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Journal of Physics A: Mathematical and Theoretical

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Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.

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Géza Tóth and Iagoba Apellaniz 2014 J. Phys. A: Math. Theor. 47 424006

We summarize important recent advances in quantum metrology, in connection to experiments in cold gases, trapped cold atoms and photons. First we review simple metrological setups, such as quantum metrology with spin squeezed states, with Greenberger–Horne–Zeilinger states, Dicke states and singlet states. We calculate the highest precision achievable in these schemes. Then, we present the fundamental notions of quantum metrology, such as shot-noise scaling, Heisenberg scaling, the quantum Fisher information and the Cramér–Rao bound. Using these, we demonstrate that entanglement is needed to surpass the shot-noise scaling in very general metrological tasks with a linear interferometer. We discuss some applications of the quantum Fisher information, such as how it can be used to obtain a criterion for a quantum state to be a macroscopic superposition. We show how it is related to the speed of a quantum evolution, and how it appears in the theory of the quantum Zeno effect. Finally, we explain how uncorrelated noise limits the highest achievable precision in very general metrological tasks.

This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

Ginestra Bianconi 2024 J. Phys. A: Math. Theor. 57 365002

We propose a theory for coupling matter fields with discrete geometry on higher-order networks, i.e. cell complexes. The key idea of the approach is to associate to a higher-order network the quantum entropy of its metric. Specifically we propose an action having two contributions. The first contribution is proportional to the logarithm of the volume associated to the higher-order network by the metric. In the vacuum this contribution determines the entropy of the geometry. The second contribution is the quantum relative entropy between the metric of the higher-order network and the metric induced by the matter and gauge fields. The induced metric is defined in terms of the topological spinors and the discrete Dirac operators. The topological spinors, defined on nodes, edges and higher-dimensional cells, encode for the matter fields. The discrete Dirac operators act on topological spinors, and depend on the metric of the higher-order network as well as on the gauge fields via a discrete version of the minimal substitution. We derive the coupled dynamical equations for the metric, the matter and the gauge fields, providing an information theory principle to obtain the field theory equations in discrete curved space.

Giuseppe Gaeta and Epifanio G Virga 2023 J. Phys. A: Math. Theor. 56 363001

In its most restrictive definition, an octupolar tensor is a fully symmetric traceless third-rank tensor in three space dimensions. So great a body of works have been devoted to this specific class of tensors and their physical applications that a review would perhaps be welcome by a number of students. Here, we endeavour to place octupolar tensors into a broader perspective, considering non-vanishing traces and non-fully symmetric tensors as well. A number of general concepts are recalled and applied to either octupolar and higher-rank tensors. As a tool to navigate the diversity of scenarios we envision, we introduce the octupolar potential , a scalar-valued function which can easily be given an instructive geometrical representation. Physical applications are plenty; those to liquid crystal science play a major role here, as they were the original motivation for our interest in the topic of this review.

Michael (Misha) Chertkov 2024 J. Phys. A: Math. Theor. 57 333001

The paper reflects on the future role of artificial intelligence (AI) in scientific research, with a special focus on turbulence studies, and examines the evolution of AI, particularly through Diffusion Models rooted in non-equilibrium statistical mechanics. It underscores the significant impact of AI on advancing reduced, Lagrangian models of turbulence through innovative use of Deep Neural Networks. Additionally, the paper reviews various other AI applications in turbulence research and outlines potential challenges and opportunities in the concurrent advancement of AI and statistical hydrodynamics. This discussion sets the stage for a future where AI and turbulence research are intricately intertwined, leading to more profound insights and advancements in both fields.

Zvi Bern et al 2024 J. Phys. A: Math. Theor. 57 333002

This review describes the duality between color and kinematics and its applications, with the aim of gaining a deeper understanding of the perturbative structure of gauge and gravity theories. We emphasize, in particular, applications to loop-level calculations, the broad web of theories linked by the duality and the associated double-copy structure, and the issue of extending the duality and double copy beyond scattering amplitudes. The review is aimed at doctoral students and junior researchers both inside and outside the field of amplitudes and is accompanied by various exercises.

