1. Introduction
The idiom "Jing Wei Is Clear", from the "Poetry Classic, Shao Feng, Gu Feng": "Jing is turbid, and it is muddy". The idiom describes the turbidity of the Jing River and the clear water of the Wei River, but when the water of the Jing River flows into the Wei River, it is clear and turbid, and the figurative boundaries are clear or clear.
In the tide of history, the distinction between the Jing River and the Wei River is exchanged from time to time because of human activities. Here, we will not study which river is clear water and which river is turbid water, that is not the direction we are good at. The introduction of "clear distinction" here is only to draw the reader's attention to an important scientific problem that we are going to present here. The topic presented in this article is the physics of magnetic excitation of quantum magnetic systems. The following will mainly use the turbidity of the Jing River and the water of the Wei River to compare high and low magnetic excitation, so as to illustrate the difference between the high and low magnetic excitation of the antiferromagnetic trimer chain.
Before explaining in detail the difference between high and low energy magnetic excitation, it is necessary to lay out some basic content so that the reader can more clearly understand what our theme is, and where the purpose and significance of our research are.
2. Magnetic excitation - spin dynamics of magnetic materials
Solid materials are composed of a large number of particles (such as electrons, ions). When there is a strong interaction between these particles, the system forms a complex quantum multibody system, showing a rich material phase and phase transition. If the electron coulomb interaction is stronger, the system will become a kind of Mott insulator. At this point, the electrons lose their cruising nature, the local area is in the orbit around the ion real, and there is a correlated interaction between the electron spin or orbital magnetic moment around the adjacent ion real. This image indicates that this is a magnetically strongly correlated system that exhibits a macroscopic magnetic or non-magnetic ground state.
At present, the means of experimental detection of material properties are based on the application of some kind of stimulus disturbance to the system (light, electricity, neutrons, etc.) to bombard the material), and the system responds to the stimulus. We invert the intrinsic properties of the material according to the input and output signals, combined with the physical laws of energy and conservation of momentum. For example, we throw a stone into a pond and a large vat filled with asphalt: the asphalt is obviously difficult to make big waves, while the pond may be magnificent. Depending on the degree of the waves, we know that the viscosity of water and asphalt is not the same. In the study of condensed matter physics, there are many such stones, one of which is inelastic neutron scattering: the use of neutrons with magnetic moments to bombard the material, apply perturbation, and then look back and forth at the magnetic excitation behavior of the magnetic system according to the experimental scattering spectrum. Physics research, is not so logical, tried and tested.
Of course, the experiment to obtain this magnetic excitation spectrum is only the first step, how to understand the magnetic system intrinsic properties behind this magnetic excitation spectrum, need the support of theoretical physicists. Theoretically, strongly interacting quantum multibody systems are so complex that it is not easy to solve them rigorously. Here's a simple example: the one-dimensional model we'll calculate later has 192 interacting spins, each = 1/2 of a spin representing two degrees of freedom (with two local degrees of freedom). If the full spectrum and wave function of the schrödinger equation of the system are strictly solved, a 2-dimensional matrix needs to be diagonized. In this way, even if the world's supercomputers are used, they will not be able to solve.
There is, of course, a solution to this problem. We can use the quantum Monte Carlo method to do random sampling to obtain the physical information of the system; or simplify the complex problem, through the Fourier transform, do the average field in the reciprocal space to study the magnetic excitators. For an explanation of reciprocal spaces, see our related popular science article "Beat Resistance And Frustration Magnetism" (click to read) [1].
With these numerical methods for quantum multibody systems, physical humans can perform theoretical simulations of the magnetic excitation spectra of materials to study the spectroscopic characteristics of various excitation modes. By comparing and cross-examining the experimental results, we can have a deeper understanding of the eigen-physical properties of the material. If so, it is a wonderful thing.
The study of magnetic excitation spectroscopy, in addition to satisfying the curiosity of physical people about physical problems and the exploration of scientific knowledge, it always has to be of some use. Of course, the practical application value of magnetic excitation spectroscopy is also great, including application expectations at different levels. For example, magnetically excited quasiparticles (such as spin waves, etc.) of magnetically ordered systems can be used to make low-power spintronic devices. Non-magnetic ground states, such as quantum spin liquids, can also exhibit fractional magnetic excitation phenomena such as "spinons". Due to its strong anti-interference ability and good coherence length, this spinon can become an ideal carrier for topological quantum computing and quantum storage in the future. Combining the two, magnetic excitation research, has always been an important frontier scientific problem in the current condensed matter physics.
