Physical review. B./Physical review. B

High-field NMR study on the topologically nontrivial 1/3 magnetization plateau state in doped Na 2 Cu 3 Ge 4 − x Si x O 12

Yuyan Han, Bocheng Yu, Zan Du, Langsheng Ling, Lei Zhang, Wei Tong, Chuanying Xi, Jinglei Zhang, Tian Shang, Li Pi, Long Ma

Published Nov 5, 2024• 2 citations

Publisher

Abstract

The spin-1/2 trimer system ${\mathrm{Na}}_{2}{\mathrm{Cu}}_{3}{\mathrm{Ge}}_{4}{\mathrm{O}}_{12}$ is proposed to host topologically nontrivial 1/3 magnetization $(M)$ plateau state. In this paper, we report a high-field nuclear magnetic resonance (NMR) study on the low-energy spin excitations of the $1/3\text{\ensuremath{-}}M$ plateau state in ${\mathrm{Na}}_{2}{\mathrm{Cu}}_{3}{\mathrm{Ge}}_{4}{\mathrm{O}}_{12}$ and Si-doped samples. First, the spin excitation gap supporting the high-field Haldane phase is indicated by the thermally activated behavior of spin-lattice relaxation rates in the $1/3\text{\ensuremath{-}}M$ plateau state. Second, by substituting ${\mathrm{Ge}}^{4+}$ with ${\mathrm{Si}}^{4+}$, the critical field for the 1/3-$M$ plateau significantly decreases, e.g., ${\ensuremath{\mu}}_{0}{H}_{c}=17$ T in the $x=1.0$ sample, which results from the suppressed intertrimer coupling ${J}_{2}$. The gapped spin excitation is observed above 17 T whose size shows the temperature-dependent behavior for ${\ensuremath{\mu}}_{0}H\ensuremath{\ge}25.72$ T. These observations support the topological nontrivial characteristic of the $1/3\text{\ensuremath{-}}M$ plateau state.

DOI: 10.1103/physrevb.110.l201102

Related papers

Topologically nontrivial $1/3$-magnetization plateau state in a spin-1/2 trimer chain

Yuyan Han, Yu, B. C., Zhijun Du +8 more

arXiv:2407.03581Jul 4, 20240 citations

Theoretical and experimental studies of melting of the 1/3 magnetization plateau in a frustrated $$S=1/2$$ antiferromagnetic trimerized quantum Heisenberg spin chain compound $$\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}$$

Sachin Kumar, Anup Kumar Bera, Amit Kumar +2 more

Abstract We report the magnetization process of $$S=1/2$$ S = 1 / 2 Heisenberg antiferromagnetic $$J_1{-}J_1{-}J_2$$ J 1 - J 1 - J 2 trimerized spin chain system (with frustration due to the next-nearest-neighbor interaction $$J_3$$ J 3 within the trimer) under applied magnetic field by means of density-matrix renormalization group method. Here, $$J'$$ J ′ s are the antiferromagnetic exchange interaction constants. Specifically, we explore the phenomenon of 1/3 magnetization plateau for the exchange coupling ratios $$ \alpha = 0.18, \beta =0.18 $$ α = 0.18 , β = 0.18 , and $$J_1=235$$ J 1 = 235 K, where $$\alpha =J_2/J_1$$ α = J 2 / J 1 , and $$\beta =J_3/J_1$$ β = J 3 / J 1 as found for the model compound $$\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}$$ Na 2 Cu 3 Ge 4 O 12 . It is found that a wide 1/3 quantum magnetization plateau state (observed in ground state magnetization) gradually disappears with increasing temperature beyond a crossover temperature $$T_{co}$$ T co due to thermal fluctuations. Based on numerical data, the width w(T) of the magnetization plateau is found to follow an exponential behave as $$\sim e^{-B/\sqrt{T_{co}-T}}$$ ∼ e - B / T co - T , where B is a constant and $$T_{co}$$ T co is estimated to be $$ \approx 60$$ ≈ 60

The European Physical Journal BMar 1, 20251 citation

Phonon anomalies and critical scaling in the spin- 1 2 trimer chain Na 2 Cu 3 Ge 4 O 12