Farhang Loran and Ali Mostafazadeh 2024 J. Phys. A: Math. Theor. 57 335205

Luca Angelani 2023 J. Phys. A: Math. Theor. 56 455003

The motion of run-and-tumble particles in one-dimensional finite domains are analyzed in the presence of generic boundary conditions. These describe accumulation at walls, where particles can either be absorbed at a given rate, or tumble, with a rate that may be, in general, different from that in the bulk. This formulation allows us to treat in a unified way very different boundary conditions (fully and partially absorbing/reflecting, sticky, sticky-reactive and sticky-absorbing boundaries) which can be recovered as appropriate limits of the general case. We report the general expression of the mean exit time, valid for generic boundaries, discussing many case studies, from equal boundaries to more interesting cases of different boundary conditions at the two ends of the domain, resulting in nontrivial expressions of mean exit times.

Francisco J Sevilla et al 2024 J. Phys. A: Math. Theor. 57 335004

Jing Liu et al 2020 J. Phys. A: Math. Theor. 53 023001

Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cramér–Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cramér–Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.

Katarzyna Siudzińska 2024 J. Phys. A: Math. Theor. 57 335302

Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete positive, operator-valued measure (POVMs) that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical two-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.

Latest articles

Michael Q May and Hong Qin 2024 J. Phys. A: Math. Theor. 57 415304

K Ziegler 2024 J. Phys. A: Math. Theor. 57 415303

We study the effect of random scattering in quantum walks on a finite graph and compare it with the effect of repeated measurements. To this end, a constructive approach is employed by introducing a localized and a delocalized basis for the underlying Hilbert space. This enables us to design Hamiltonians whose eigenvectors are either localized or delocalized. By presenting some specific examples we demonstrate that the localization of eigenvectors restricts the transition probabilities on the graph and leads to a removal of energy states from the quantum walk in the monitored evolution. We conclude that repeated measurements as well as random scattering provide efficient tools for controlling quantum walks.

Andrea Cavagna et al 2024 J. Phys. A: Math. Theor. 57 415002

Experiments on bird flocks and midge swarms reveal that these natural systems are well described by an active theory in which conservation laws play a crucial role. By building a symplectic structure that couples the particles' velocities to the generator of their internal rotations (spin), the Inertial Spin Model (ISM) reinstates a second-order temporal dynamics that captures many phenomenological traits of flocks and swarms. The reversible structure of the ISM predicts that the total spin is a constant of motion, the central conservation law responsible for all the novel dynamical features of the model. However, fluctuations and dissipation introduced in the original model to make it relax, violate the spin conservation law, so that the ISM aligns with the biophysical phenomenology only within finite-size regimes, beyond which the overdamped dynamics characteristic of the Vicsek model takes over. Here, we introduce a novel version of the ISM, in which the irreversible terms needed to relax the dynamics strictly respect the conservation of the spin. We perform a numerical investigation of the fully conservative model, exploring both the fixed-network case, which belongs to the equilibrium class of Model G, and the active case, characterized by self-propulsion of the agents and an out-of-equilibrium reshuffling of the underlying interaction network. Our simulations not only capture the correct spin wave phenomenology of the ordered phase, but they also yield dynamical critical exponents in the near-ordering phase that agree very well with the theoretical predictions.

Francesco Iachello et al 2024 J. Phys. A: Math. Theor. 57 415302

Shubham Garg and Kirankumar R Hiremath 2024 J. Phys. A: Math. Theor. 57 415202

Many interesting phenomena in applications are based on interactions between their constituent sub-systems. The first principle exact models of these phenomena can be quite complicated. Therefore, many practitioners prefer to use so-called phenomenological models, which are generally known as models based on coupled mode theory (CMT). This type of reduced-order model captures the dominant behavior of the system under appropriate conditions. Quite often, these validity conditions are qualitatively described, but no detailed mathematical analysis is provided. This work addresses this issue and presents improvements in the traditional phenomenological models. Although an LC circuit model is used for illustration due to its simplicity, the results in this work are equally applicable to a wide variety of coupled models. A detailed mathematical analysis is carried out to quantify the order of approximation involved in the model-based CMT. Using it, the validity of the model in the regime from weak coupling to strong coupling is analytically investigated. An improved reduced-order model is proposed, which gives better results than the traditional phenomenological model. The analytical studies are verified with numerical simulations, which clearly show better validity of the proposed improved model of coupled systems.

