnetization. The experimental data is extracted from Fig. 1d in ref. ¹². ({{mathcal{S}}}^{xx}(q,omega )) of spin model related to experimental materials Na₂Cu₃Ge₄O₁₂ in (e) the case without magnetic field, (f) XY-I phase, (g, h) 1/3 magnetization plateau phase, (i) XY-II phase, and (j) Ferromagnetic phase obtained by DMRG-TDVP calculation for L = 120. The color coding of ({{mathcal{S}}}^{xx}(q,omega )) uses a piecewise function with a boundary value U₀ = 2.

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In Fig. 6b, the application of a magnetic field results in the splitting of the three energy levels of a single trimer into eight distinct levels. When H_z ≤ 1.5, the ground state is (leftvert 0rightrangle) with an energy of E₀ = −0.955J₁ − H_z/2. Although the antiferromagnetic interaction J₃ competes with the interaction J₁ and induces frustration within the spin system, when J₃ ≤ J₁, introducing J₃ does not change the relative sequence of energy levels and their quantum numbers. There are only minor shifts in their eigenvalues, as depicted in Fig. 6c. Thus, the spin excitations can still be characterized by the quasiparticles doublons and quartons. The trimer chain subjected to the J₃ interaction continues to display a 1/3 magnetization plateau, as shown in Fig. 6d. Experimental measurements have confirmed the presence of 1/3 magnetization plateau above 28 Tesla¹², which aligns with our DMRG calculations. Although the phase diagram and 1/3 magnetization plateau have been elucidated¹², significantly less is understood regarding the evolution of intermediate-energy and high-energy excitations under the influence of a magnetic field. In this subsection, we present the excitation spectrum of the model depicted in Fig. 6a, which is pertinent to the material Na₂Cu₃Ge₄O₁₂. Due to the weak J₃, the spin excitations in four phases displayed in Fig. 6f–j exhibit similarities to those of a trimer chain devoid of J₃. This includes the separation of high-energy spectra, the presence of gapless excitations in the XY-I and XY-II phases, and the emergence of a gap at the edges of Brillouin zones in the Ferromagnetic phase. Our theoretical results concerning the high-energy quasiparticles excitations under the magnetic field can be directly validated through the inelastic neutron scattering experiments on the material Na₂Cu₃Ge₄O₁₂.

The realization of doublon and quarton in specific experimental materials drives us to think deeper into how we might manipulate these excitations and explore their possible applications. Among that, the BEC of doublon or quarton is quite an important topic. BEC represents a compelling state of matter that has been observed in bosonic atoms and cold gases. Quasiparticles associated with magnetic excitations, which possess integer spin and adhere to Bose statistics, such as the magnon and triplon, are integral to the study of BEC^60,61,62,63. In particular, in the dimerized antiferromgnets like TlCuCl₃, the intradimer interaction is stronger than the interdimer interaction, thereby an isolated dimer exhibits a singlet ground state characterized by a total spin S = 0 and a triplet excited state with spin S = 1. Due to the relatively weak interdimer interaction, the magnetic excitations are predominantly governed by triplons. After applying a magnetic field, the Zeeman term controls the density of triplons, resulting in a decrease in energy for the triplon with a magnetic quantum number S^z = 1. At a critical magnetic field H_C1, the energy of the triplons reaches zero, leading to their gradual condensation into the ground state until a second critical magnetic field H_C2 is attained. Beyond H_C2, all spins become polarized.

Inspired by the triplon BEC phenomenon, it is pertinent to inquire that whether quarton BEC can be observed in trimerized systems. In this context, we present a preliminary analysis of quarton BEC based on our findings. Firstly, the high-energy quartons arise from the internal trimer excitations and possess an integer spin quantum number S = 1, thereby conforming the Bose statistics. Secondly, the application of a magnetic field causes the lower branch of quartons to approach zero energy, as show in Fig. 5 and Fig. 6g, h. At the critical point H_C1, which delineates the transition between the 1/3 magnetization plateau phase and the XY-II phase, the lower quartons commence condensation and accumulation as the magnetic field intensifies within the XY-II phase.

