The document discusses Nambu-Goldstone modes that arise from supersymmetry breaking in quantum chromodynamics (QCD) and Bose-Fermi cold atom systems. It contains the following key points:
1) Supersymmetry breaking results in a Nambu-Goldstone fermion known as the "goldstino". This arises due to the relationship between supersymmetry breaking and the order parameter in these systems.
2) In relativistic systems like the Wess-Zumino model and QCD, the goldstino appears as a pole in fermion propagators and its properties are calculated.
3) In cold atom systems of Bose-Fermi mixtures, supersymmetry breaking and the resulting
Quantum Annealing for Dirichlet Process Mixture Models with Applications to N...Shu Tanaka
Our paper entitled “Quantum Annealing for Dirichlet Process Mixture Models with Applications to Network Clustering" was published in Neurocomputing. This work was done in collaboration with Dr. Issei Sato (Univ. of Tokyo), Dr. Kenichi Kurihara (Google), Professor Seiji Miyashita (Univ. of Tokyo), and Prof. Hiroshi Nakagawa (Univ. of Tokyo).
http://www.sciencedirect.com/science/article/pii/S0925231213005535
The preprint version is available:
http://arxiv.org/abs/1305.4325
佐藤一誠さん(東京大学)、栗原賢一さん(Google)、宮下精二教授(東京大学)、中川裕志教授(東京大学)との共同研究論文 “Quantum Annealing for Dirichlet Process Mixture Models with Applications to Network Clustering" が Neurocomputing に掲載されました。
http://www.sciencedirect.com/science/article/pii/S0925231213005535
プレプリントバージョンは
http://arxiv.org/abs/1305.4325
からご覧いただけます。
Quantum Annealing for Dirichlet Process Mixture Models with Applications to N...Shu Tanaka
Our paper entitled “Quantum Annealing for Dirichlet Process Mixture Models with Applications to Network Clustering" was published in Neurocomputing. This work was done in collaboration with Dr. Issei Sato (Univ. of Tokyo), Dr. Kenichi Kurihara (Google), Professor Seiji Miyashita (Univ. of Tokyo), and Prof. Hiroshi Nakagawa (Univ. of Tokyo).
http://www.sciencedirect.com/science/article/pii/S0925231213005535
The preprint version is available:
http://arxiv.org/abs/1305.4325
佐藤一誠さん(東京大学)、栗原賢一さん(Google)、宮下精二教授(東京大学)、中川裕志教授(東京大学)との共同研究論文 “Quantum Annealing for Dirichlet Process Mixture Models with Applications to Network Clustering" が Neurocomputing に掲載されました。
http://www.sciencedirect.com/science/article/pii/S0925231213005535
プレプリントバージョンは
http://arxiv.org/abs/1305.4325
からご覧いただけます。
Unconventional phase transitions in frustrated systems (March, 2014)Shu Tanaka
Presentation file using the workshop which was held at the University of Tokyo (March 26, 2014). The presentation was based on two papers:
- Physical Review B Vol. 87, 214401 (2013)
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.214401
(preprint: http://arxiv.org/abs/1209.2520)
(A brief explanation: http://www.slideshare.net/shu-t/prb-87214401slideshare)
- Physical Review E Vol. 88, 052138 (2013)
http://journals.aps.org/pre/abstract/10.1103/PhysRevE.88.052138
(preprint: http://arxiv.org/abs/1308.2467)
(A brief explanation: http://www.slideshare.net/shu-t/interlayerinteraction-dependence-of-latent-heat-in-the-heisenberg-model-on-a-stacked-triangular-lattice-with-competing-interactions)
2014年3月26日に東京大学で開催された「統計物理学の新しい潮流」での講演スライドです。この講演は、以下の2つの論文に関係するものです。
- Physical Review B Vol. 87, 214401 (2013)
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.214401
(preprint: http://arxiv.org/abs/1209.2520)
(A brief explanation: http://www.slideshare.net/shu-t/prb-87214401slideshare)
- Physical Review E Vol. 88, 052138 (2013)
http://journals.aps.org/pre/abstract/10.1103/PhysRevE.88.052138
(preprint: http://arxiv.org/abs/1308.2467)
(A brief explanation: http://www.slideshare.net/shu-t/interlayerinteraction-dependence-of-latent-heat-in-the-heisenberg-model-on-a-stacked-triangular-lattice-with-competing-interactions)
Entanglement Behavior of 2D Quantum ModelsShu Tanaka
I gave an oral presentation at YITP Workshop on Quantum Information Physics (YQIP2014).
http://www2.yukawa.kyoto-u.ac.jp/~yitpqip2014.ws/index.php
This presentation is based on the following papers.
http://iopscience.iop.org/1751-8121/43/25/255303
http://prb.aps.org/abstract/PRB/v84/i24/e245128
http://pra.aps.org/abstract/PRA/v86/i3/e032326
https://www.jstage.jst.go.jp/article/iis/19/1/19_IIS190115/_article
Preprint version are available via
http://arxiv.org/abs/1003.2007
http://arxiv.org/abs/1107.3888
http://arxiv.org/abs/1207.6752
http://arxiv.org/abs/1307.1939
京都大学基礎物理学研究所で開催されたワークショップ、"YITP Workshop on Quantum Information Physics (YQIP2014)"で口頭講演を行いました。
http://www2.yukawa.kyoto-u.ac.jp/~yitpqip2014.ws/index.php
本発表は以下の論文に基づいています。
http://iopscience.iop.org/1751-8121/43/25/255303
http://prb.aps.org/abstract/PRB/v84/i24/e245128
http://pra.aps.org/abstract/PRA/v86/i3/e032326
https://www.jstage.jst.go.jp/article/iis/19/1/19_IIS190115/_article
プレプリントバージョンは、以下のサイトから閲覧できます。
http://arxiv.org/abs/1003.2007
http://arxiv.org/abs/1107.3888
http://arxiv.org/abs/1207.6752
http://arxiv.org/abs/1307.1939
Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a...Shu Tanaka
Our paper entitled “Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular Lattice with Competing Interactions" was published in Physical Review E. This work was done in collaboration with Dr. Ryo Tamura (NIMS).
http://pre.aps.org/abstract/PRE/v88/i5/e052138
NIMSの田村亮さんとの共同研究論文 “Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular Lattice with Competing Interactions" が Physical Review E に掲載されました。
http://pre.aps.org/abstract/PRE/v88/i5/e052138
Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with ...Shu Tanaka
Our paper entitled “Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with Competing Interactions" was published in Physical Review B. This work was done in collaboration with Dr. Ryo Tamura (NIMS) and Professor Naoki Kawashima (ISSP).
http://prb.aps.org/abstract/PRB/v87/i21/e214401
NIMSの田村亮さん、物性研の川島直輝教授との共同研究論文 “Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with Competing Interactions" が Physical Review B に掲載されました。
http://prb.aps.org/abstract/PRB/v87/i21/e214401
On the existence properties of a rigid body algebraic integralsMagedHelal1
In this article, we consider kinematical considerations of a rigid body rotating around a
given fixed point in a Newtonian force field exerted by an attractive center with a rotating
couple about their principal axes of inertia. The kinematic equations and their well-known
three elementary integrals of the problem are introduced. The existence properties of the
algebraic integrals are considered. Besides, we search as a special case of the fourth algebraic
integral for the problem of the rigid body’s motion around a fixed point under the action of a
Newtonian force field with an orbiting couple. Lagrange’s case and Kovalevskaya’s one are
obtained. The large parameter is used for satisfying the existing conditions of the algebraic
integrals. The comparison between the obtained results and the previous ones is arising. The
numerical solutions of the regulating system of motion are obtained utilizing the fourth order
Runge-Kutta method and plotted in some figures to illustrate the positive impact of the
imposed forces and torques on the behavior of the body at any time.