P. Srikanth Patnaik, Anup Kumar Bera, Srishti Bhardwaj +4 more

arXiv:2608.12061Physical review. B./Physical review. BJul 13, 20260 citations

Frustration effects on the magnetization plateau physics in a trimerized quantum spin-1/2 chain

L. M. Ramos, M. Schmidt, Fábio M. Zimmer

arXiv:2606.13265Jun 11, 20260 citations

Trimerized Spin-$1/2$ Chain: Emergent Low-Energy Hamiltonian, Higher-Energy Excitations, and Magnetic and Thermodynamic Responses

Snehasish Sen, Sudhansu S. Mandal

arXiv:2605.25960May 25, 20260 citations

Observation of a Spin Gap in Sr Cu 2 O 3 Comprising Spin-½ Quasi-1D Two-Leg Ladders

Masaki Azuma, Zenji Hiroi, M. Takano +2 more

Physical Review LettersDec 19, 1994732 citations

Physical review. B./Physical review. B

High-field NMR study on the topologically nontrivial 1/3 magnetization plateau state in doped Na 2 Cu 3 Ge 4 − x Si x O 12

Yuyan Han, Bocheng Yu, Zan Du, Langsheng Ling, Lei Zhang, Wei Tong, Chuanying Xi, Jinglei Zhang, Tian Shang, Li Pi, Long Ma

Published Nov 5, 2024• 2 citations

Publisher

Abstract

The spin-1/2 trimer system

{\mathrm{Na}}_{2}{\mathrm{Cu}}_{3}{\mathrm{Ge}}_{4}{\mathrm{O}}_{12}

is proposed to host topologically nontrivial 1/3 magnetization

(M)

plateau state. In this paper, we report a high-field nuclear magnetic resonance (NMR) study on the low-energy spin excitations of the

1/3\text{\ensuremath{-}}M

plateau state in

{\mathrm{Na}}_{2}{\mathrm{Cu}}_{3}{\mathrm{Ge}}_{4}{\mathrm{O}}_{12}

and Si-doped samples. First, the spin excitation gap supporting the high-field Haldane phase is indicated by the thermally activated behavior of spin-lattice relaxation rates in the

1/3\text{\ensuremath{-}}M

plateau state. Second, by substituting

{\mathrm{Ge}}^{4+}

with

{\mathrm{Si}}^{4+}

, the critical field for the 1/3-

M

plateau significantly decreases, e.g.,

{\ensuremath{\mu}}_{0}{H}_{c}=17

T in the

x=1.0

sample, which results from the suppressed intertrimer coupling

{J}_{2}

. The gapped spin excitation is observed above 17 T whose size shows the temperature-dependent behavior for

{\ensuremath{\mu}}_{0}H\ensuremath{\ge}25.72

T. These observations support the topological nontrivial characteristic of the

1/3\text{\ensuremath{-}}M

plateau state.

DOI: 10.1103/physrevb.110.l201102

Related papers

Topologically nontrivial $1/3$ -magnetization plateau state in a spin-1/2 trimer chainTopologically nontrivial $1/3$-magnetization plateau state in a spin-1/2 trimer chain