Furthermore, although the BEC is more possible to be observed in the real materials exhibiting 3D spin systems, the 1D and 2D limits can serve as effective starting points for comprehending the field-induced quasiparticle BEC^62,63,64,65. In the context of 1D systems, it is well-established that BEC does not occur due to the significant quantum fluctuations. For 2D systems, the presence of a finite density of states at zero energy poses a barrier to form the BEC. Nevertheless, the critical exponents associated with the 1D or 2D quantum critical points can be observed over a substantial temperature range⁶². In the 2D quantum dimer magnets, both the critical field and the critical temperature of the BEC dome can be accurately characterized⁶⁵. Consequently, the studies of 1D and 2D systems are instrumental in the exploration of quasiparticle BEC. Our recent investigation into spin dynamics in 2D trimer systems has revealed the emergence of high-energy quartons²². These 2D trimer systems offer valuable platforms for further examination of the quarton BEC. Moreover, a small but finite interlayer coupling in a quasi-2D magnet stabilizes marginal BEC⁶³. Therefore, a field-induced quarton BEC may emerge in the quantum material Na₂Cu₃Ge₄O₁₂. However, a technical challenge arises due to the fact that the critical magnetic field H_C1 (see Fig. 6d) may exceed the limits of experimental accessibility. Therefore, only the transition between the XY-I phase and the 1/3 magnetization plateau phase can be realized in Na₂Cu₃Ge₄O₁₂ through the currently available magnetic field. A quantum material characterized by trimer structures and relatively weak intratrimer interactions is essential to experimentally investigate the quarton BEC. Theoretically, it is of interest to seek further evidence of quarton BEC by examining the (T, H_z) phase diagram and the power-law temperature dependence of thermodynamic properties in both 2D and 3D trimer systems in future studies.

In summary, we have utilized the ED, CPT, and DMRG-TDVP methods to investigate the quantum phase transition and spin dynamics of the antiferromagnetic trimer chain subjected to a longitudinal magnetic field. Our findings reveal that the interplay of magnetic field and interaction leads to the emergence of four distinct phases: XY-I, 1/3 magnetization plateau, XY-II, and ferromagnetic phases. By mapping the entanglement entropy onto the parameter space (g, H_z), we obtain a comprehensive phase diagram. Furthermore, it has been confirmed that the critical phases XY-I and XY-II phases are both characterized by the conformal field theory with a central charge c ≃ 1. The transitions between these phases are identified as the second-order quantum phase transitions.

In the context of transverse and longitudinal excitations of a trimer chain subjected to a magnetic field, we have determined that the incommensurate wave numbers corresponding to zero energy are dependent of the magnetization in the XY-I and XY-II phases. In addition, we have observed the presence of gapped excitations within both the 1/3 magnetization plateau and ferromagnetic phases. Specifically, a continuum of excitations is observed in the high-energy regime of the 1/3 magnetization plateau phase. In the ferromagnetic phases, the excitations continue to be characterized by spin waves, however, magnons located at the edges of the Brillouin zone exhibit two diverse energies for the same wave vector.

Furthermore, we have identified the intermediate-energy and high-energy excitations for small g, and have elucidated their excitation mechanisms within the XY-I and 1/3 magnetization plateau phases by analyzing their dispersion relations. In these phases, the intermediate-energy and high-energy modes correspond to the propagating internal trimer excitations, referred to as doublons and quartons, respectively. In comparison to the trimer chain in the absence of a magnetic field, the high-energy spectrum exhibits a splitting two branches, which correspond to the upper quarton and lower quarton, respectively. As the magnetic field increases, the gap between these two branches widens, resulting in the lower quarton becoming the low-energy spectrum.

Experimentally, there are existing examples of coupled-trimer quantum magnets, such as A₃Cu₃(PO₄)₄ (A = Ca, Sr, Pb)^27,28,29,30 and Na₂Cu₃Ge₄O₁₂¹². Although the trimers in Pb₃Cu₃(PO₄)₄ do not exhibit a linear arrangement, two flat excitations at approximately ω ~ 9 meV and ω ~ 13.5 meV have been observed in the inelastic neutron-scattering spectra measured at 8K²⁷. These excitations are closely associated with the intermediate-energy (at ω ~ J₁) and the high-energy (at ω ~ 1.5J₁) excitations in the trimer chain without magnetic field¹¹. Additionally, for the quantum magnet Na₂Cu₃Ge₄O₁₂, an additional next-nearest neighbor interaction is present within the trimers; however, its strength is relatively weak, and the wave functions and quantum numbers of a single trimer remain invariant. Consequently, the doublons and quartons have been observed in the inelastic neutron-scattering experiments¹². We have theoretically demonstrated that doublons and quartons remain observable in the trimer chain under a magnetic field, even with the introduction of the interaction J₃. These findings can be directly investigated through the inelastic neutron-scattering experiments conducted on the aforementioned quantum materials. Moreover, based on the results obtained from the 1D trimer chain under a magnetic field, it is probable that the quarton BEC may be identified in experiments once suitable materials are found. Our results will be instrumental in interpreting inelastic neutron scattering and other experiments that probe the high-energy excitations beyond spin waves and spinons, as well as in facilitating detailed investigations of coexisting exotic excitations.