Unconventional phase transitions in frustrated systems (March, 2014)Shu Tanaka
Presentation file using the workshop which was held at the University of Tokyo (March 26, 2014). The presentation was based on two papers:
- Physical Review B Vol. 87, 214401 (2013)
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.214401
(preprint: http://arxiv.org/abs/1209.2520)
(A brief explanation: http://www.slideshare.net/shu-t/prb-87214401slideshare)
- Physical Review E Vol. 88, 052138 (2013)
http://journals.aps.org/pre/abstract/10.1103/PhysRevE.88.052138
(preprint: http://arxiv.org/abs/1308.2467)
(A brief explanation: http://www.slideshare.net/shu-t/interlayerinteraction-dependence-of-latent-heat-in-the-heisenberg-model-on-a-stacked-triangular-lattice-with-competing-interactions)
2014年3月26日に東京大学で開催された「統計物理学の新しい潮流」での講演スライドです。この講演は、以下の2つの論文に関係するものです。
- Physical Review B Vol. 87, 214401 (2013)
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.214401
(preprint: http://arxiv.org/abs/1209.2520)
(A brief explanation: http://www.slideshare.net/shu-t/prb-87214401slideshare)
- Physical Review E Vol. 88, 052138 (2013)
http://journals.aps.org/pre/abstract/10.1103/PhysRevE.88.052138
(preprint: http://arxiv.org/abs/1308.2467)
(A brief explanation: http://www.slideshare.net/shu-t/interlayerinteraction-dependence-of-latent-heat-in-the-heisenberg-model-on-a-stacked-triangular-lattice-with-competing-interactions)
Entanglement Behavior of 2D Quantum ModelsShu Tanaka
I gave an oral presentation at YITP Workshop on Quantum Information Physics (YQIP2014).
http://www2.yukawa.kyoto-u.ac.jp/~yitpqip2014.ws/index.php
This presentation is based on the following papers.
http://iopscience.iop.org/1751-8121/43/25/255303
http://prb.aps.org/abstract/PRB/v84/i24/e245128
http://pra.aps.org/abstract/PRA/v86/i3/e032326
https://www.jstage.jst.go.jp/article/iis/19/1/19_IIS190115/_article
Preprint version are available via
http://arxiv.org/abs/1003.2007
http://arxiv.org/abs/1107.3888
http://arxiv.org/abs/1207.6752
http://arxiv.org/abs/1307.1939
京都大学基礎物理学研究所で開催されたワークショップ、"YITP Workshop on Quantum Information Physics (YQIP2014)"で口頭講演を行いました。
http://www2.yukawa.kyoto-u.ac.jp/~yitpqip2014.ws/index.php
本発表は以下の論文に基づいています。
http://iopscience.iop.org/1751-8121/43/25/255303
http://prb.aps.org/abstract/PRB/v84/i24/e245128
http://pra.aps.org/abstract/PRA/v86/i3/e032326
https://www.jstage.jst.go.jp/article/iis/19/1/19_IIS190115/_article
プレプリントバージョンは、以下のサイトから閲覧できます。
http://arxiv.org/abs/1003.2007
http://arxiv.org/abs/1107.3888
http://arxiv.org/abs/1207.6752
http://arxiv.org/abs/1307.1939
Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a...Shu Tanaka
Our paper entitled “Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular Lattice with Competing Interactions" was published in Physical Review E. This work was done in collaboration with Dr. Ryo Tamura (NIMS).
http://pre.aps.org/abstract/PRE/v88/i5/e052138
NIMSの田村亮さんとの共同研究論文 “Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular Lattice with Competing Interactions" が Physical Review E に掲載されました。
http://pre.aps.org/abstract/PRE/v88/i5/e052138
Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with ...Shu Tanaka
Our paper entitled “Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with Competing Interactions" was published in Physical Review B. This work was done in collaboration with Dr. Ryo Tamura (NIMS) and Professor Naoki Kawashima (ISSP).
http://prb.aps.org/abstract/PRB/v87/i21/e214401
NIMSの田村亮さん、物性研の川島直輝教授との共同研究論文 “Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with Competing Interactions" が Physical Review B に掲載されました。
http://prb.aps.org/abstract/PRB/v87/i21/e214401
On the existence properties of a rigid body algebraic integralsMagedHelal1
In this article, we consider kinematical considerations of a rigid body rotating around a
given fixed point in a Newtonian force field exerted by an attractive center with a rotating
couple about their principal axes of inertia. The kinematic equations and their well-known
three elementary integrals of the problem are introduced. The existence properties of the
algebraic integrals are considered. Besides, we search as a special case of the fourth algebraic
integral for the problem of the rigid body’s motion around a fixed point under the action of a
Newtonian force field with an orbiting couple. Lagrange’s case and Kovalevskaya’s one are
obtained. The large parameter is used for satisfying the existing conditions of the algebraic
integrals. The comparison between the obtained results and the previous ones is arising. The
numerical solutions of the regulating system of motion are obtained utilizing the fourth order
Runge-Kutta method and plotted in some figures to illustrate the positive impact of the
imposed forces and torques on the behavior of the body at any time.
Quantum gravitational corrections to particle creation by black holesSérgio Sacani
We calculate quantum gravitational corrections to the amplitude for the emission of a Hawking particle
by a black hole. We show explicitly how the amplitudes depend on quantum corrections to the exterior
metric (quantum hair). This reveals the mechanism by which information escapes the black hole. The
quantum state of the black hole is reflected in the quantum state of the exterior metric, which in turn
influences the emission of Hawking quanta.
Justification of canonical quantization of Josephson effect in various physic...Krzysztof Pomorski
Quantum devices based on the Josephson effect in superconductors are usually described by a Hamiltonian obtained by commonly used canonical quantization.