Yuyan Han, Yu, B. C., Zhijun Du +8 more

Topologically nontrivial Haldane phase is theoretically proposed to be realized in the 1/3-magnetization ( $M$ ) plateau of spin-1/2 trimer systems. However, the spin excitation gap, typical characteristic of Haldane phase, is not yet experimentally verified. Here, we report the nuclear magnetic resonance investigations into the low-energy spin dynamics in the $S=1/2$ spin-trimer antiferromagnetic chain compound Na $_2$ Cu $_3$ Ge $_{4-x}$ Si $_{x}$ O $_{12}$ ( $x=0, 0.1\sim1.5$ ). In the parent compound ( $x=0$ ), the spin-lattice relaxation rate (1/ $T_1$ ) shows significantly different temperature dependence when the external magnetic field is increased above the critical field of $μ_0$ $H_{c}$ = 29 T. The spin excitation gap is evidenced from the thermally activated behavior of $1/T_1(T)$ in the 1/3- $M$ plateau state. By substituting Ge $^{4+}$ with Si $^{4+}$ , the critical field for the 1/3- $M$ plateau significantly decreases, e.g. $μ_0H_{c}=17$ T in $x=1.0$ samples, which results from the suppressed inter-trimer coupling $J_2$ . The gapped spin excitation is confirmed again above 17 T, whose size shows temperature-dependent behavior for $μ_0H\geq25.72$ T. These observations provide further insights into the Haldane physics.Topologically nontrivial Haldane phase is theoretically proposed to be realized in the 1/3-magnetization ($M$) plateau of spin-1/2 trimer systems. However, the spin excitation gap, typical characteristic of Haldane phase, is not yet experimentally verified. Here, we report the nuclear magnetic resonance investigations into the low-energy spin dynamics in the $S=1/2$ spin-trimer antiferromagnetic chain compound Na$_2$Cu$_3$Ge$_{4-x}$Si$_{x}$O$_{12}$ ($x=0, 0.1\sim1.5$). In the parent compound ($x=0$), the spin-lattice relaxation rate (1/$T_1$) shows significantly different temperature dependence when the external magnetic field is increased above the critical field of $μ_0$$H_{c}$ = 29 T. The spin excitation gap is evidenced from the thermally activated behavior of $1/T_1(T)$ in the 1/3-$M$ plateau state. By substituting Ge$^{4+}$ with Si$^{4+}$, the critical field for the 1/3-$M$ plateau significantly decreases, e.g. $μ_0H_{c}=17$ T in $x=1.0$ samples, which results from the suppressed inter-trimer coupling $J_2$. The gapped spin excitation is confirmed again above 17 T, whose size shows temperature-dependent behavior for $μ_0H\geq25.72$ T. These observations provide further insights into the Haldane physics.

arXiv:2407.03581Jul 4, 20240 citations

Theoretical and experimental studies of melting of the 1/3 magnetization plateau in a frustrated $S=1/2$ antiferromagnetic trimerized quantum Heisenberg spin chain compound $\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}$ Theoretical and experimental studies of melting of the 1/3 magnetization plateau in a frustrated $$S=1/2$$ antiferromagnetic trimerized quantum Heisenberg spin chain compound $$\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}$$

Sachin Kumar, Anup Kumar Bera, Amit Kumar +2 more

Abstract We report the magnetization process of $S=1/2$ S = 1 / 2 Heisenberg antiferromagnetic $J_1{-}J_1{-}J_2$ J 1 - J 1 - J 2 trimerized spin chain system (with frustration due to the next-nearest-neighbor interaction $J_3$ J 3 within the trimer) under applied magnetic field by means of density-matrix renormalization group method. Here, $J'$ J ′ s are the antiferromagnetic exchange interaction constants. Specifically, we explore the phenomenon of 1/3 magnetization plateau for the exchange coupling ratios $\alpha = 0.18, \beta =0.18$ α = 0.18 , β = 0.18 , and $J_1=235$ J 1 = 235 K, where $\alpha =J_2/J_1$ α = J 2 / J 1 , and $\beta =J_3/J_1$ β = J 3 / J 1 as found for the model compound $\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}$ Na 2 Cu 3 Ge 4 O 12 . It is found that a wide 1/3 quantum magnetization plateau state (observed in ground state magnetization) gradually disappears with increasing temperature beyond a crossover temperature $T_{co}$ T co due to thermal fluctuations. Based on numerical data, the width w(T) of the magnetization plateau is found to follow an exponential behave as $\sim e^{-B/\sqrt{T_{co}-T}}$ ∼ e - B / T co - T , where B is a constant and $T_{co}$ T co is estimated to be $\approx 60$ ≈ 60Abstract We report the magnetization process of $$S=1/2$$ S = 1 / 2 Heisenberg antiferromagnetic $$J_1{-}J_1{-}J_2$$ J 1 - J 1 - J 2 trimerized spin chain system (with frustration due to the next-nearest-neighbor interaction $$J_3$$ J 3 within the trimer) under applied magnetic field by means of density-matrix renormalization group method. Here, $$J'$$ J ′ s are the antiferromagnetic exchange interaction constants. Specifically, we explore the phenomenon of 1/3 magnetization plateau for the exchange coupling ratios $$ \alpha = 0.18, \beta =0.18 $$ α = 0.18 , β = 0.18 , and $$J_1=235$$ J 1 = 235 K, where $$\alpha =J_2/J_1$$ α = J 2 / J 1 , and $$\beta =J_3/J_1$$ β = J 3 / J 1 as found for the model compound $$\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}$$ Na 2 Cu 3 Ge 4 O 12 . It is found that a wide 1/3 quantum magnetization plateau state (observed in ground state magnetization) gradually disappears with increasing temperature beyond a crossover temperature $$T_{co}$$ T co due to thermal fluctuations. Based on numerical data, the width w(T) of the magnetization plateau is found to follow an exponential behave as $$\sim e^{-B/\sqrt{T_{co}-T}}$$ ∼ e - B / T co - T , where B is a constant and $$T_{co}$$ T co is estimated to be $$ \approx 60$$ ≈ 60