Review articles

D B Milošević et al 2024 J. Phys. A: Math. Theor. 57 393001

Norbert Büttgen and Hans-Albrecht Krug von Nidda 2024 J. Phys. A: Math. Theor. 57 313001

Based on a previous review on magnetic resonance in quantum spin chains (Krug von Nidda et al 2010 Eur. Phys. J. Spec. Top. 180 161–89) we report on further development in this field with special focus on transition–metal oxides and halogenides consisting of quasi one–dimensional spin systems, where both intra–and inter–chain exchange interaction may give rise to frustration effects and higher–order anisotropic exchange contributions like the Dzyaloshinskii–Moriya interaction become decisive for the formation of the magnetic ground state. Selected examples show how NMR and ESR contribute valuable information on the magnetic phases and exchange interactions involved: LiCuVO 4 with competing nearest neighbour and next–nearest neighbour intra–chain exchange, LiCu 2 O 2 with complex zig–zag chains, and Cs 2 CuCl 4 where the chains form a triangular lattice with the inter–chain interaction weaker but of the same order of magnitude than the intra–chain interaction. The so called paper–chain compound Ba 3 Cu 3 In 4 O 12 , where each successive pair of CuO 4 plaquettes is rotated by 90° with respect to its predecessor along the c –direction like in a paper–chain, provides an interesting topology of frustrated intra–chain exchange interactions. Finally, a few dimer systems are considered.

Iddo Eliazar 2024 J. Phys. A: Math. Theor. 57 233002

Diffusion is a generic term for random motions whose positions become more and more diffuse with time. Diffusion is of major importance in numerous areas of science and engineering, and the research of diffusion is vast and profound. This paper is the first in a stochastic 'intro series' to the multidisciplinary field of diffusion. The paper sets off from a basic question: how to quantitatively measure diffusivity? Having answered the basic question, the paper carries on to a follow-up question regarding statistical behaviors of diffusion: what further knowledge can the diffusivity measure provide, and when can it do so? The answers to the follow-up question lead to an assortment of notions and topics including: persistence and anti-persistence; aging and anti-aging; short-range and long-range dependence; the Wiener–Khinchin theorem and its generalizations; spectral densities, white noise, and their generalizations; and colored noises. Observing diffusion from a macro level, the paper culminates with: the universal emergence of power-law diffusivity; the three universal diffusion regimes—one regular, and two anomalous; and the universal emergence of 1/f noise. The paper is entirely self-contained, and its prerequisites are undergraduate mathematics and statistics.

Featured articles

Tim Adamo and Sumer Jaitly 2020 J. Phys. A: Math. Theor. 53 055401

Keith Alexander et al 2020 J. Phys. A: Math. Theor. 53 045001

We probe the character of knotting in open, confined polymers, assigning knot types to open curves by identifying their projections as virtual knots. In this sense, virtual knots are transitional, lying in between classical knot types, which are useful to classify the ambiguous nature of knotting in open curves. Modelling confined polymers using both lattice walks and ideal chains, we find an ensemble of random, tangled open curves whose knotting is not dominated by any single knot type, a behaviour we call weakly knotted. We compare cubically confined lattice walks and spherically confined ideal chains, finding the weak knotting probability in both families is quite similar and growing with length, despite the overall knotting probability being quite different. In contrast, the probability of weak knotting in unconfined walks is small at all lengths investigated. For spherically confined ideal chains, weak knotting is strongly correlated with the degree of confinement but is almost entirely independent of length. For ideal chains confined to tubes and slits, weak knotting is correlated with an adjusted degree of confinement, again with length having negligible effect.

Yongchao Lü and Joseph A Minahan 2020 J. Phys. A: Math. Theor. 53 024001

Accepted manuscripts

Rouhi et al 

An exceptional point of degeneracy (EPD) occurs when both the eigenvalues and the corresponding eigenvectors of a square matrix coincide and the matrix has a nontrivial Jordan block structure. It is not easy to achieve an EPD exactly. In our prior studies, we synthesized simple conservative (lossless) circuits with evolution matrices featuring EPDs by using two LC loops coupled by a gyrator. In this paper, we advance even a simpler circuit with an EPD consisting of only two LC loops with one capacitor shared. Consequently, this circuit involves only four elements and it is perfectly reciprocal. The shared capacitance and parallel inductance are negative with values determined by explicit formulas which lead to EPD. This circuit can have the same Jordan canonical form as the nonreciprocal circuit we introduced before. This implies that the Jordan canonical form does not necessarily manifest systems' nonreciprocity. It is natural to ask how nonreciprocity is manifested in the system's spectral data. Our analysis of this issue shows that nonreciprocity is manifested explicitly in: (i) the circuit Lagrangian and (ii) the breakdown of certain symmetries in the set of eigenmodes. All our significant theoretical findings were thoroughly tested and confirmed by numerical simulations using commercial circuit simulator software.