3. Heisenberg model – a basic physical model for the study of magnetism
In 1981, Ludwig D. Faddeev and Leon A. Takhtajan used the Bethe ansatz to reveal that the metaexcitation of a one-dimensional spin = 1/2 antiferromagnetic Heisenberg chain was a spinon [2]. In 1983, Nobel laureate F. Duncan M. Haldane discovered the main difference between integer spin chains and half integer spin chains [3]. These achievements have led to a surge of interest in new quasi-one-dimensional magnetic materials, which has greatly promoted the development of corresponding theoretical and experimental technologies.
Later, high-temperature superconductivity and neutron scattering brought new vitality to the study of low-dimensional materials. The Heisenberg model, with one-dimensional spin = 1/2, is an important example of describing low-dimensional quantum magnetism. For example, E8 particles [4] were observed in BaCoVO, a quasi-one-dimensional antiferromagnetic material with a horizontal field. The Hamiltonian quantities of the isotropic Heisenberg model are:
where the interaction strength, the system satisfies SU(2) and translational symmetry. By betted, a continuous spectrum of two spinons whose element excitation is an impotent gap can be obtained [2]. Figure 1 shows the simulation results of Quantum Monte Carlo with the nether ω= π · J |sin(q)| / 2 and upper bound ω= π · J |sin(q / 2)|。 The double spintron continuous spectrum here is derived from a spin flip in the antiferromagnetic ground state to form | ΔM| Excited of = 1. Since one-dimensional systems are not affected by inter-chain interactions, | ΔM | The excitation of = 1 splits into two domain walls, i.e. two spinons. Each spindle, carrying a spin of 1/2, can move freely on the chain [5].
The motion of the spinon is an important basis for studying the spin-charge separation predicted by the Luttinger liquid theory [6]. Figure 7(a) below shows the formation of two spinons. The two spinners are like twins out of the palms of their parents, running freely in both directions on a narrow path.
Figure 1. Quantum Monte Carlo simulation of a double spinon continuous spectrum.
4. Kinetic structural factors - a bridge to communicate inelastic neutron scattering experiments
Spin dynamic structural factors contain spatial and temporal correlation information about system fluctuations, which is a very important link between theoretical and experimental research on magnetic excitation. Its definition can be expressed as:
where wave vector and ω is frequency. Usually in theoretical calculations, we all set Planck's constant = 1, so there is E = ω = ω, where ω reflects the magnitude of the energy. This dynamic structural factor is obtained by doing a double Fourier transform (real space momentum space, time energy) on the spin correlation function. From this, it can also have another definition:
where |ψ> represents the first eigenstat, and the corresponding eigenvalue is; |ψ> is the ground state, and the corresponding energy is . Refers to the spin operator in the momentum space.
The theoretical calculation results of the dynamic structural factor can be directly compared with the results of inelastic neutron scattering and resonance inelastic X-ray scattering experiments, so they are favored by both theoretical and experimental physicists. It is worth mentioning that the high-energy direct geometric inelastic neutron scattering time-of-flight spectrometer (referred to as "Zhongda spectrometer") built by Sun Yat-sen University is the first high-energy inelastic neutron scattering spectrometer in China, and the first material dynamic properties research spectrometer in China with the ability to detect medium and high-energy magnetic excitation spectra and phonon spectroscopy [7]. The ZHONGDA spectrometer is expected to be put into use this year, and experimental and theoretical physicists can't wait for the day when this scientific research device will start, looking forward to new research discoveries in research such as medium and high energy magnetic excitation.
Figure 2. Schematic diagram of the structure of the medium and large spectrometer [6].
5. Medium and high energy magnetic excitation - colorful quasiparticles
Magnetic excitation embodies the spin dynamics of matter, which plays an important role in understanding the mechanism of high-temperature superconductivity, exploring quantum entanglement between spins, and guiding the development of magnetic electronic devices. Conventional wisdom holds that the magnetic excitation of a magnetically ordered system is primarily a spin wave (Magnon). However, for some high-energy excitation spectra with clustered quantum spin systems, the spin wave description always lacks credibility, and the corresponding high-energy quasiparticle excitation mechanism is always unclear.
