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L.A. Kulak
The quantum world conceals a geometry whose foundation is essentially universal. This geometry governs and synchronizes all processes in the quantum world. Based on this geometry, Nature further constructs itself, creating stable blocks—atoms—from which the observable material world is constructed. The observable world, like a fractal, reflects all processes occurring in the quantum world. Based on this logic, the geometry of the quantum world can be comprehended both theoretically and experimentally, opening the door to an understanding of the original geometric foundation, subsequently the deepest mysteries of Nature, resolving the difficulties of physics itself and giving new impetus to technical development, studying what Nature shares with us.
Quantum materials, whose electronic, magnetic, and optical properties are determined not only by the structure of the crystal lattice but also by the quantum mechanical effects that arise within them, help us understand the geometry of the quantum world. [154]
In quantum materials (superconductors and quantum Hall effect systems), quantum mechanical effects manifest themselves on a macroscopic scale as a collective action in the crystal lattice of the matter at low temperatures or high magnetic fields.
«The Berry curvature is part of the quantum geometry of a material. The QHE is the archetypical demonstration that quantum geometry is one of the key quantities that make a material a quantum material. As we will discuss in the remainder of the review, the Berry curvature turns out to be the anti-symmetric part of a tensor Q, the quantum geometric tensor (QGT) [2]. In recent years it has become apparent that the symmetric part of this tensor, the quantum metric, g, also plays a key role in making a material, quantum. In a loose sense, the quantum metric appears to be the key quantity to understand the properties of materials in which both interactions and quantum geometry lead to macroscopic manifestations of quantum mechanics». [5]
Quantum geometry plays a key role in various physical phenomena in quantum materials, particularly optical responses, Landau levels, and fractional Chern insulators. Its influence on superfluid weight, spin stiffness, exciton condensates and electron-phonon coupling is also studied and discussed. Taking these aspects into account, quantum geometry has a pervasive significance for understanding the emergent behavior of quantum material structures, which explains its rapid development in the current period. [5]
The tool of quantum geometry, characterized by the quantum geometric tensor, is a set of mathematical rules for describing physical phenomena (changes in the shape of internal structures and properties in a material) which occur on sub-micro-scales, but in its macro-effects, when exposed to external fields. From a practical standpoint, the laws of change studied provide the key to controlling the properties of semiconductor materials and to creating new ones with superconductivity. These aspects are also important for both energy supply and the creation of quantum computers. Quantum geometry also serves as the basis for the theory of quantum gravity, which aims to unite quantum mechanics and general relativity into a single, unifying theory of everything.
A group of scientists from the Massachusetts Institute of Technology (MIT), Seoul National University, and their colleagues have proposed a new approach to measuring quantum geometry in crystalline materials [1]. Their method can be considered groundbreaking. Based on photoelectron spectroscopy (ARPES), it opens new horizons in the study of quantum states. A quantum state can be represented as a landscape. The map of this landscape is the quantum geometric tensor (QGT), which shows how the landscape changes when various factors influence a substance, reveals the behavior of the material, and allows insight into the structural essence of matter and the factors that shape its properties. This new method opens up promising possibilities for materials research and enables the linking of theoretical calculations with real experimental data.
In the news [2] Elizabeth Thomson introduces the details of the experiment [1] and provides a number of quotes, in particular from the project leader Riccardo Comin, an associate professor of physics at the Massachusetts Institute of Technology (MIT).
The work done is based on the description of the wave hypostasis of the electron."You can think of it like a surface in a three-dimensional space, - says Comin. - There are different types of wave functions, ranging from the simple to the complex. Think of a ball. That is analogous to a simple, or trivial wave function. Now picture a Mobius strip. That's analogous to a complex, or non-trivial wave function. And the quantum world is filled with materials composed of the latter”. [2]
“Until now, the quantum geometry of wave functions could only be inferred theoretically, or sometimes not at all. And the property is becoming more and more important as physicists find more and more quantum materials with potential applications in everything from quantum computers to advanced electronic and magnetic devices”. [2]
The results of the previous experiment by the same group are published in [3][4]. Other detailed information about this experiment [1] can also be found in the publication Shalma Wegsman in Quanta Magazine. [10]
“Topological materials are no longer so mysterious: Physicists commonly use them to discover new phases of matter and are exploring their potential for quantum computing. But only recently have they come to appreciate the fuller picture of quantum geometry that includes not only the Berry curvature but also the quantum metric – the shape of the rugged landscape that can exist on top of a torus-shaped map. A couple of years ago, the quantum metric helped researchers understand what was going on in a 2D crystal that hosted an exotic new form of superconductivity – the flow of electricity with no resistance.