Mariji et al 

We develop a self-consistent theoretical formalism to model the dynamics of heat transfer in
dissipative, dispersive, anisotropic nanoscale media, such as metamaterials. We employ our envelope
dyadic Green's function method to solve Maxwell's macroscopic equations for the propagation of
fluctuating electromagnetic fields in the media. We assume that the photonic radiative heat transfer
mechanism in the media is complemented by phononic mechanisms of heat storage and conduction,
accounting for effects of local heat generation and material phase transitions. By employing the Poynting
theorem and the fluctuation-dissipation theorem, we derive novel closed-form expressions for the coupling
term of photonic and phononic subsystems, which contains the heating rate and the radiative heat power
terms, as well as for the radiative heat flux. We apply our formalism to the paraxial heat transfer
in uniaxial media and present relevant closed-form expressions. By considering a Gaussian transverse
temperature profile, we also obtain and solve the integro-differential heat diffusion equations to model
the paraxial heat transfer in uniaxial reciprocal media.

Aizawa et al 

A superfield formalism for the minimal Z 2 2 -graded version of supersymmetry is developed. This is done by using the recently introduced definition of integration on the minimal Z 2 2 -superspace. It is shown that one may construct Z 2 2 -supersymmetric action by the procedure similar to the standard supersymmetry. 
However, the Lagrangian obtained has very general interaction terms, which give rise to a Z 2 2 -graded extension of many known theories defined in two-dimensional spacetime. 
As an illustration, we will give a Z 2 2 -extension of the sine-Gordon model different from the one already discussed in the literature.

WEI et al 

We describe how the methods of group theory (symmetry) are used to optimize the problem of exact diagonalization of a quantum system on a 16-site pyrochlore lattice. By analytically constructing a complete set of symmetrized states, we completely block-diagonalize the Hamiltonian. As an example, we consider a spin-1/2 system with nearest neighbour exchange interactions.

Takács et al 

Quasicondensation in one dimension is known to occur for equilibrium systems of hard-core bosons (HCBs) at zero temperature. This phenomenon arises due to the off-diagonal long-range order in the ground state, characterized by a power-law decay of the one-particle density matrix $g_1(x,y)\sim |x-y|^{-1/2}$~--~a well-known outcome of Luttinger liquid theory. 
Remarkably, HCBs, when allowed to freely expand from an initial product state (i.e., characterized by initial zero correlation), exhibit quasicondensation and demonstrate the emergence of off-diagonal long-range order during nonequilibrium dynamics. 
This phenomenon has been substantiated by numerical and experimental investigations in the early 2000s.
In this work, we revisit the dynamical quasicondensation of HCBs, providing a fully analytical treatment of the issue. 
In particular, we derive an exact asymptotic formula for the equal-time one-particle density matrix by borrowing ideas from the framework of quantum Generalized Hydrodynamics. 
Our findings elucidate the phenomenology of quasicondensation and of dynamical fermionization occurring at different stages of the time evolution, as well as the crossover between the two. 

More Accepted manuscripts

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Open access.

Marco Benedetti and Enrico Ventura 2024 J. Phys. A: Math. Theor. 57 415001

The beneficial role of noise-injection in learning is a consolidated concept in the field of artificial neural networks, suggesting that even biological systems might take advantage of similar mechanisms to optimize their performance. The training-with-noise (TWN) algorithm proposed by Gardner and collaborators is an emblematic example of a noise-injection procedure in recurrent networks, which can be used to model biological neural systems. We show how adding structure to noisy training data can substantially improve the algorithm performance, allowing the network to approach perfect retrieval of the memories and wide basins of attraction, even in the scenario of maximal injected noise. We also prove that the so-called Hebbian Unlearning rule coincides with the TWN algorithm when noise is maximal and data are stable fixed points of the network dynamics.

