Magnon, doublon and quarton excitations in 2D <i>S</i>=1/2 trimerized Heisenberg models

doi:10.1007/s11467-024-1418-3

Front. Phys.

2024, Vol. 19

Issue (6) : 63202 https://doi.org/10.1007/s11467-024-1418-3

Magnon, doublon and quarton excitations in 2D S=1/2 trimerized Heisenberg models

Yue-Yue Chang¹, Jun-Qing Cheng²(

), Hui Shao^3,⁴, Dao-Xin Yao^1,⁵(

), Han-Qing Wu¹(

)

¹. Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, State Key Laboratory of Optoelectronic Materials and Technologies, Center for Neutron Science and Technology, School of Physics, Sun Yat-sen University, Guangzhou 510275, China
². School of Physical Sciences, Great Bay University, Dongguan 523000, China & Great Bay Institute for Advanced Study, Dongguan 523000, China
³. Center for Advanced Quantum Studies, Department of Physics, Beijing Normal University, Beijing 100875, China
⁴. Key Laboratory of Multiscale Spin Physics, Ministry of Education, Beijing 100875, China
⁵. International Quantum Academy, Shenzhen 518048, China

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Abstract

We investigate the magnetic excitations of the two-dimensional (2D) $S$ = 1/2 trimerized Heisenberg models with intratrimer interaction $J 1$ and intertrimer interaction $J 2$ on four different lattices using a combination of stochastic series expansion quantum Monte Carlo (SSE QMC) and stochastic analytic continuation methods (SAC), complemented by cluster perturbation theory (CPT). These models exhibit quasi-particle-like excitations when $g = J 2 / J 1$ is weak, characterized by low-energy magnons, intermediate-energy doublons, and high-energy quartons. The low-energy magnons are associated with the magnetic ground states. They can be described by the linear spin wave theory (LSWT) of the effective block spin model and the original spin model. Doublons and quartons emerge from the corresponding internal excitations of the trimers with distinct energy levels, which can be effectively analyzed using perturbative calculation when the ratio of exchange interactions $g$ is weak. In this weak $g$ regime, we observe a clear separation between the magnon and higher-energy spectra. As $g$ increases, doublon and quarton gradually merge into the magnon modes or some continua. Notably, in the Collinear II and trimerized Hexagon lattice, a broad continuum emerges above the single-magnon spectrum, originating from the quasi-1D physics due to the dilute connections between chains. In addition, we also compare our numerical results to the experimental RIXS spectrum and analyze the difference. Our numerical analysis of these 2D trimers yields valuable theoretical predictions and explanations for the inelastic neutron scattering (INS) spectra of 2D magnetic materials featuring trimerized lattices.

Keywords quantum Monte Carlo trimerized Heisenberg model

Corresponding Author(s): Jun-Qing Cheng,Dao-Xin Yao,Han-Qing Wu

Issue Date: 16 July 2024

Cite this article:

Yue-Yue Chang,Jun-Qing Cheng,Hui Shao, et al. Magnon, doublon and quarton excitations in 2D S=1/2 trimerized Heisenberg models[J]. Front. Phys. , 2024, 19(6): 63202.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-024-1418-3
https://academic.hep.com.cn/fop/EN/Y2024/V19/I6/63202

Fig.1 Four 2D trimerized lattices and their full Brillouin zones: (a1) Collinear I lattice, corresponding to a 2D square lattice in the

g = J 2 / J 1 = 1

limit. (b1) Collinear II lattice, derived from the structure of CaNi

3

2

7

)

2

[22]. (c1) Trimerized Lieb lattice exhibiting a ferrimagnetic ground state. (d1) Trimerized Hexagon lattice, derived from the structure of Ba

4

3

10

[23-27]. (e1) A topological equivalent lattice of trimerized Hexagon lattice. The green quadrilaterals in panels (a1)−(e1) represent unit cells. The Collinear I lattice comprises three sites and six bonds within a cell, while the other lattices feature three sites and four bonds per cell. The red bonds denote the intratrimer interactions, while the dark blue ones represent the intertrimer interactions. It is worth noting that all these neighbor bonds are antiferromagnetic interactions. For (a1)−(e1), the length of red bonds is set to be one (as the length unit), and the length of blue dashed bonds in (a1)−(c1) is also equal to one, while in (d1) and (e1), we set the length of blue bonds to be two. Besides, the lattices shown in (a1)−(c1) are also the clusters used in the CPT calculation with 24 sites (a1, b1) and 27 sites (c1). For panels (a2)−(e2), we illustrate the corresponding full Brillouin zone (BZ), and the shadow areas denote the reduced BZ. The point