The physicist Riccardo Comin and collaborators recently mapped the full quantum geometry of a crystal for the first time. Päivi Törmä helped show how quantum geometry can explain exotic material behaviors such as superconductivity”. [10]
Recent publications on quantum geometry in the literature: [5]-[8][11][152]-[155]
The tutorial review [5] introduces the concept of quantum geometry and highlights its implications in various fields, including optical responses, Landau levels, fractional Chern insulators, superfluid mass, spin stiffness, exciton condensates, and electron–phonon interactions. "By integrating these topics, we emphasize the broad significance of quantum geometry in understanding emergent behaviors in quantum systems and conclude with an outlook on open questions and future directions". [5]
The review [6] examines how quantum geometry influences materials properties and describes recent experimental advances that have paved the way for the study of quantum geometric effects in various condensed matter platforms. Scale separation, which allows one to appreciate the significance of quantum geometry in various response functions, is also discussed.
In the publication [7], it is demonstrated within the framework of density functional perturbation theory “that the Berry curvature of electron-phonon coupling (EPC) plays a pivotal role in the interatomic force matrix. A subsequent model analysis shows that external strain can reverse the polarity of the EPC Berry curvature in (quasi)degenerate electronic subsystems through band inversion, thereby directly leading to phonon softening. The general theory is then applied to the BiOCl monolayer as a benchmark, which offers an accurate description of the density-functional theory calculations. This mechanism is further observed across a broad range of materials through ab initio calculations, providing an insightful perspective on EPC quantum geometry in lattice dynamics and FE phase transitions”.
The review [8] examines the influence of quantum geometry on the properties of materials and describes recent experimental advances that have paved the way for the study of quantum geometric effects in various condensed matter platforms. Scale separation, which allows one to assess the significance of quantum geometry in various response functions, is also discussed.
“Electronic properties of quantum material solids are often well understood via the low-energy dispersion of Bloch bands, motivating single-band approximations in many metals and semiconductors. However, a closer look reveals length scales and timescales introduced by quantum dipole fluctuations due to interband mixing, which are reflected in the momentum-space textures of the electronic wavefunctions. This structure is usually referred to as quantum geometry. These new scales not only qualitatively modify the linear and nonlinear responses of a material but can also have a vital role in determining the many-body ground state at low temperatures. In this Perspective, we explore how quantum geometry affects properties of materials and outline recent experimental advances that have begun to explore quantum geometric effects in various condensed matter platforms. We discuss the separation of scales that can allow us to estimate the significance of quantum geometry in various response functions.”[8]
In [153] researchers focus on the next: “Topological quantum materials (TQMs) have symmetry-protected band structures with useful electronic properties that have applications in information, sensing, energy and other technologies. In this Perspective, we present an overview of recent progress in topological quantum catalysis. We describe the open problems and the potential applications of TQMs in water splitting, batteries, supercapacitors and other prospects in energy conversion and storage”.
In a recent publication [152] “How quantum is quantum geometry?” the authors mentioned: “The quantum geometric tensor – the Berry curvature together with the quantum metric – now underlies a long list of observables, from the anomalous Hall effect to the superfluid weight of a flat band. We ask which of these observables actually require quantum mechanics”. [152]
Their final conclusion is: “Two consequences of quantum geometry are inherently quantum mechanical. The first is the magnetization as a thermodynamic property. The single-particle orbital moment of Eq. (12) is classical, but by the Bohr–van Leeuwen theorem [25] the equilibrium magnetization is quantum mechanical. The second is superconductivity as a Bose condensate of pairs. Our two particles acquire a joint inertial mass and hence a Drude weight in the absence of scattering, but nothing in Eq. (20) produces off-diagonal long-range order, phase rigidity, or a Meissner effect. The flat-band statement that the superfluid weight is bounded by the integrated metric thus splits into a classical part – the inverse mass – and a quantum part – the condensation. Beyond these two consequences, quantum mechanics also introduces the periodic Brillouin-zone structure to momentum space, which underlies the topological consequences of quantum geometry.