However, this recipe has not yet been rigorously and systematically justified. We show that this approach is indeed correct in a certain range of parameters. We find the condition of validity of such quantization and the systematic corrections to the Josephson energy EJ: namely, the capacitance
energy Ec = e^2/2C must be much smaller than the superconducting gap Δ. Moreover, we find an experimentally testable modification of Josephson energy at large capacitance energy also with nonlinear capacitance.
Reference
[1]. K.Pomorski, A.Bednorz, "Justification of canonical quantization of Josephson effect and its modification due to large capacitance energy", J. Phys. A: Math. Theor. 49, 2016
(http://iopscience.iop.org/article/10.1088/1751-8113/49/12/125002/meta )
[2]. K.Pomorski, A.Bednorz, "Justification of canonical quantization of Josephson junction", 2015 (http://arxiv.org/abs/1502.00511)
Invited Seminar presented at the VIA Forum Astroparticle Physics Forum COSMOVIA
21 March 2020
http://viavca.in2p3.fr/2010c_o_s_m_o_v_i_a__forum_sd24fsdf4zerfzef4ze5f4dsq34sdteerui45788789745rt7yr68t4y54865h45g4hfg56h45df4h86d48h48t7uertujirjtiorjhuiofgrdsqgxcvfghfg5h40yhuyir/viewtopic.php?f=73&t=3705&sid=c56cbf76f87536fc4c3ff216d9edaba2
Author: O.M. Lecian
Speaker: O.M. Lecian
Abstract: The LHAASO experiment is aimed at detecting highly-energetic particles of cosmological origin within a large
range of energies.
The sensitivity of the experimental apparatus can within the frameworks of statistical fluctuations of the
background.
Acceleration and lower-energy particles can be analyzed.
The anisotropy mass composition of cosmic rays can analytically described.
The LHAASO Experiment is also suited for detecting particles of cosmological origin originated from the breach
(and/or other kinds of modifications) of particle theories paradigms comprehending other symmetry groups.
Some physical implications of anisotropies can be looked for.
The study of anisotropy distribution for particles of cosmological origin as well as the anisotropies of their velocities
both in the case of a flat Minkowskian background as well as in the case of curved space-time can be investigated,
as far as the theoretical description of the cross-section is concerned, as well as for the theoretical expressions of
such quantities to be analyzed.
The case of a geometrical phase of particles can be schematized by means of a geometrical factor.
Particular solutions are found under suitable approximations.
A comparison with the study of ellipsoidal galaxies is achieved.
The case of particles with anisotropies in velocities falling off faster than dark matter (DM) is compared.
The study of possible anisotropies in the spatial distribution of cosmological particles can therefore be described
also deriving form the interaction of cosmic particles with the gravitational field, arising at quantum distances, at
the semiclassical level and at the classical scales, within the framework of the proper description of particles
anisotropies properties.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
Uma equipe de astrônomos norte-americanos descobriu o par de buracos negros supermassivos mais próximos até agora no universo.
O par de buracos negros está localizado no centro do quasar chamado de PKS 1302-102, a aproximadamente 3.5 bilhões de anos-luz de distância.
Esses dois buracos negros estão separados de apenas uma semana-luz e estão num movimento espiral um em direção ao outro que deve acabar com uma colisão cataclísmica.
Em contraste, o par de buracos negros mais próximos descoberto até então estava separado de aproximadamente 20 anos-luz.
We apply the recently derived constraintless Clairaut-type formalism (by S. Duplij) to the Cho-Duan-Ge decom- position in SU(2) QCD. We find nontrivial corrections to the physical equations of motion and that the contribution of the topological degrees of freedom is qualitatively different from that found by treating the monopole potential as though it were dynamic. We also find alterations to the field commutation relations that undermine the particle interpretation in the presence of the chromomonopole condensate.
Exact Sum Rules for Vector Channel at Finite Temperature and its Applications...Daisuke Satow
Slides used in presentation at:
“International School of Nuclear Physics 38th Course Nuclear matter under extreme conditions -Relativistic heavy-ion collisions”, in September, 2016 @ Erice, Italy
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
3. 3
Symmetry related to interchanging
a boson and a fermion
Supersymmetry (SUSY)
fb fb
=
Same energy
4. 4
Supercharge operator: annihilate one fermion
and create one boson (and its inverse process)
SUSY and supercharge
Mathematically, [Q, H]=0
fb
Q~bf: supercharge
fb
5. 5
SUSY breaking
As long as the system is not at vacuum,
the states always contain double degeneracy.
examples: High Temperature (QCD), Fermi sea+BEC (cold atom)
ground state
Q
Q2=0Q2=0
6. 6
NG “fermion” which is related to SUSY breaking
Generally, degeneracy generates zero energy
excitation (Nambu-Goldstone (NG) mode).
degeneracy
order parameter
NG mode
We expect that NG mode appears in SUSY case.
No NG mode
7. 7
“When the order parameter is finite, the propagator
in the left-hand side has a pole at p→0.”
Order parameterNG mode
Broken symmetry
Nambu-Goldstone theorem:
(fermion ver.)
Jµ: supercurrent
Q=J0: supercharge
NG “fermion” which is related to SUSY breaking
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989)
ipµ
Z
d4
xeip·(x y)
hTJµ
(x)O(y)i = h{Q, O}i
8. 8
If we set O=Q, order parameter is energy-momentum
tensor (T µν) in the relativistic system.
single fermion intermediate state. Hence, we can write, in momentum space
Γµν
JJ (k) ≡ d4
x eik(x−y)
⟨TJµ
(x)Jν
(y)⟩
= Γµ
JΨ(k)S(k)Γν
ΨJ (k).
By making use of the Ward-Takahashi identity (3.39), we find
−ikµΓµν
JJ (k) = −im⟨A⟩Γν
ΨJ (k).
This can now be calculated by using our earlier results for Γµ
JΨ(k).
First, in the limit of high temperature, we make use of (3.37) and obtai
the left hand side of the Ward-Takahashi identity as
−ikµΓµν
JJ (k) = m2
⟨A⟩2 π2
g2
γ0
−1
3 γi
=
π2
4
T4
γ0
−1
3 γi
.
For the right hand side, we must evaluate the thermal expectation valu
of the energy-momentum tensor. In a thermal equilibrium state, it reads
⟨Tµν
⟩ = diag(ρ, p, p, p),
where in the high-temperature limit the pressure is given by p = 1
3ρ, an
the energy density can be calculated as
NG “fermion” which is related to SUSY breaking
In the presence of medium, SUSY is always broken.
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989)
Order parameterNG mode
Broken symmetry
ipµ
Z
d4
xeip·(x y)
hTJµ
(x)O(y)i = h{Q, O}i
9. 9
SUSY breaking
SUSY is fermionic symmetry,
so the NG mode is fermion (Goldstino).
Rare fermionic zero mode
fb
Q
fb
NG mode appears in <QQ>.