Figure 3. (a) Schematic of a spin model of a 3 x 3 checkerboard structure, magnetic excitation spectrum at (b) = / = 0.1 [9].
Compared with low-energy excitation, high-energy excitation research on quantum magnets is still relatively lacking. High-energy excitation is not a simple spin wave, and sometimes abnormal excitation characteristics appear, so it attracts more and more physical interest. For example, in LaCuO, a high-temperature superconducting antiferromagnetic material, the high-energy excitation observed by inelastic neutron scattering is anomalous: the high-energy spin wave is multiplied into a spinon in the inverted space (1/2, 0). In 2019, Professor Yao Daoxin's team studied spin excitation spectra of two-dimensional models of different checkerboard structures [9], as shown in Figure 3. The excitation spectrum of the 3 x 3 structure (also called the Tian Zi Lattice) is voidless, forming a continuous spectrum with a periodic structure at low energies. This is because each cluster contains an odd number of spins that can be reformulated to a single spin = 1/2, with a clear energy gap between the high and low spectrum. The high-energy part is no longer the spin wave excitation, and what the specific quasiparticles are is still inconclusive.
In order to explore the high-energy excitation mechanism, we may wish to simplify the above two-dimensional model to a one-dimensional trimeric antiferromagnetic spin chain to find the physical laws in high-energy excitation.
Figure 4. Schematic of a one-dimensional antiferromagnetic trimer spin chain system [12].
Figure 5. Momentum spatial magnetic excitation energy spectra obtained by the quantum Monte Carlo method [12].
6. Antiferromagnetic trimer spin chain - fractional combined excitation
Using quantum Monte Carlo, precision diagonalization, and perturbation theory analysis, we systematically studied the magnetic excitation properties of antiferromagnetic tripolean spin chains (see Figure 4).
(1) By changing the ratio of the inter-trimer coupling intensity to the trimer coupling intensity = /, we show that the system has a very rich magnetic excitation spectrum (see Figure 5).
(2) In the efficient Hilbert space formed by the trimeric degenerate ground state, an efficient model describing the low-energy fraction can be obtained, which is the Heisenberg spin chain = 1/2. Thus, when = 1, the double spintron continuous spectrum with width ~ evolves into a similar continuous spectrum with width ~ at g0.
(3) Due to the weaker trimer interaction, the system excitation is a localized excitation in the trimer, and the medium energy and high energy modes cannot be described by spin waves or spintrons. According to the corresponding excitation mechanism, medium-energy and high-energy quasiparticles are called Doublon and quadreticons (Quarton), respectively. The "heavy" reading chóng, the second sound, represents degeneration, is not the heavy zòng in particle physics, reading the fourth sound. These novel high-energy magnetic excitations go beyond traditional renormalization descriptions.
Figure 6. The Jing River meets the Wei River [10].
Now, back to our theme: high- and low-energy magnetic excitation.
When = 0.1, as shown in Figure 5(a), the low-energy excitation presents a periodic structure corresponding to an efficient antiferromagnetic Heisenberg chain with uniform interaction. Therefore, low-energy quasiparticles are still spinons, carrying 1/2 of the spin. The low-energy excitation here, because it contains only spinons, is clear about the corresponding excitation mechanism, as clear as the Wei River (as shown in Figure 6).
The two approximate flat bands of medium and high energy are derived from the excitation inside the trimer, and the corresponding quasiparticles are no longer common magnetic oscillators. According to the excitation mechanism, the medium-energy and high-energy quasiparticles are named "double-son" and "quadruplet", respectively, carrying spin = 1. Figure 7 shows a diagram of the real spaces of spinons, doubles, and quadratons.
The so-called spintron, carrying a spin 1/2, is formed by flipping a spin. They usually appear in pairs, moving freely on a one-dimensional spin chain. In the schematic of a double and quadruplet, the arrows represent the effective spin of a trimer. The flip of the effective spin means a jump in the energy level of the trimer. As shown in Figure 7(b), the effective spin in the middle blue flips upwards, representing the transition of the excited trimer from the ground state to the first excited state, resulting in a |ΔM | Excited of = 1. The effective spin that is flipped and the active spin around it form two domain walls, two spinons.
The excited trimer, which has some binding effect on these two spinons, is no longer the free spinon shown in Figure 7(a). Eventually, a trimer is formed around the excited trimer, accompanied by two spinons that travel together over the spin chain. The double son is not fractionated , so the spin carried is = 1.