To summarize, our analysis shows that most phenomena associated with quantum geometry are in fact of a classical origin, and originate from the interplay between slow and fast degrees of freedom of a dynamical system.” [152]
In a recent essay, “Where Can Quantum Geometry Lead Us?” [9], Päivi Törmä, a professor of physics at Aalto University, Finland, notes many important aspects of quantum geometry and also identifies current issues in this new science:
“It has been theoretically predicted that quantum geometry governs a variety of physical phenomena.
Once a few smoking-gun experiments on quantum geometry effects have been achieved, the focus should swiftly shift from showing that “quantum geometry is there” to how can we utilize this concept to better understand nature and, eventually, to create new technologies.
To proceed on the quantum geometry road, we should implement the extraction – and smart visualization – of quantum geometric concepts as a standard functionality of these tools.
Despite challenges, our goal should be that the most powerful numerical methods of condensed matter physics will provide the essentials of the quantum geometry of the bands, properly visualized, as easily as they give the energies. Once this becomes a routine, we will start seeing things from the perspective of quantum geometry, like we now do from the band structure’s, and this will be a source of understanding and discovery.
A whole new theory framework is needed to capture the physics-relevant quantum geometry aspects even in cases where the standard concepts cannot be used. Inspiration for this search can be found from the effective mass tensor of a two-body bound state in a system of multiple nonisolated bands where the quantum geometric contributions and band dispersions intertwine [30].
Finally, widening means also that we condensed-matter–quantum-geometry enthusiasts should make friends with the communities that have been studying quantum geometry for a long time—high energy physics, cosmology, and quantum information—and dive into their literature [31]. The danger of drowning there is of course considerable, but the embarrassment of reinventing too many wheels is an even more daunting prospect.
It remains to be seen whether quantum geometry, including both quantum metric and the Berry curvature and thus topological physics, will play an equally large role in our world. What is clear is that the world needs out-of-the-box discoveries: world energy production and consumption are literally burning questions, as are scarcity of materials and several other gut-wrenching problems. Innovations as big as the transistor are much more urgently needed now than at the time transistor was invented. Therefore, we should try all possibilities.
Perhaps the most notable prospect from quantum geometry is to guide the way to room temperature superconductivity. The interplay of light, electronic transport, and quantum geometry intrigues me as well, because photovoltaics is another area where major technological advances can have world-changing impact. There may be other equally important goals for which quantum geometry can provide guidance, but my imagination stops here.
I give the floor to the reader who probably has a different background and can thus come up with different ideas”. [9]
Well, if the floor is given to a reader with a different background and the potential to offer new ideas, then perhaps it's simply impossible not to take advantage of it. My thanks to Professor Päivi Törmä for this suggestion. And if I'm not alone in this intention, then we can expect a kind of brain injection of other experiences and ideas that undoubtedly exist and could be a worthy contribution to the overall perspective.
Whether my ideas and experiences are directly related to the format of quantum geometry and the experiments guiding its development, as well as to the construction of a general theory, will naturally be answered at the end of this essay. I believe that the model of the atom I proposed, based on the nontrivial topology of magnetic dipoles, their geometry, and the interrelationships that hold the atomic structure of atom together, is directly relevant to the issues of quantum geometry, which studies macro-effects in quantum materials.
First of all, my ideas and research results concern the issues of atom structure, which is geometrically determined, and the reasoning begins from the structure of the cosmic medium and the initial geometry of its matrix unit as a physical point of space, and, accordingly, the geometry of quanta, and then the subatomic geometry that determines the properties of matter. In accordance with this, logic moves toward questions of energy extraction, as well as fundamental theoretical generalizations. It's worth discussing how all of this is connected.