NG mode
ipµ
Z
d4
xeip·(x y)
hTJµ
(x)O(y)i = h{Q, O}i
11. Goldstino in Wess-Zumino model
11
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989).
In Wess-Zumino model, the goldstino exists.
3.4. The phonino
a b
ψ
Q~(A,B)×Ψ
A, B
(Weyl fermion+Scalar and Pseudo-scalar bosons)
<QQ>~
Pole appears in the Green function.
12. Goldstino in Wess-Zumino model
12
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989).
Through the fermion-boson coupling,
it appears also in fermion propagator.
3.4. The phonino
a b
Figure 3.6: Typical one- and two-loop co
energy
ψ
gg
dispersion relation Reω=p/3
Residue g2/π2
13. 13
(Quasi) SUSY in QCD
V. V. Lebedev and A. V. Smilga, Annals Phys. 202, 229 (1990)
There is SUSY approximately if we neglect the
interaction.
q g
=
Both of the quark and the gluon are regarded as
massless at high T
14. Quasi-goldstino in QCD
14
In weak coupling regime, we established the
existence of the (quasi) goldstino in QCD.
Dispersion relation Reω=p/3
Damping rate Imω=ζq+ζg=O(g2T)
Residue
as follows: We expand the vertex function in terms of ˜p/g T as in Eq. (2.26). The zeroth
order solution is
A =
1
1 + Cf λ
1, B(k) =
1
k0
2Cf λ
1 + Cf λ
γ0
, C(k) = 0. (2.49)
Here A, B(k), and C(k) are defined by Eq. (2.27), and λ is defined by Eq. (2.7). The
self-consistent equation which determines the vertex function at the first order δΓµ(p, k)
is written as
δΓµ
(p, k) = Cf
d4k′
(2π)4
˜X(k′
)
kνγµ + γνk′µ
k · k′
/k′
PT
νρ(k′
) −2
˜p · k′
δm2
Aγρ
+ δΓρ
(p, k′
) .
(2.50)
Then δΠ(p) defined by Eq. (2.33) becomes
δΠ(p) =
16π2A2λ3Cf
g2
(γ0
(p0
+ iζ) + vp · γ). (2.51)
Owing to Eq. (2.43), the retarded quark self-energy is found to be
ΣR
(p) = Cf
d4k
(2π)4
˜X(k)
γµ/kPT
µν(k)Γν(p, k)
1 + 2˜p · k/δm2
= −
1
Z
(γ0
(p0
+ iζ) + vp · γ),
(2.52)
where the expression of the residue Z will be given shortly. We note that this expression
is the same as that in QED except for the numerical factor. Thus, the expression for the
pole position of the ultrasoft mode in QCD is the same as in QED, while the residue of
that mode is not: The residue is
Z =
g2
16π2λ2Cf
(1 + Cf λ)2
=
g2N
8π2(N2 − 1)
5
6
N +
1
2N
+
2
3
Nf
2
. (2.53)
We can also show that the analytic solution of the self-consistent equation satisfies
the WT identity in the leading order as in QED, by checking that the counterparts of
Eqs. (2.40) and (2.41) are satisfied.
16. Experimental detection
16
How can we detect goldstino in experiment?
So far, detecting quark spectrum in heavy ion collision
has not been done…
Picture: UrQMD group, Frankfurt.
17. 17
SUSY in Cold Atom System
Picture: Ferlaino group, Innsbruck.
cf: T. Ozawa, Nature Physics 11, 801 (2015),
Wess-Zumino model: Y. Yu, and K. Yang, PRL 105, 150605 (2010),
Dense QCD: K. Maeda, G. Baym and T. Hatsuda, PRL 103, 085301 (2009),
Relativistic QED: Kapit and Mueller, PRA 83, 033625 (2011).
Cold atom system is
easier to realize, and its experiment is cleaner.
(Test site of many-body physics)
19. 19
SUSY in Cold Atom System Y. Yu and K. Yang, PRL 100, 090404.
Possible candidates: 6Li-7Li, 173Yb-174Yb
(mb/mf =1.17, U is easy to tune) (Ubb/Ubf =1.32, m is almost same)
m=mb=mf
Ubb =Ubf
(Q=∫dx b†
xfx, Q†=∫dx bx f †
x)
[Q, H]=0
S. Endo, private communication.
20. 20
If we set O=Q†, NG mode appears in <QQ†>.
Order parameter is density (<{Q, Q†}>=ρ) in this case.
Goldstino in cold atom systems
Q =bf †
Q† =b†f
SUSY is always broken when ρ is finite.
Order parameterNG mode
Broken symmetry
ipµ
Z
d4
xeip·(x y)
hTJµ
(x)O(y)i = h{Q, O}i
21. 21
Explicit breaking of SUSY Y. Yu and K. Yang, PRL 100, 090404.
Finite density causes explicit SUSY breaking.
[Q, H-µf Nf -µb Nb]=-ΔµQ
µf - µbGrand Canonical Hamiltonian
Q
Q†
Δµ
Gapped Goldstino
(ω=-Δµ)
23. Goldstino spectrum
23
Calculate the spectrum of the Goldstino.
2
he generic
(2.1)
(2.2)
(2.3)
x)nf (x) ,
(2.4)
x) are the
s hamilto-
wavelength
[12]. Note
where the angular brakets denote an average over the
ground state of the system. Its Fourier transform is writ-
ten as
GR
(p) = i
Z
dt
Z
d3
x ei!t ip·x
✓(t)h{q(t, x), q†
(0)}i.
(2.8)
Here we have introduced a 4-vector notation, to be used
throughout: xµ
⌘ (t, x) and pµ
⌘ (!, p). The frequency
! is assumed to contain a small positive imaginary part
✏ (! ! ! + i✏) in order to take into account the retarded
condition. Such a small imaginary part will not be indi-
cated explicitly in order to simplify the formulae. In fact,
we shall also most of the time drop the superscript R, and
indicate it only when necessary to avoid confusion.
Let us recall some general features of this Green’s func-
tion by looking at its spectral representation in terms of
the excited states n and m that can be reached from
conservation law and the canonical (anti-) commuta-
n relations [12, 13]. It does not depend on the details
the Hamiltonian. This is the Goldstino’s counter part
gapped Nambu-Goldstone modes [20–22]. In the fol-
wing, we shall often refer to G(!, p) as the Goldstino
pagator.
We shall also be interested in the associated Goldstino
ctral function
(!, p) = 2Im G(!, p). (2.12)
is spectral function obeys simple sum rules [13]. The
t sum rule determines the zeroth moment of the spec-
l function. It is valid regardless of the details of the
miltonian, and reads
Z
d¯!