The European Physical Journal BMar 1, 20251 citation

Phonon anomalies and critical scaling in the spin- 1 2 trimer chain Na 2 Cu 3 Ge 4 O 12

P. Srikanth Patnaik, Anup Kumar Bera, Srishti Bhardwaj +4 more

Low-dimensional quantum magnets provide an ideal platform to explore spin-lattice coupling-mediated quantum correlations, which give rise to emergent quasiparticle excitations. The antiferromagnetically coupled spin-1/2 trimer chain of copper ions in Na $_2$ Cu $_3$ Ge $_4$ O $_{12}$ (NCGO) hosts high-energy spin excitations of different species, whose energy scales overlap with those of lattice vibrations. Here, we report a comprehensive temperature-dependent Raman spectroscopic study performed between 80 and 400 K. The dynamic spin susceptibility, as obtained from the analysis of the broad spectral background, reveals the emergence of quasiparticle excitations below 170 K. We further identify an unusual crossover of phonon dynamics when the material transits from a normal paramagnetic state to a correlated quantum magnetic state. A power law dependence of the integrated Raman susceptibility of the phonon modes, $I_{χ^{\prime\prime}}^{i}\sim|T-T_{c}|^β$ , is observed with the critical temperature $T_c$ =167$\pm$1 K, and critical exponent $β= 0.24\pm 0.02$ . The combined results obtained from the broad spectral background and sharp phonon peaks further indicate that the phonon renormalization observed across the crossover is driven by dynamic spin states. Additionally, statistical correlations among phonon energy eigenvalues, quantified through matrix-norm and power-test analyses of 200 spectra recorded at 80 K, reveal an unexpected linear correlation among phonon modes, also indicating that the collective lattice response is governed by spin correlations. These findings establish NCGO as a model system for investigating cooperative spin-lattice coupling and critical scaling behavior of phonon dynamics in low-dimensional magnetic materials.Low-dimensional quantum magnets provide an ideal platform to explore spin-lattice coupling-mediated quantum correlations, which give rise to emergent quasiparticle excitations. The antiferromagnetically coupled spin-1/2 trimer chain of copper ions in Na$_2$Cu$_3$Ge$_4$O$_{12}$ (NCGO) hosts high-energy spin excitations of different species, whose energy scales overlap with those of lattice vibrations. Here, we report a comprehensive temperature-dependent Raman spectroscopic study performed between 80 and 400 K. The dynamic spin susceptibility, as obtained from the analysis of the broad spectral background, reveals the emergence of quasiparticle excitations below 170 K. We further identify an unusual crossover of phonon dynamics when the material transits from a normal paramagnetic state to a correlated quantum magnetic state. A power law dependence of the integrated Raman susceptibility of the phonon modes, $I_{χ^{\prime\prime}}^{i}\sim|T-T_{c}|^β$, is observed with the critical temperature $T_c$=167$\pm$1 K, and critical exponent $β= 0.24\pm 0.02$. The combined results obtained from the broad spectral background and sharp phonon peaks further indicate that the phonon renormalization observed across the crossover is driven by dynamic spin states. Additionally, statistical correlations among phonon energy eigenvalues, quantified through matrix-norm and power-test analyses of 200 spectra recorded at 80 K, reveal an unexpected linear correlation among phonon modes, also indicating that the collective lattice response is governed by spin correlations. These findings establish NCGO as a model system for investigating cooperative spin-lattice coupling and critical scaling behavior of phonon dynamics in low-dimensional magnetic materials.