Wave geometries are mediated by the structure of the cosmic medium and are determined by the processes occurring within it. If we take into account the work of our predecessors regarding the structure of the cosmic medium, as well as the achievements of our contemporaries [17][18][19][20][21][22][23][24][25][26][79][80][83][84][85], then the cosmic medium can be called the Quantum Medium with all its contents, the basis of which is a matrix of dynamic spinning units [19-21][22][23-25]. The nature of these units is magnetic and represents a living cosmic plasma according to how Nobel laureate H. Alfvén defined it [12][13][14][15], and Nobel laureate F. Wilczek also expressed related thoughts about the magnetic ingredients of the cosmic medium. [16]
In my proposed self-consistent concept [89][90][91][92][93], such a living plasma is called magneto-plasma, the magnetic units of which perform continuous and unceasing oscillations (spinning), caused by the geometry of their complementary structure and its properties. [90][31]
According to my new ideas presented in the proposed concept based on the ontology of the principle of complementarity, these magnetic units are magnetic loops that include inversion, in other words, a 180-degree reversal. Such a geometric representation of the original magnetic loop in the format of the nontrivial topology of the Möbius strip as a starting point for an entire conceptual framework has not yet been proposed by any scientist.
The basis for this concept is the scientifically established fact of the all-encompassing nature of the principle of complementarity, which is the primary principle. The ontological essence of this primary principle is the dual complementary interrelationship of two mirror-asymmetric opposites within a single whole, traditionally illustrated by the ancient Yin-Yang symbol. As the ancient pictorial symbol visualizes, the complementary relationship is the original natural duality, organized by a mirror-asymmetrical interrelationship of interconnected components in their mutual cyclical movement.
Magnetism is the physical embodiment of the principle of complementarity and is also comprehensive in nature. The magnetic loop must be displayed with the inclusion of the designated complementary duality in its representation. This is achieved geometrically using a Möbius strip with one 180-degree turn initially.
The complementary principle of combining the geometric image of a cyclical closed movement of energy with its mirror-asymmetric image, which underlies the construction of a one-sided Möbius strip, allows it to be considered an analogy to the Yin-Yang symbol, which reflects the principle of complementarity in a two-dimensional format, and the Möbius strip, respectively, in a three-dimensional format.
In the concept I propose, the ontology of the complementarity principle is the starting point from which all subsequent conceptual reasoning proceeds, and its geometric analogue – the Möbius strip – and its topological properties provide the necessary visualization of all transformations of a dynamic physical unit in space on such a nontrivial topology.
What is the connection between this proposal and the General Theory of Relativity? General Relativity (GTR) considers not the acting forces, but the curvature of space itself, and motion in such space occurs along its curved geodesic lines.
The Möbius strip is the geodesic line of the shortest distance in which a magnetic loop (magnetic flux) makes a two-cycle revolution, creating in space the dynamics of the interaction of complementary opposites, as a result of which portions of energy are released, in other words, quanta are emitted. The magnetic circuit, closed in such a geodesic line, mediates the original dual complementary relationship of magnetism, which underlies all movement and construction of the manifested world.
"The one-sided Mobius strip and its derivative, the two-sided strip with a node connecting two loops, are material symbols that reflect the fundamental principle of creation – the interaction of primordial energy with its mirror image. This principle underlies the construction of a cosmological model of the Universe of Light, where the dual nature of the Universe is predetermined." [101]
A geodesic line in the form of a magnetic loop with inversion is a line of movement of two mirror-asymmetric flows of magneto-plasma with the effect of generating energy quanta. In other words, magnetic geometry induces quantum geometry. As a consequence, the spatial geometry of emitted quanta replicates the shape of their emitter. And this is a direct connection between General Relativity and the concept of Quantum Mechanics.
Thus, the geometric approach based on the Möbius strip topology provides General Relativity with the initial geometry of space-time curvature and visualization of the form of a closed elemental as a space-time matrix unit, and Quantum Mechanics with the initial form of a quantum phenomenon determined by the geometric structure and magnetoelectric properties of such a matrix unit, inducing quanta in vortex motion. In this regard, within a single physical unit of space, in the initial form of the nontrivial Möbius strip topology, it is possible to combine the formats of two fundamental theories at the very starting point of the discussion and carry out the process of geometrization.
If we look at the graphic representation of the ancient Yin-Yang symbol from the discussed perspective, it geometrically visualizes a closed circuit of two complementary opposites and the release of an energy quantum at the points of their junction (the wave function in the middle of the circle). When the sinusoidal front of the emitted discrete wave quantum closes, a Möbius pattern is always obtained.
Ïîëíûé òåêñò äîñòóïåí â ôîðìàòå PDF (1941Êá)
L.A. Kulak, Quantum geometry. Nontrivial topology. Geometrization of physics // «Àêàäåìèÿ Òðèíèòàðèçìà», Ì.,
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