T

(2 π / 3, 0)

, point

M 2

(3 π / 6, π / 6)

Y

(0, π)

and

Z

(0, π / 2)

Fig.2 The energy levels and the corresponding wave functions of an isolated trimer block. Its ground state is a doublet with effective spin-

1 / 2

| n ? S, m

denotes an eigenstate, where

n = 0, 1, a n d 2

represent the ground state, the first-excited state, and the second-excited state, respectively.

S

is the total spin quantum number, and

m

is the magnetic quantum number. The dashed lines denote the

| Δ m | = 1

excitations, and the solid lines are

| Δ S | = 1

excitations. The orange lines denote the excitations from

| 0 ? 12, 12

ground state and the blue lines are the excitations from

| 0 ? 12, − 12

ground state.

Fig.3 The extrapolations of square staggered magnetization on four lattices with (a)

g

= 0.1 and (b)

g

=1. Error bars are much smaller than the size of the symbols. The extrapolated values are shown directly on the horizontal axis. To avoid text overlap, we represented the intercepts of the trimerized Hexagon using arrows.

Fig.4 The dynamic spin structure factors of the Collinear I lattice, obtained through QMC and CPT methods, are presented with varying ratios

g = J 2 / J 1

. We follow a high-symmetry path

Γ (0, 0)

→

T (2 π / 3, 0)

→

X (π, 0)

→

M (π, π)

→

Γ (0, 0)

, as shown in Fig.1(a2), to show the dynamic spin structure factors. The solid lines are the LSWT results of its low-energy effective block spin model. In contrast, the white dotted lines correspond to the LSWT results of the original spin model. The dashed lines in the higher-energy parts denote the optimal dispersions obtained by the PA. Panels (a1)−(e1) display results from QMC-SAC, while panels (a2)−(e2) feature CPT results. For clarity, in the QMC-SAC results, a piecewise function is utilized where the intensity is bifurcated at

U 0 = 5

. When the value is less than 5, the low-intensity section follows a linear distribution. Beyond this threshold, a logarithmic scale is applied, expressed as

U = U 0 + log 10 S (q, ω) − log 10 U 0

. However, no additional transformation is applied to the CPT spectrum. This same piecewise function is also used in the subsequent excitation spectra of other lattices.

Fig.5 Dynamic spin structure factors of the Collinear II lattice for different

g

values. The pink solid lines represent the magnon dispersion of the corresponding effective model formed by the trimer block spins, while the white dotted lines illustrate the LSWT results for the original model. The yellow and green dashed lines represent the dispersions of quarton and doublon, respectively, obtained from the PA in weak

g

. Panels (a1)−(e1) show results obtained from the SAC method, and (a2)−(e2) are the results obtained from the CPT. The high-symmetry path is the same as the Collinear I model as shown in Fig.1(b2), and we use the same logarithmic scale in SAC results when

S (q, ω) > U 0 = 5

, expressed as

U = U 0 + log 10 S (q, ω) −

log 10 U 0

Fig.6 Dynamic spin structure factors of the trimerized Lieb lattice at various

g

values. The pink solid lines represent the magnon dispersions of the corresponding effective model formed by the trimer block spins. The white dotted lines illustrate the LSWT results of the original model. The yellow and green dashed lines represent the dispersions of quarton and doublon, respectively, obtained from the PA in weak

g

. Panels (a1)−(e1) show results obtained from the QMC-SAC method, while (a2)−(e2) present results obtained from CPT. The high-symmetry path is shown in Fig.1(a2), and we use the same logarithmic scale in QMC-SAC results when