CHEN WEI and S H Curnoe 2024 J. Phys. A: Math. Theor.

Federico Corberi and Luca Smaldone 2024 J. Phys. A: Math. Theor.

We study the non-equilibrium response function $R_{ij}(t,t')$, namely the variation of the local magnetization $\langle S_i(t)\rangle$ on site $i$ at time $t$ as an effect of a perturbation applied at the earlier time $t'$ on site $j$, in a class of solvable spin models characterized by the vanishing of the so-called {\it asymmetry}.
This class encompasses both systems brought out of equilibrium by the variation of a thermodynamic control parameter, as after a temperature quench, or intrinsically out of equilibrium models with violation of detailed balance. The one-dimensional Ising model and the voter model (on an arbitrary graph) are prototypical examples of these two situations which are used here as guiding examples. Defining the fluctuation-dissipation ratio $X_{ij}(t,t')=\beta R_{ij}/(\partial G_{ij}/\partial t')$, where $G_{ij}(t,t')=\langle S_i(t)S_j(t')\rangle$ is the spin-spin correlation function and $\beta$ is a parameter regulating the strength of the perturbation (corresponding to the inverse temperature when detailed balance holds), we show that, in the quite general case of a kinetics obeying dynamical scaling, on equal sites this quantity has a universal form
$X_{ii}(t,t') = (t+t')/(2t)$, whereas $
\lim _{t\to \infty}X_{ij}(t,t')=1/2$ for any $ij$ couple. The specific case of voter models with long-range interactions is thoroughly discussed.

Andrew N W Hone et al 2024 J. Phys. A: Math. Theor. 57 415201

Maurice Duits et al 2024 J. Phys. A: Math. Theor. 57 405202

We study the limiting behavior of random lozenge tilings of the hexagon with a q -Racah weight as the size of the hexagon grows large. Based on the asymptotic behavior of the recurrence coefficients of the q -Racah polynomials, we give a new proof for the fact that the height function for a random tiling concentrates near a deterministic limit shape and that the global fluctuations are described by the Gaussian free field. These results were recently proved using (dynamic) loop equation techniques. In this paper, we extend the recurrence coefficient approach that was developed for (dynamic) orthogonal polynomial ensembles to the setting of q -orthogonal polynomials. An interesting feature is that the complex structure is easily found from the limiting behavior of the (explicitly known) recurrence coefficients. A particular motivation for studying this model is that the variational characterization of the limiting height function has an inhomogeneous term. The study of the regularity properties of the minimizer for general variational problems with such inhomogeneous terms is a challenging open problem. In a general setup, we show that the variational problem gives rise to a natural complex structure associated with the same Beltrami equation as in the homogeneous situation. We also derive a relation between the complex structure and the complex slope. In the case of the q -Racah weighting of lozenge tilings of the hexagon, our representation of the limit shape and their fluctuations in terms of the recurrence coefficients allows us to verify this relation explicitly.

Anastasia Doikou et al 2024 J. Phys. A: Math. Theor. 57 405203

Albert Rico Andres and Karol Życzkowski 2024 J. Phys. A: Math. Theor.

A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators summing to identity. This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. Discrete transformations in the set of quantum measurements can be described by blockwise stochastic matrices , composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices (also called quantum magic squares ) and study their properties. Analyzing the dynamics they induce we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in this set.

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• 2007-present Journal of Physics A: Mathematical and Theoretical doi: 10.1088/issn.1751-8121 Online ISSN: 1751-8121 Print ISSN: 1751-8113

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• 2007-present Journal of Physics A: Mathematical and Theoretical
• 1975-2006 Journal of Physics A: Mathematical and General
• 1973-1974 Journal of Physics A: Mathematical, Nuclear and General
• 1968-1972 Journal of Physics A: General Physics

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Mathematical Physics

A section of Mathematics (ISSN 2227-7390).

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The section Mathematical Physics will publish articles that apply mathematics to any area of physics, with the aim of bridging these two disciplines. We welcome original research of the highest quality in all active areas of mathematical physics. Review articles which are of common interest to practitioners in both fields are also welcome.