(p) = ⇢. (2.13)
in ord
Th
depen
sume
two p
the b
U. T
the fe
shall
ble or
have
Ferm
for th
to th
U⇢ h
can b
with
24. 24
Goldstino spectrum (free case)
Free case (U=0)
qp~fkb†
k+p
f
b
q†
p~f†
kbk+p
⌘ k2
/(2m), ¯! = !+ µ0 with µ0 = k2
F /(2m),
= ⌦⇢b, while np ⌘ ✓(kF p) denotes the fermion
on number.
q†q
q†
q
The one-loop diagrams contributing to G0
. The full
line represents a fermion (boson) propagator. The
n used for these diagrams, and those below that con-
ws is as follows. The time flows from left to right.
pointing to the right indicates a “particle”, while an
nting to the left indicates a “hole”. The boson hole
Cut
(ω, p)
25. 2525
Landau damping
Continuum.
Width is Δω=pkF/m
-1~10~kF
Goldstino spectrum (free case)
E
nf
E
nb
E
nb
εk+p
E
nf
εk
qp~Σk fkb†
k+p
εk=k2/2m, |k|<kF
(-ω, -p) (-ω+k2/2m, -p+k)
should be on-shell: ω=-p2/2m-p k cosθ/m
26. 26
Pole, No width.
εk=0, k=0
(ω, p)
(ω, p)
should be on-shell: ω=p2/2m
Other value of ω is not allowed!
Dispersion Relation ω=p2/2m
Strength ρb
Goldstino spectrum (free case)
q†
p~Σk f†
k-pbk
E
nf
E
nb
E
nb
E
nf
ε-p
k=0
nf(1+nb)+(1-nf)nb
PoleContinuum
27. 27
Goldstino spectrum (free case)
Continuum+Pole.
is
4.
n-
ds
he
ve
1
d-
it
m
ee
ce
is
by
st
he
n-
4)
he
stino. The process in the left represents q†
, which replace a
boson with a fermion, while the one in the right represents
q, which replaces a fermion with a boson. The blue square
represents the Fermi sea.
-2
-1
0
1
2
0 0.2 0.4 0.6 0.8 1 1.2
ω
-
/εF
p/kF
FIG. 3: The continuum (red shaded area) and the pole (blue
! =
p2
2m
! =
p2
2m
⇢f
⇢b
28. 28
These pole and continuum satisfies the NG theorem.
Goldstino spectrum (free case)
tly related to the mag-
F . The range of this
d its shape is in Fig. 4.
nit of momentum (en-
Eq. (4.1) corresponds
h turns a boson in the
0) into a fermion above
m |p| kF (see Figs. 1
pole contribution hid-
t line of Eq. (4.1): it
sea with a momentum
lls the condensate (see
plified by the presence
N0 accompanying this
bution is cancelled by
first term in the first
s the first term of the
lds the following con-
p2
2m
◆
. (4.4)
FIG. 2: Particle-hole excitations contributing to the Gold-
stino. The process in the left represents q†
, which replace a
boson with a fermion, while the one in the right represents
q, which replaces a fermion with a boson. The blue square
represents the Fermi sea.
-2
-1
0
1
2
0 0.2 0.4 0.6 0.8 1 1.2
ω
-
/εF
p/kF
! =
p2
2m
! =
p2
2m
⇢f
⇢b
29. Goldstino spectrum (interacting case)
29
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
Switch on the interaction.
For simplicity, we start with two-dimension case, in which there are no BEC.
Mean field approximation
fermion: Uρb boson: Uρf +2Uρb
Q, Q†
Uρ
Different MF correction
Gap in goldstino spectrum?
30. 30
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
Goldstino spectrum (interacting case)
G0
(p) =
Z
d2
k
(2⇡)2
nF (✏f
k) + nB(✏b
k+p)
! + [2k · p + p2]/2m + U⇢
implies
ution of
pectrum
entually
an su↵er
densities
eraction
calcula-
ch occur
we note
plies
(3.3)
where ✏0
k ⌘ k2
/(2m), ¯! = !+ µ0 with µ0 = k
and N0 = ⌦⇢b, while np ⌘ ✓(kF p) denotes the
occupation number.
q
q† q†q
q†
q
FIG. 1: The one-loop diagrams contributing to G0
.
(dashed) line represents a fermion (boson) propaga
convention used for these diagrams, and those below
tain arrows is as follows. The time flows from left
An arrow pointing to the right indicates a “particle”
arrow pointing to the left indicates a “hole”. The b
propagator is disconnected, and represented by the
Actually, it is the case in Green function
It contradicts with the exact result (Gapless NG mode).
We should have missed something…
At p=0
~U -1
⇢
! + U⇢
31. 31
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
All ring diagrams contributes at the same order.
We need to sum up infinite ring diagrams.
Goldstino spectrum (interacting case)
Random Phase Approximation
U-1 ×U ×U-1=U-1U-1
7
+ + +...=
+= + +...q†
+= + +...q
: The ring diagrams that are summed in the RPA
tion of GRPA
, Eq. (4.12). Note that the propagators
mean field propagators. The interaction joining two
ive bubbles is the one in the second line of Eq. (4.11),
H4.
32. Goldstino spectrum (interacting case)
32
Result 1. Goldstino Pole
At p=0
GRPA(p) =
1
[G0(p)] 1 + U
G0
=
⇢
! + U⇢
Gap disappears!
GRPA(!, 0) =
⇢
!
33. Goldstino spectrum (interacting case)
33
Dispersion Relation ω=-Δµ +αp2/2m
Strength
(p=0: maximum value allowed by sum rule.
The sum rule is saturated by the pole)
Z = ⇢ p2 1
4⇡
✓
✏F
U⇢
◆2
Expression at finite p
10
h
Z = ⇢
4
5
⇢f
✓
|p|
kF
✏F
U⇢
◆2
, (4.29)
↵ ⌘
⇢b ⇢f
⇢
+
4
5
⇢f
⇢
"F
U⇢
= ↵s +
4
5
⇢f
⇢
"F
U⇢
. (4.30)
se formulae reduce to Eqs. (4.17) and (4.18) [(4.19)]
n ⇢b = 0 [⇢f = 0], as they should. Also that the ex-
sion for ↵ is the same as that obtained in the absence
EC [13].
he location of the Goldstino pole obtained numer-
y is plotted in Fig. 11, and compared to the ap-
ximate expression ¯! = ↵p2
/(2m). The interaction
ngth is set to a small value, U⇢f /✏F = 0.1, or kF a =
/4 ' 0.24 in terms of a, for which the weak-coupling
ysis is reliable. One sees on Fig. 11 that the approxi-
e expression is accurate as long as |p| . 0.16kF . This
deed the expected range of validity of the expansion,
ely U⇢ kF |p|/m, as can be seen from the denomi-
or of the expression of GMF
cont, Eq. (4.9). This condition
s to |p| ⌧ U⇢m/kF ' 0.15kF for the current values
he parameters. Note that because the continuum is
ted down by the MF correction U⇢, as compared to
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
FIG. 11: The range of the continuum (red shed area), the
numerical result for the pole position of the Goldstino prop-
agator (black solid line), and the pole position obtained from
the small |p| expansion, Eq. (4.30) (blue long-dashed line).