arXiv:2608.12061Physical review. B./Physical review. BJul 13, 20260 citations

Frustration effects on the magnetization plateau physics in a trimerized quantum spin-1/2 chain

L. M. Ramos, M. Schmidt, Fábio M. Zimmer

We investigate frustration-induced instabilities in a trimerized quantum spin chain motivated by recent experimental findings for the compound Na $_2$ Cu $_3$ Ge $_4$ O $_{12}$ . Employing a cluster mean-field approach combined with Lanczos exact diagonalization, we analyze the ground-state and quantum-information properties of a Heisenberg model with competing interactions in a magnetic field. In the weakly frustrated regime, the system exhibits a robust $1/3$ magnetization plateau associated with a collective ferrimagnetic-like trimer state. Increasing the next-nearest-neighbor intratrimer coupling drives a pronounced reorganization of spin correlations, leading to a crossover toward a doublon-like correlation regime and providing a static ground-state picture consistent with the composite excitations observed dynamically in trimerized chains. The resulting low-energy behavior can be interpreted in terms of weakly interacting emergent spins, offering a microscopic explanation for the extended stability of the magnetization plateau. Furthermore, through finite-size scaling analyses of the energy gap, von Neumann entanglement entropy, and fidelity susceptibility, we characterize the zero-field criticality of the model. Ultimately, our results suggest that frustration gives rise to qualitatively distinct quantum states and provide a microscopic framework for understanding the emergence of fractionalized excitations in trimerized quantum spin systems.We investigate frustration-induced instabilities in a trimerized quantum spin chain motivated by recent experimental findings for the compound Na$_2$Cu$_3$Ge$_4$O$_{12}$. Employing a cluster mean-field approach combined with Lanczos exact diagonalization, we analyze the ground-state and quantum-information properties of a Heisenberg model with competing interactions in a magnetic field. In the weakly frustrated regime, the system exhibits a robust $1/3$ magnetization plateau associated with a collective ferrimagnetic-like trimer state. Increasing the next-nearest-neighbor intratrimer coupling drives a pronounced reorganization of spin correlations, leading to a crossover toward a doublon-like correlation regime and providing a static ground-state picture consistent with the composite excitations observed dynamically in trimerized chains. The resulting low-energy behavior can be interpreted in terms of weakly interacting emergent spins, offering a microscopic explanation for the extended stability of the magnetization plateau. Furthermore, through finite-size scaling analyses of the energy gap, von Neumann entanglement entropy, and fidelity susceptibility, we characterize the zero-field criticality of the model. Ultimately, our results suggest that frustration gives rise to qualitatively distinct quantum states and provide a microscopic framework for understanding the emergence of fractionalized excitations in trimerized quantum spin systems.

arXiv:2606.13265Jun 11, 20260 citations

Trimerized Spin- $1/2$ Chain: Emergent Low-Energy Hamiltonian, Higher-Energy Excitations, and Magnetic and Thermodynamic ResponsesTrimerized Spin-$1/2$ Chain: Emergent Low-Energy Hamiltonian, Higher-Energy Excitations, and Magnetic and Thermodynamic Responses