Similarly, high-energy excitation can be explained by energy level transitions in trimers. The difference is that there are two cases during quadratic formation: with spinon accompaniment and without spinon accompaniment (as shown in Figure 7(c) and (d), respectively). The spin carried by the quadruplet produced by high-energy excitation is also = 1. Therefore, the dual-electron excitation of medium energy and the quadretic excitation of high energy are both combined excitations.
When = 0.2, as shown in Figure 5(b), medium and high energy first fuse to form a new high energy spectrum. At this time, the high-energy spectrum contains double, quadruplet and spinons, which are as turbid as the Jing River and are difficult to understand. As it increases to around 0.7, as shown in Figure 5(f), the energy gap between the high and low energy excitation spectra closes, the water of the Jing River flows into the Wei River, and various quasiparticles in the high-energy part begin to fractionate into spinons. This continues until = 1, as shown in Figure 5(h), the system evolves into a uniform Heisenberg spin chain. At this time, all the high-energy quasi-particles are fractionated into spinons, and the waters of the two rivers are finally integrated into one, and there is no longer a distinction between them.
Of course, we expect the water to be clear at this point, because this would perfectly match the description of only a quasiparticle with a spinon at = 1. It should be emphasized that the introduction of clear distinction here is only to illustrate the difference between high and low energy excitation, not to illustrate that the boundaries are still clear after the fusion of high and low energy.
Figure 7. Schematic of spinons, doubles, and quadreticons in real space [12].
In the material, if it is possible to identify having a linear trimer structure and
7. Conclusion
Quantum magnetism is an extremely important scientific frontier in the current condensed matter physics. It is closely related to new states of matter, high-temperature superconductivity, quantum computing, etc., and has important value in both basic and application.
In quantum magnetic systems, the competition between quantum fluctuations and interactions will not only produce novel magnetic ground states, such as quantum spin liquids, spin glass, valence solids, etc., but also produce extraordinary magnetic excitation behaviors, such as fractional excitation, chord excitation, multimagnetic oscillator excitation, topological excitation, etc. These physical phenomena, and their deep understanding, have potentially important value in areas such as physics and information science.
In this study, we found a non-simple high-energy excitation mechanism in a seemingly simple physical model. High-energy excitation is like the world under the ice, and it is difficult to find out whether it is colorful without breaking the ice. The ice-breaking process, similar to the road of exploration, is long and arduous. But the desire for the unknown always brings with it an unquenchable zeal, driving researchers to discover new continents.
This work, which is currently available here, was recently published in NPJ Quantum Materials 7, 3 (2022) under the title "Fractional and composite excitations of antiferromagnetic quantum spin trimer chains". Interested readers should click on the "Read the original" link at the end of the article, or copy the bracketed link (https://www.nature.com/articles/s41535-021-00416-4).
bibliography
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remark:
(1) The author, Yao Daoxin, is a professor at the School of Physics, Sun Yat-sen University. Professor Yao Daoxin's research group website http://spe.sysu.edu.cn/node/324 (Chinese version), http://spe.sysu.edu.cn/yao/ (English version). The author, Cheng Junqing, once served as a postdoctoral fellow in Professor Yao Daoxin's research group, and now works at the University of the Greater Bay Area. Thanks to Associate Professor Wu Hanqing, he has made revisions and useful additions to this article.
(2) Other authors of the papers recommended here include: Li Jun, who was an associate researcher in Professor Yao Daoxin's research group and now works at Yanshan University; Xiong Zijian was a doctoral student in Professor Yao Daoxin's research group and is now working at Chongqing University; Wu Hanqing, an associate professor at the School of Physics, Sun Yat-sen University; W. Sandvik is a professor at Boston University.
(3) The animation and cover image at the beginning of the article are from a presentation of the work (20210331) at Oak Ridge National Laboratory in the United States: Spin chains in a quantum system undergo a collective twisting motion as the result of quasiparticles clustering together. Demonstrating this KPZ dynamics concept are pairs of neighboring spins, shown in red, pointing upward in contrast to their peers, in blue, which alternate directions. Credit: Michelle Lehman / ORNL, U.S. Dept. of Energy. The link is: https://www.ornl.gov/news/quantum-materials-subtle-spin-behavior-proves-theoretical-predictions.
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