For illustration of the “level repulsion”, the pole positions of
GMF
pole (green dashed line) and GRPA
(magenta dotted line) are
also plotted. Note that at p = 0 the tip of the continuum
corresponds to the fictitious pole at ¯! = U⇢, carrying no
spectral weight. The densities are the same as in Sec. IV A,
i.e., ⇢b = 2⇢f , and the interaction strength is U⇢f /✏F = 0.1.
↵ =
⇢b ⇢f
⇢
+
✏F
U⇢
⇢f
⇢
ysis is reliable. One sees on Fig. 11 that the approxi-
expression is accurate as long as |p| . 0.16kF . This
deed the expected range of validity of the expansion,
ely U⇢ kF |p|/m, as can be seen from the denomi-
r of the expression of GMF
cont, Eq. (4.9). This condition
to |p| ⌧ U⇢m/kF ' 0.15kF for the current values
e parameters. Note that because the continuum is
ed down by the MF correction U⇢, as compared to
ree case (4.2), the Goldstino pole remains out of the
nuum as long as |p| is smaller than ⇠ 0.21kF .
so plotted in Fig. 11 are the dispersion relations cor-
nding to the poles of GMF
pole (Eq. (4.21)) and GRPA
(4.12)). This illustrates the e↵ect of the level re-
on already discussed in the case |p| = 0, yielding
ually the distribution of spectral weight between
ontinuum and the Goldstino pole. Of course, the
rsymmetry plays a crucial role here in putting the
stino pole at ¯! = 0 for p = 0.
e spectral function is analyzed in more details in
12. The contributions to the zeroth moment of
the pole and the continuum are displayed in the up-
anel of this figure. At small momenta, |p| . 0.11kF ,
are well accounted for by the expansion (4.29). In
ame plot, we see that the continuum contribution
ppressed for small momentum, with all the spec-
weight being carried there by the Goldstino. The
panel of Fig. 12 reveals large cancellation between
the small |p| expansion, Eq. (4.30) (blue lo
For illustration of the “level repulsion”, the
GMF
pole (green dashed line) and GRPA
(magenta
also plotted. Note that at p = 0 the tip o
corresponds to the fictitious pole at ¯! =
spectral weight. The densities are the same
i.e., ⇢b = 2⇢f , and the interaction strength is
contribution vanishes since the pole is a
continuum (see Fig. 11). The sum of th
continuum contributions to the zeroth
⇢, as it should because of the sum rule
plies that the spectral weight of the conti
rapidly around the momentum at which
sorbed, which is demonstrated in Fig. 12
iors, namely the suppression (enhanceme
tinuum at small |p| (above |p| ' 0.21kF
also from the spectral function plotted
V. PHENOMENOLOGICAL IMP
The strong coupling between the fermio
stino may o↵er a possibility to infer the p
f F F
ms of a, for which the weak-coupling
One sees on Fig. 11 that the approxi-
curate as long as |p| . 0.16kF . This
d range of validity of the expansion,
m, as can be seen from the denomi-
n of GMF
cont, Eq. (4.9). This condition
kF ' 0.15kF for the current values
Note that because the continuum is
MF correction U⇢, as compared to
e Goldstino pole remains out of the
|p| is smaller than ⇠ 0.21kF .
. 11 are the dispersion relations cor-
oles of GMF
pole (Eq. (4.21)) and GRPA
lustrates the e↵ect of the level re-
ussed in the case |p| = 0, yielding
bution of spectral weight between
he Goldstino pole. Of course, the
s a crucial role here in putting the
= 0 for p = 0.
tion is analyzed in more details in
butions to the zeroth moment of
e continuum are displayed in the up-
e. At small momenta, |p| . 0.11kF ,
ted for by the expansion (4.29). In
e that the continuum contribution
all momentum, with all the spec-
rried there by the Goldstino. The
2 reveals large cancellation between
uum contributions to the first mo-
function. This can be understood
momentum, the pole contribution is
m) by using Eq. (4.29). On the other
(2.14) requires the sum of the pole
numerical result for the pole position of the Goldstino prop-
agator (black solid line), and the pole position obtained from
the small |p| expansion, Eq. (4.30) (blue long-dashed line).
For illustration of the “level repulsion”, the pole positions of
GMF
pole (green dashed line) and GRPA
(magenta dotted line) are
also plotted. Note that at p = 0 the tip of the continuum
corresponds to the fictitious pole at ¯! = U⇢, carrying no
spectral weight. The densities are the same as in Sec. IV A,
i.e., ⇢b = 2⇢f , and the interaction strength is U⇢f /✏F = 0.1.
contribution vanishes since the pole is absorbed by the
continuum (see Fig. 11). The sum of the pole and the
continuum contributions to the zeroth moment equals
⇢, as it should because of the sum rule (2.13). It im-
plies that the spectral weight of the continuum increases
rapidly around the momentum at which the pole is ab-
sorbed, which is demonstrated in Fig. 12. These behav-
iors, namely the suppression (enhancement) of the con-
tinuum at small |p| (above |p| ' 0.21kF ), can be seen
also from the spectral function plotted in Fig. 13.
V. PHENOMENOLOGICAL IMPLICATION
The strong coupling between the fermion and the Gold-
stino may o↵er a possibility to infer the properties of the
Goldstino from the study of the fermion propagator. This
is what we explore in this section.
34. Goldstino spectrum (interacting case)
34
Result 2. Continuum is shifted.
Cut
shift: Uρ
10
4
5
⇢f
✓
|p|
kF
✏F
U⇢
◆2
, (4.29)
⇢f
⇢
+
4
5
⇢f
⇢
"F
U⇢
+
4
5
⇢f
⇢
"F
U⇢
. (4.30)
to Eqs. (4.17) and (4.18) [(4.19)]
as they should. Also that the ex-
me as that obtained in the absence
Goldstino pole obtained numer-
ig. 11, and compared to the ap-
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
(εk=k2/2m+Uρb, k)
(-ω, -p) (-ω+k2/2m+Uρb, -p+k)
should be on-shell: ω=-p2/2m-p k cosθ/m+Uρ
35. Goldstino spectrum (interacting case)
35
• Continuum+Pole (as U=0 case), but the continuum is
shifted so that the pole is out of the continuum at small p.
• At p=0, all the spectral weights are given to the pole.