Snehasish Sen, Sudhansu S. Mandal

We investigate a spin-1/2 antiferromagnetic chain with trimerized exchange couplings relevant to Na $_2$ Cu $_3$ Ge $_4$ O $_{12}$ . We show that the low-energy Hilbert space maps onto an effective Heisenberg chain of composite spin-1/2 trimer degrees of freedom with positive coupling, enabling an exact spinon excitations via the Bethe ansatz. To access higher-energy dynamics, we develop a self-consistent Jordan-Wigner mean-field theory that yields three fermionic bands reflecting the underlying trimer structure. Remarkably, this approach reproduces the exact low-energy spinon continuum while predicting two additional higher-energy excitation bands consistent with the experimental observations and previous numerical simulations. The theory further captures the 1/3 magnetization plateau under an applied magnetic field, and provides testable predictions in magnetic susceptibility and specific heat. Our results establish a unified framework connecting low- and high-energy excitations in trimerized quantum spin chains.We investigate a spin-1/2 antiferromagnetic chain with trimerized exchange couplings relevant to Na$_2$Cu$_3$Ge$_4$O$_{12}$. We show that the low-energy Hilbert space maps onto an effective Heisenberg chain of composite spin-1/2 trimer degrees of freedom with positive coupling, enabling an exact spinon excitations via the Bethe ansatz. To access higher-energy dynamics, we develop a self-consistent Jordan-Wigner mean-field theory that yields three fermionic bands reflecting the underlying trimer structure. Remarkably, this approach reproduces the exact low-energy spinon continuum while predicting two additional higher-energy excitation bands consistent with the experimental observations and previous numerical simulations. The theory further captures the 1/3 magnetization plateau under an applied magnetic field, and provides testable predictions in magnetic susceptibility and specific heat. Our results establish a unified framework connecting low- and high-energy excitations in trimerized quantum spin chains.

arXiv:2605.25960May 25, 20260 citations

Observation of a Spin Gap in Sr Cu 2 O 3 Comprising Spin-½ Quasi-1D Two-Leg Ladders

Masaki Azuma, Zenji Hiroi, M. Takano +2 more

Magnetic properties of Sr ${\mathrm{Cu}}_{2}$ ${\mathrm{O}}_{3}$ and ${\mathrm{Sr}}_{2}$ ${\mathrm{Cu}}_{3}$ ${\mathrm{O}}_{5}$ , containing two-leg and three-leg $S=\frac{1}{2}$ ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, Sr ${\mathrm{Cu}}_{2}$ ${\mathrm{O}}_{3}$ , is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of ${\mathrm{Sr}}_{2}$ ${\mathrm{Cu}}_{3}$ ${\mathrm{O}}_{5}$ , containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate, $\frac{1}{{T}_{1}}$ , of $^{63}\mathrm{Cu}$ NMR also indicates the existence of a spin gap only for Sr ${\mathrm{Cu}}_{2}$ ${\mathrm{O}}_{3}$ . The spin gap we observed for Sr ${\mathrm{Cu}}_{2}$ ${\mathrm{O}}_{3}$ confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.Magnetic properties of Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$ and ${\mathrm{Sr}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{5}$, containing two-leg and three-leg $S=\frac{1}{2}$ ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$, is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of ${\mathrm{Sr}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{5}$, containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate, $\frac{1}{{T}_{1}}$, of $^{63}\mathrm{Cu}$ NMR also indicates the existence of a spin gap only for Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$. The spin gap we observed for Sr${\mathrm{Cu}}_{2}$${\mathrm{O}}_{3}$ confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.

Physical Review LettersDec 19, 1994732 citations

The spin-1/2 trimer system

{\mathrm{Na}}_{2}{\mathrm{Cu}}_{3}{\mathrm{Ge}}_{4}{\mathrm{O}}_{12}

is proposed to host topologically nontrivial 1/3 magnetization

(M)

plateau state. In this paper, we report a high-field nuclear magnetic resonance (NMR) study on the low-energy spin excitations of the

1/3\text{\ensuremath{-}}M

plateau state in

{\mathrm{Na}}_{2}{\mathrm{Cu}}_{3}{\mathrm{Ge}}_{4}{\mathrm{O}}_{12}

and Si-doped samples. First, the spin excitation gap supporting the high-field Haldane phase is indicated by the thermally activated behavior of spin-lattice relaxation rates in the