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Theoretical physics articles from across Nature Portfolio

Theoretical physics is the development of mathematical formalisms and computational protocols for describing all aspects of objects found in the world around us and their interaction. This can involve both providing models for understanding empirical results or constructing self-logical theories for explain phenomena beyond current experiments.

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News and Comment

Photon losses create tension for Gaussian boson sampling

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The tangled state of quantum hypothesis testing

Quantum hypothesis testing—the task of distinguishing quantum states—enjoys surprisingly deep connections with the theory of entanglement. Recent findings have reopened the biggest questions in hypothesis testing and reversible entanglement manipulation.

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Scientific illustration: striking the balance between creativity and accuracy

A misleading image in a medical textbook could have life and death implications, but some disciplines can deploy myth and metaphor to convey their science through art.

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The gravity of quantum thermalization

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Supersymmetric renormalization group flow

Supersymmetric quantum field theories have special properties that make them easier to study. This Comment discusses how the constraints that supersymmetry places on renormalization group flows have been used to study strongly coupled field theories.

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Journal of Mathematical Physics

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• Mathematical Physics
• Statistical and Nonlinear Physics

American Institute of Physics

Publication type

00222488, 10897658

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The set of journals have been ranked according to their SJR and divided into four equal groups, four quartiles. Q1 (green) comprises the quarter of the journals with the highest values, Q2 (yellow) the second highest values, Q3 (orange) the third highest values and Q4 (red) the lowest values.

The SJR is a size-independent prestige indicator that ranks journals by their 'average prestige per article'. It is based on the idea that 'all citations are not created equal'. SJR is a measure of scientific influence of journals that accounts for both the number of citations received by a journal and the importance or prestige of the journals where such citations come from It measures the scientific influence of the average article in a journal, it expresses how central to the global scientific discussion an average article of the journal is.

Evolution of the number of published documents. All types of documents are considered, including citable and non citable documents.

This indicator counts the number of citations received by documents from a journal and divides them by the total number of documents published in that journal. The chart shows the evolution of the average number of times documents published in a journal in the past two, three and four years have been cited in the current year. The two years line is equivalent to journal impact factor ™ (Thomson Reuters) metric.

Evolution of the total number of citations and journal's self-citations received by a journal's published documents during the three previous years. Journal Self-citation is defined as the number of citation from a journal citing article to articles published by the same journal.

Evolution of the number of total citation per document and external citation per document (i.e. journal self-citations removed) received by a journal's published documents during the three previous years. External citations are calculated by subtracting the number of self-citations from the total number of citations received by the journal’s documents.

International Collaboration accounts for the articles that have been produced by researchers from several countries. The chart shows the ratio of a journal's documents signed by researchers from more than one country; that is including more than one country address.

Not every article in a journal is considered primary research and therefore "citable", this chart shows the ratio of a journal's articles including substantial research (research articles, conference papers and reviews) in three year windows vs. those documents other than research articles, reviews and conference papers.

Ratio of a journal's items, grouped in three years windows, that have been cited at least once vs. those not cited during the following year.

Evolution of the percentage of female authors.

Evolution of the number of documents cited by public policy documents according to Overton database.

Evoution of the number of documents related to Sustainable Development Goals defined by United Nations. Available from 2018 onwards.

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Mathematics (since February 1992)

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Theoretical and Mathematical Physics

• Covers quantum field theory, theory of elementary particles, nuclear physics, many-body problems and statistical physics, nonrelativistic quantum mechanics, and gravitation theory.
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Latest issue

Volume 220, Issue 3

Latest articles

On positive fixed points of operator of hammerstein type with degenerate kernel and gibbs measures.

• I. M. Mavlonov
• Kh. N. Khushvaktov
• F. H. Haydarov

Boundedness-below conditions for a general scalar potential of two real scalar fields and the Higgs boson

• Yisheng Song

Exact solution of the problem of the interaction between a point charge and an insulator with nonlinear susceptibility

• A. A. Belov
• M. A. Tintul
• P. A. Polyakov

Quasi-Grammian loop dynamics of a multicomponent semidiscrete short pulse equation

• M. ul Hassan

Explicit multiple solitons of the mixed Chen–Lee–Liu equation derived from the Riemann–Hilbert approach

• Yumin Zheng
• Yunqing Yang

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