Matrix product states

DSF is a significant physical quantity for studying spin dynamics, a process that been effectively facilitated by the DMRG in conjunction with the time evolution algorithms^{37,66,67,68,69,70}. In this article, we primarily employ the TDVP method to handle with the time evolution of many-body systems^39,40. Specifically, we conduct DMRG-TDVP calculations on a finite chain with open boundary conditions to analyze the spectrum. We denote the ground state of the trimer chain in the presence of a magnetic field as (leftvert {mathcal{G}}rightrangle), which allows us to compute the real time evolution of the correlation function,

for various times t and distances j. E₀ represents the ground state energy. We select the site at the center of the chain designated as the site index 0. Firstly, we obtain the ground state (leftvert {mathcal{G}}rightrangle) by employing the DMRG method. Subsequently, we introduce a local perturbation ({hat{S}}_{0}^{beta }) at the center of the spin chain to generate the initial state

for real-time evolution. The real-time evolution state

is carried out using the single-site TDVP with a time step of (dt=0.05{J}_{1}^{-1}) and a maximum time ({t}_{max }=200{J}_{1}^{-1}). Finally, a Fourier transformation is performed to obtain ({{mathcal{S}}}^{alpha beta }(q,omega ))

Technically, to mitigate the limitations imposed by the finite-time limit constraint on the resolution of the spectral functions in frequency space, a Gaussian windowing function, expressed as (exp left[-4{(t/{t}_{max })}^{2}right]), is incorporated in the reconstruction of the DSF³⁷. During the DMRG calculations, we set ε_SVD = 10⁻¹¹ and retained a maximum of 6000 states. The time evolution is executed on a chain with open boundary conditions and N = 120 spins, which is sufficiently large to mitigate finite-size effects, with the maximum bond dimension set to 2000. All MPS simulations are performed using the ITensor library⁷¹.

Cluster perturbation theory

CPT is a theoretical framework used to study the electronic and magnetic properties of strongly correlated electrons^41,42,43,44, especially for calculating the single-particle spectral functions of Hubbard-type fermionic models and the dynamical spin structure factors of Heisenberg models. The fundamental concept of CPT involves partitioning a large system into smaller clusters, accurately calculating the properties of these clusters, and then use the mean-field and perturbation theory to infer the properties of the entire system. Here, we employ ED as a computational method to determine the dynamical spin structure factor within the cluster framework. Following ref. ⁴⁴, we provide a concise overview of the procedural steps involved in applying CPT to spin models.

Firstly, we transform the spin model into a hard-core boson model through the application of the following mapping,

Then the Hamiltonian can be rewrited as,

where the ({b}_{i}^{dagger }), b_i and ({n}_{i}={b}_{i}^{dagger }{b}_{i}) are the bosonic operators with hard-core constraint n_i = 0 or 1.

Secondly, we split the system into clusters. In our calculations, the cluster size is chosen to be N = 8, L = 24 which is sufficiently large to yield the accurate results. For the interaction bonds connecting adjacent clusters. We use self-consistent mean-field treatment to decouple the interactions between the clusters,

Thirdly, we employ ED to self-consistently obtain the mean-field potentials of the two end sites, (leftlangle {n}_{1,c}rightrangle) and (leftlangle {n}_{N,c}rightrangle). And after the convergence, we run a ED simulation to obtain the real-frequency single-particle Green function matrix ({{bf{G}}}_{ij}^{C}(omega )) using the Lanczos iteration method, where C denotes the Green function matrix of the cluster.

Fourthly, the original lattice Green function matrix can be derived from the cluster Green function matrix by disregarding the nonlocal self-energy contributions between clusters.

where L denotes the Green function matrix of the original lattice, (tilde{{bf{q}}}) is the wave vector within the Brillouin zone of the supercell formed by the cluster.

Fifthly, we perform the reperiodization of the Green function matrix to restore translational invariance.

Then the transverse dynamical spin structure factor can be obtained via

The CPT applied to spin models has been successfully applied to investigate the J₁ − J₂ and J₁ − J₃ models on a 2D square lattice^44,72, the J₁ − J₂ model on honeycomb lattice⁷³, as well as 2D trimer models²². This methodology demonstrates efficacy in characterizing the continua present in quantum spin liquid phases, as well as the magnon and triplon excitations observed in conventional Néel and valence bond solid phases. The method is exact in two limiting scenarios: one in which the interactions between clusters approach zero or are exceedingly weak, and the other in witch cluster size approaches infinity or is significantly large. In the context of our trimer chain model, when g = J₂/J₁ is small, accurate results can be achieved even with elatively small clusters, such as N = 2, 4. The selection of a cluster size of N = 8 in our study ensures accuracy across both small and large g regimes. This cluster size is sufficiently large to guarantee precision across a wide range of parameter values.