1/3\text{\ensuremath{-}}M

plateau state. Second, by substituting

{\mathrm{Ge}}^{4+}

with

{\mathrm{Si}}^{4+}

, the critical field for the 1/3-

M

plateau significantly decreases, e.g.,

{\ensuremath{\mu}}_{0}{H}_{c}=17

T in the

x=1.0

sample, which results from the suppressed intertrimer coupling

{J}_{2}

. The gapped spin excitation is observed above 17 T whose size shows the temperature-dependent behavior for

{\ensuremath{\mu}}_{0}H\ensuremath{\ge}25.72

T. These observations support the topological nontrivial characteristic of the

1/3\text{\ensuremath{-}}M

plateau state.

Topologically nontrivial

1/3

-magnetization plateau state in a spin-1/2 trimer chain

Topologically nontrivial Haldane phase is theoretically proposed to be realized in the 1/3-magnetization (

M

) plateau of spin-1/2 trimer systems. However, the spin excitation gap, typical characteristic of Haldane phase, is not yet experimentally verified. Here, we report the nuclear magnetic resonance investigations into the low-energy spin dynamics in the

S=1/2

spin-trimer antiferromagnetic chain compound Na

_2

_3

_{4-x}

_{x}

_{12}

(

x=0, 0.1\sim1.5

). In the parent compound (

x=0

), the spin-lattice relaxation rate (1/

T_1

) shows significantly different temperature dependence when the external magnetic field is increased above the critical field of

μ_0

H_{c}

= 29 T. The spin excitation gap is evidenced from the thermally activated behavior of

1/T_1(T)

in the 1/3-

M

plateau state. By substituting Ge

^{4+}

with Si

^{4+}

, the critical field for the 1/3-

M

plateau significantly decreases, e.g.

μ_0H_{c}=17

T in

x=1.0

samples, which results from the suppressed inter-trimer coupling

J_2

. The gapped spin excitation is confirmed again above 17 T, whose size shows temperature-dependent behavior for

μ_0H\geq25.72

T. These observations provide further insights into the Haldane physics.

Theoretical and experimental studies of melting of the 1/3 magnetization plateau in a frustrated

S=1/2

antiferromagnetic trimerized quantum Heisenberg spin chain compound

\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}

Abstract We report the magnetization process of

S=1/2

S = 1 / 2 Heisenberg antiferromagnetic

J_1{-}J_1{-}J_2

J 1 - J 1 - J 2 trimerized spin chain system (with frustration due to the next-nearest-neighbor interaction

J_3

J 3 within the trimer) under applied magnetic field by means of density-matrix renormalization group method. Here,

J'

J ′ s are the antiferromagnetic exchange interaction constants. Specifically, we explore the phenomenon of 1/3 magnetization plateau for the exchange coupling ratios

\alpha = 0.18, \beta =0.18

α = 0.18 , β = 0.18 , and

J_1=235

J 1 = 235 K, where

\alpha =J_2/J_1

α = J 2 / J 1 , and

\beta =J_3/J_1

β = J 3 / J 1 as found for the model compound

\hbox {Na}_2\hbox {Cu}_3\hbox {Ge}_4\hbox {O}_{12}

Na 2 Cu 3 Ge 4 O 12 . It is found that a wide 1/3 quantum magnetization plateau state (observed in ground state magnetization) gradually disappears with increasing temperature beyond a crossover temperature

T_{co}

T co due to thermal fluctuations. Based on numerical data, the width w(T) of the magnetization plateau is found to follow an exponential behave as

\sim e^{-B/\sqrt{T_{co}-T}}

∼ e - B / T co - T , where B is a constant and

T_{co}

T co is estimated to be

\approx 60

≈ 60

_2

_3

_4

_{12}

(NCGO) hosts high-energy spin excitations of different species, whose energy scales overlap with those of lattice vibrations. Here, we report a comprehensive temperature-dependent Raman spectroscopic study performed between 80 and 400 K. The dynamic spin susceptibility, as obtained from the analysis of the broad spectral background, reveals the emergence of quasiparticle excitations below 170 K. We further identify an unusual crossover of phonon dynamics when the material transits from a normal paramagnetic state to a correlated quantum magnetic state. A power law dependence of the integrated Raman susceptibility of the phonon modes,