Summary
10
⇢
4
5
⇢f
✓
|p|
kF
✏F
U⇢
◆2
, (4.29)
⇢b ⇢f
⇢
+
4
5
⇢f
⇢
"F
U⇢
↵s +
4
5
⇢f
⇢
"F
U⇢
. (4.30)
ce to Eqs. (4.17) and (4.18) [(4.19)]
0], as they should. Also that the ex-
same as that obtained in the absence
he Goldstino pole obtained numer-
Fig. 11, and compared to the ap-
n ¯! = ↵p2
/(2m). The interaction
mall value, U⇢f /✏F = 0.1, or kF a =
ms of a, for which the weak-coupling
One sees on Fig. 11 that the approxi-
ccurate as long as |p| . 0.16kF . This
ed range of validity of the expansion,
/m, as can be seen from the denomi-
MF
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
FIG. 11: The range of the continuum (red shed area), the
numerical result for the pole position of the Goldstino prop-
agator (black solid line), and the pole position obtained from
the small |p| expansion, Eq. (4.30) (blue long-dashed line).
For illustration of the “level repulsion”, the pole positions of
GMF
pole (green dashed line) and GRPA
(magenta dotted line) are
also plotted. Note that at p = 0 the tip of the continuum
corresponds to the fictitious pole at ¯! = U⇢, carrying no
37. 37
What happens in BEC phase?
In free case, no difference. Let us consider interacting case.
-Uρb Uρf +2Uρb
Goldstino spectrum (interacting case)
Mean field approximation
Fermion particle-Boson hole
excitation (Continuum)
E
nf
E
nb
E
nb
εk+p
E
nf
εk
qp~Σk fkb†
k+p
At p=0
G0
(p) =
⇢f
! + U⇢
38. Goldstino spectrum (interacting case)
38
Fermion hole-Boson particle excitation (Pole)
At p=0
q†
p~Σk f†
k-pbk
E
nf
E
nb
E
nb
E
nf
ε-p
k=0
GMF
pole(p) =
⇢b
! + U⇢f
Uρb -Uρ
39. Goldstino spectrum (interacting case)
39
RPA
At p=0
The gap remains!
Inconsistent with the NG theorem,
so we should have missed something again…
GRPA(p) =
1
[G0(p)] 1 + U
G0
(p) =
⇢f
! + U⇢
GRPA(p) =
⇢f
! + U⇢b
40. Goldstino spectrum (interacting case)
40
Three-point coupling due to BEC
Mixing between
Fermion particle-Boson hole excitation (Continuum)
and Fermion hole-Boson particle excitation (Pole)
+ +
FIG. 8: The diagrams containing the mixing b
RPA MF
b !
p
⇢b
+ +
+ +
FIG. 8: The diagrams containing the mixing betw
the RPA diagrams (GRPA
) and GMF
pole contributin
blob represents the RPA diagrams.
41. Goldstino spectrum (interacting case)
41
Taking into account the mixing
+ + +...
+ + +...
+ + +...
FIG. 8: The diagrams containing the mixing between between
the RPA diagrams (GRPA
) and GMF
pole contributing to ˜G. The
blob represents the RPA diagrams.
case, it is corrected by the diagrams in Fig. 8. The sec-
ond diagram in this figure is of order U2
⇥ U 2
⇥ U 1
,
where the factor U2
comes from the two vertices, the fac-
tor U 2
from the two bubbles, and U 1
from the mean
field propagator near ¯! = |p| = 0, i.e.,
FIG
RPA
FIG
+ + +...case, it is corrected by the diagrams in Fig. 8. The sec-
ond diagram in this figure is of order U2
⇥ U 2
⇥ U 1
,
where the factor U2
comes from the two vertices, the fac-
tor U 2
from the two bubbles, and U 1
from the mean
field propagator near ¯! = |p| = 0, i.e.,
GMF
pole(p) =
⇢b
¯! ✏0
p + U⇢f
⇡
⇢b
U⇢f
. (4.21)
This diagram has the same order of magnitude ⇠ U 1
as the RPA diagrams, and the same holds for the entire
family of diagrams displayed in Fig. 8. Their sum yields
˜G(p) =
1
[GRPA
(p)] 1 U2GMF
pole(p)
, (4.22)
where GRPA
(p) is given by Eq. (4.12). At zero momen-
tum, it reduces to
˜G(!, 0) =
"
⇢2
f
⇢
1
¯!
+
⇢f ⇢b
⇢
1
¯! + U⇢
#
. (4.23)
Here we have one pole with no gap, and another one
with a finite gap (¯! = U⇢), whose existence is due to
the presence of a BEC. One may interpret this result
+
FIG. 10: The d
tween the RPA
G3. There is an
tex (the black
identical contrib
The remaini
is G3(p). It is
are connected
finite momentu
The resulting e
which reduces
G3(!, 0)
At p=0
Gap disappears!
GRPA(p) =
⇢f
! + U⇢b
GMF
pole(p) =
⇢b
! + U⇢f
˜G(p) '
1
⇢
⇢2
f
!
42. Goldstino spectrum (interacting case)
42
Level Repulsion
10
= ⇢
4
5
⇢f
✓
|p|
kF
✏F
U⇢
◆2
, (4.29)
↵ ⌘
⇢b ⇢f
⇢
+
4
5
⇢f
⇢
"F
U⇢
= ↵s +
4
5
⇢f
⇢
"F
U⇢
. (4.30)
reduce to Eqs. (4.17) and (4.18) [(4.19)]
= 0], as they should. Also that the ex-
the same as that obtained in the absence
of the Goldstino pole obtained numer-
in Fig. 11, and compared to the ap-
ssion ¯! = ↵p2
/(2m). The interaction
a small value, U⇢f /✏F = 0.1, or kF a =
terms of a, for which the weak-coupling
e. One sees on Fig. 11 that the approxi-
is accurate as long as |p| . 0.16kF . This
ected range of validity of the expansion,
|p|/m, as can be seen from the denomi-
ession of GMF
cont, Eq. (4.9). This condition
⇢m/kF ' 0.15kF for the current values
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
FIG. 11: The range of the continuum (red shed area), the
numerical result for the pole position of the Goldstino prop-
agator (black solid line), and the pole position obtained from
the small |p| expansion, Eq. (4.30) (blue long-dashed line).
For illustration of the “level repulsion”, the pole positions of
GMF
pole (green dashed line) and GRPA
(magenta dotted line) are
also plotted. Note that at p = 0 the tip of the continuum
corresponds to the fictitious pole at ¯! = U⇢, carrying no
spectral weight. The densities are the same as in Sec. IV A,
RPA
Pole
Result at finite p
At small p, 2-peak structure due to level repulsion
(Goldstino+continuum)
43. Goldstino spectrum (interacting case)
43
Mixing from the other point of view
9
...
...
+...
ween
The
sec-
U 1
,
fac-
mean
+ + +...
+ + +...
+ + +...