S (q, ω) > U 0 = 5

, expressed as

U = U 0 + log 10 S (q, ω) − log 10 U 0

Lattice	Excitation	Dispersions

Collinear I	Doublon	$? = − E 0 + E 1 + 7 g / 9$
	Doublon	$? = − E 0 + E 1 + 5 g / 9$
	Quarton	$? = − E 0 + E 2 + 4 g / 9 + 2 g cos ? (3 q x) / 9 + 2 g cos ? (q y) / 3$
		$? = − E 0 + E 2 + 5 g / 9 + g cos ? (3 q x) / 18 + 2 g cos ? (q y) / 3$
		$? = − E 0 + E 2 + g / 9 + g cos ? (3 q x) / 18 + 2 g cos ? (q y) / 9$
Collinear II	Doublon	$? = − E 0 + E 1 + g / 9 + g cos ? (q x) / 33$
	Quarton	$? = − E 0 + E 2 + g / 9 + g cos ? (3 q x) / 9 + 4 g cos ? (q y) / 9$
	Quarton	$? = − E 0 + E 2 + g / 9 − g cos ? (3 q x) / 63 − 2 g cos ? (q y) / 33$
Trimerized Lieb lattice	Doublon	$? = − E 0 + E 1 − 2 g / 9$
Trimerized Lieb lattice	Quarton	$? = − E 0 + E 2 + 7 g / 18 − g cos ? (2 q x) / 3 − g cos ? (2 q y) / 3$

Tab.1 The optimal dispersions in the different lattices.

Fig.7 Dynamic spin structure factors in the reduced BZ of the trimerized Hexagon lattice at different

g

values: (a)

g = 0.1

, (b)

g = 0.5

, (c)

g = 0.8

, and (d)

g = 1

. The pink lines illustrate the magnon dispersion obtained from LSWT using the low-energy effective block spin model. The white lines represent the LSWT results of the original model. The high-symmetry path is shown in Fig.1(d2), and we use the same logarithmic scale in SAC results when

S r e d z z (q, ω) >

U 0 = 5

, expressed as

U = U 0 + log 10 S r e d z z (q, ω) − log 10 U 0

Fig.8 Dynamic spin structure factors of spin model shown in Fig.1(e1) along the path

Γ → Y (0, π)

at different

g

values: (a)

g = 0.1

, (b)

g = 0.3

, (c)

g = 0.5

, (d)

g = 0.7

, (e)

g = 1

, (f)

g ′

= 0.1, (g)

g ′

= 0.3, (h)

g ′

= 0.5, (i)

g ′

= 0.7, and (j)

g ′

= 1. The white lines represent the LSWT results of the original model. The high-symmetry path is shown in Fig.1(e2), and we use the same logarithmic scale in SAC results when

S (q, ω) > U 0 = 30

, expressed as

U = U 0 + log 10 S (q, ω) − log 10 U 0

Fig.9 The low-energy effective block spin models of each 2D trimerized lattice when the value

g

is small. (a) Collinear I lattice; (b) Collinear II lattice; (c) Trimerized Lieb lattice; (d) Trimerized Hexagon lattice. The light blue and orange lines represent the effective interactions between trimer blocks along two primitive vectors of unit cells. Here,

J h

represents an interaction along the horizontal direction of the primitive cell, while

J v

is associated with the vertical direction. For the trimerized Lieb lattice, the effective interaction is ferromagnetic.

⇑

represents the effective block spin

S = 1 / 2

of each trimer. The arrays of

⇑

and

⇓

show the magnetically ordered ground state of each effective model.

Fig.10 The perturbation dispersion relations for

g

=0.1. (a) Collinear I lattice; (b) Collinear II lattice; (c) Trimerized Lieb lattice; (d) Trimerized Hexagon lattice. The green lines denote the doublons, and the orange ones denote the quartons.

Fig.A1 Test of a synthetic spectral function (black curves) using correlation data with error levels from

10 − 3

10 − 6

. The sampling temperature is decided according to Eq. (A1) with different values of

a

Fig.A2 Spectral functions for the Collinear II model with momenta along the

Γ

−T path and close to the

T

point.

Fig.A3 Spectral functions for the Collinear II model with

q = (15 π / 24, 0)

and different sampling temperatures.

Fig.A4 Dynamic spin structure factors of the trimerized Collinear II lattice at different vertical interchain

J 2 ′

interactions: (a)

J 2 = 1

and

J 2 ′ = 0.1

, we show the interactions in the inset. (b)

J 2 = 1

and

J 2 ′ = 0.3

, (c)

J 2 = 1

and

J 2 ′ = 0.5

, (d)

J 2 = 1

and

J 2 ′ = 0.7

and (e)

J 2 = 1

and

J 2 ′ = 1

. The white lines represent the LSWT results. The high-symmetry path is shown in Fig.1(a2), and we use the same logarithmic scale in the SAC results when

S (q, ω) > 5

, expressed as

U = U 0 + log 10 S (q, ω) − log 10 U 0

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