I_{χ^{\prime\prime}}^{i}\sim|T-T_{c}|^β

, is observed with the critical temperature

T_c

=167$\pm$1 K, and critical exponent

β= 0.24\pm 0.02

. The combined results obtained from the broad spectral background and sharp phonon peaks further indicate that the phonon renormalization observed across the crossover is driven by dynamic spin states. Additionally, statistical correlations among phonon energy eigenvalues, quantified through matrix-norm and power-test analyses of 200 spectra recorded at 80 K, reveal an unexpected linear correlation among phonon modes, also indicating that the collective lattice response is governed by spin correlations. These findings establish NCGO as a model system for investigating cooperative spin-lattice coupling and critical scaling behavior of phonon dynamics in low-dimensional magnetic materials.

We investigate frustration-induced instabilities in a trimerized quantum spin chain motivated by recent experimental findings for the compound Na

_2

_3

_4

_{12}

. Employing a cluster mean-field approach combined with Lanczos exact diagonalization, we analyze the ground-state and quantum-information properties of a Heisenberg model with competing interactions in a magnetic field. In the weakly frustrated regime, the system exhibits a robust

1/3

magnetization plateau associated with a collective ferrimagnetic-like trimer state. Increasing the next-nearest-neighbor intratrimer coupling drives a pronounced reorganization of spin correlations, leading to a crossover toward a doublon-like correlation regime and providing a static ground-state picture consistent with the composite excitations observed dynamically in trimerized chains. The resulting low-energy behavior can be interpreted in terms of weakly interacting emergent spins, offering a microscopic explanation for the extended stability of the magnetization plateau. Furthermore, through finite-size scaling analyses of the energy gap, von Neumann entanglement entropy, and fidelity susceptibility, we characterize the zero-field criticality of the model. Ultimately, our results suggest that frustration gives rise to qualitatively distinct quantum states and provide a microscopic framework for understanding the emergence of fractionalized excitations in trimerized quantum spin systems.

Trimerized Spin-

1/2

Chain: Emergent Low-Energy Hamiltonian, Higher-Energy Excitations, and Magnetic and Thermodynamic Responses

We investigate a spin-1/2 antiferromagnetic chain with trimerized exchange couplings relevant to Na

_2

_3

_4

_{12}

. We show that the low-energy Hilbert space maps onto an effective Heisenberg chain of composite spin-1/2 trimer degrees of freedom with positive coupling, enabling an exact spinon excitations via the Bethe ansatz. To access higher-energy dynamics, we develop a self-consistent Jordan-Wigner mean-field theory that yields three fermionic bands reflecting the underlying trimer structure. Remarkably, this approach reproduces the exact low-energy spinon continuum while predicting two additional higher-energy excitation bands consistent with the experimental observations and previous numerical simulations. The theory further captures the 1/3 magnetization plateau under an applied magnetic field, and provides testable predictions in magnetic susceptibility and specific heat. Our results establish a unified framework connecting low- and high-energy excitations in trimerized quantum spin chains.

Magnetic properties of Sr

{\mathrm{Cu}}_{2}

{\mathrm{O}}_{3}

and

{\mathrm{Sr}}_{2}

{\mathrm{Cu}}_{3}

{\mathrm{O}}_{5}

, containing two-leg and three-leg

S=\frac{1}{2}

ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, Sr

{\mathrm{Cu}}_{2}

{\mathrm{O}}_{3}

, is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of

{\mathrm{Sr}}_{2}

{\mathrm{Cu}}_{3}

{\mathrm{O}}_{5}

, containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate,

\frac{1}{{T}_{1}}

, of

^{63}\mathrm{Cu}

NMR also indicates the existence of a spin gap only for Sr

{\mathrm{Cu}}_{2}

{\mathrm{O}}_{3}

. The spin gap we observed for Sr

{\mathrm{Cu}}_{2}

{\mathrm{O}}_{3}

confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.