FIG. 9: The diagrams containing the mixing between the
RPA diagrams (GRPA
) and GMF
pole contributing to GS.
+ + +...
+ + +...
+ + +...
FIG. 10: The diagrams containing the mixing between be-
tween the RPA diagrams (GRPA
) and GMF
pole contributing to
Fermion spectrum is also significantly affected
by the mixing with the supercharge!
Novel feature in BEC phase.
Very similar to QCD!
3.4. The phonino
a
ψ
gg
44. Goldstino spectrum (interacting case)
44
12
fermion self-energy at the two-loop order, which
ke into account.
ve the comparable spectral weights, as can be
ig. 17. This is quite di↵erent compared with
oldstino propagator that we discussed in the
ction, in which the continuum is suppressed
|. Here the continuum ends at p = 0 in a
carries a fraction ⇢f /⇢ of the spectral weight.
r” pole carries a fraction ⇢b/⇢, as can be de-
Eq. (5.3). The total spectral weight is equal
agreement with the well-known sum rule,
Z
d¯!
2⇡
S(p) = 1. (5.5)
momentum exceeds ⇠ 0.21kF , the pole is ab-
he continuum, and the whole spectral weight
ed by the continuum. The width of the peak
nuum is decreasing function of |p| for |p| &
s is to be expected since, when |p| becomes
0
0.2
0.4
-0.4
-0.2
0
0.2
20
40
p/kF
ω
-
/εF
FIG. 16: The fermion spectral function S as a function of
|p| and ¯!. The unit of S is 1/✏F . Densities and coupling are
the same as in Fig. 11. At p = 0, the continuum ends in a
pole with spectral weight ⇢f /⇢, while the spectral weight of
the other pole is ⇢b/⇢.
0.8
1
Similar result to the goldstino spectrum
Small p: Goldstino pole + Continuum
Large p: Free particle pole
10
.29)
.30)
19)]
ex-
ence
mer-
ap-
tion
a =
pling
roxi-
This
sion,
omi-
tion
lues
m is
d to
the
cor-
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.05 0.1 0.15 0.2 0.25
ω
-
/εF
p/kF
FIG. 11: The range of the continuum (red shed area), the
numerical result for the pole position of the Goldstino prop-
agator (black solid line), and the pole position obtained from
the small |p| expansion, Eq. (4.30) (blue long-dashed line).
For illustration of the “level repulsion”, the pole positions of
GMF
pole (green dashed line) and GRPA
(magenta dotted line) are
also plotted. Note that at p = 0 the tip of the continuum
corresponds to the fictitious pole at ¯! = U⇢, carrying no
spectral weight. The densities are the same as in Sec. IV A,
i.e., ⇢b = 2⇢f , and the interaction strength is U⇢f /✏F = 0.1.
contribution vanishes since the pole is absorbed by the
! =
p2
2m
µb Z = 1
! = ↵
p2
2m
Z =
⇢b
⇢
Z =
⇢f
⇢
45. Possible Experimental Detection
45
1. Fermion spectrum
At small p, it is quite different from the free result.
(2-peak structure: Goldstino+continuum)
12
y at the two-loop order, which
spectral weights, as can be
te di↵erent compared with
or that we discussed in the
e continuum is suppressed
nuum ends at p = 0 in a
⇢f /⇢ of the spectral weight.
raction ⇢b/⇢, as can be de-
tal spectral weight is equal
he well-known sum rule,
p) = 1. (5.5)
0
0.2
0.4
-0.4
-0.2
0
0.2
20
40
p/kF
ω
-
/εF
FIG. 16: The fermion spectral function S as a function of
|p| and ¯!. The unit of S is 1/✏F . Densities and coupling are
the same as in Fig. 11. At p = 0, the continuum ends in a
pole with spectral weight ⇢f /⇢, while the spectral weight of
the other pole is ⇢b/⇢.It can be detected via the spectroscopy?
46. Possible Experimental Detection
46
2. Fermion distribution
-2
-1
0
1
2
ω/εF
0
0.5
1
0 0.5 1 1.5 2
nf
p/kF
-2
-1
0
1
2
ω/εF
0
0.5
1
0
nf
2 2
Because the fermion spectrum is modified, the fermion
distribution in momentum space is also changed.
Free case
Only one branch in the spectrum
ω<0 states are occupied (p<kF).
47. Possible Experimental Detection
47
pproxi-
F . This
ansion,
enomi-
ndition
values
uum is
ared to
t of the
F .
ons cor-
d GRPA
evel re-
yielding
etween
se, the
ing the
tails in
nt of
the up-
agator (black solid line), and the pole position obtained from
the small |p| expansion, Eq. (4.30) (blue long-dashed line).
For illustration of the “level repulsion”, the pole positions of
GMF
pole (green dashed line) and GRPA
(magenta dotted line) are
also plotted. Note that at p = 0 the tip of the continuum
corresponds to the fictitious pole at ¯! = U⇢, carrying no
spectral weight. The densities are the same as in Sec. IV A,
i.e., ⇢b = 2⇢f , and the interaction strength is U⇢f /✏F = 0.1.
contribution vanishes since the pole is absorbed by the
continuum (see Fig. 11). The sum of the pole and the
continuum contributions to the zeroth moment equals
⇢, as it should because of the sum rule (2.13). It im-
plies that the spectral weight of the continuum increases
rapidly around the momentum at which the pole is ab-
sorbed, which is demonstrated in Fig. 12. These behav-
iors, namely the suppression (enhancement) of the con-
tinuum at small |p| (above |p| ' 0.21kF ), can be seen
also from the spectral function plotted in Fig. 13.
-2
-1
0
1
2
ω/εF
0
0.5
1
0 0.5 1 1.5 2
nf
p/kF
-2
-1
0
1
2
ω/εF
0
0.5
1
0 0.5 1 1.5 2
nf
p/kF
Weak coupling case
Goldstino pole+Continuum
Near kF, the pole energy becomes positive.
Almost same as free case,
because the weight of the pole near kF
is almost one.
48. Possible Experimental Detection
48
0
0.5
0 0.5 1 1.5 2
p/kF
-3
-2
-1
0
1
2
ω/εF
0
0.5
1
0 0.5 1 1.5 2
nf
p/kF
nf
U⇢f /✏F = 2/3
Strong coupling case
Goldstino pole and Continuum is
separated, since the distance (Uρ)
becomes large in strong coupling.
The energy at which the pole energy
becomes positive is different from kF.
Fermi sea is distorted.
49. • We analyzed the spectral properties of the goldstino in the
absence/presence of interaction with RPA.
• We observed the crossover from small p to large p region
(from interaction dominant to free case).
• In BEC phase, the importance of the mixing process between
Fermion particle-Boson hole excitation and the Fermion
hole-Boson particle excitation was emphasized.
• We discussed the possibility for experimental detection of
the goldstino.
Summary
49