Journal of Physics A: Mathematical and Theoretical

In a multi-level energy system with energy transitions, dark states are eigenstates of a Hamiltonian that consist entirely of ground states, with zero amplitude in the excited states. We present several criteria which allows one to deduce the presence of dark states in a general multi-level system based on the submatrices of the Hamiltonian. The dark states can be shown to be the right-singular vectors of the submatrix that connect the ground states to the excited states. Furthermore, we show a simple way of finding the dark state involving the determinant of a matrix constructed from the same submatrix.

Some new Hamiltonian systems of quasi-Painlevé type are presented and the analogue of Okamoto's space of initial conditions computed. Using the geometric approach that was introduced originally for the identification problem of Painlevé equations, comparing the irreducible components of the inaccessible divisors arising in the blow-up process, we find bi-rational coordinate changes between some of these systems that give rise to the same global Hamiltonian structure. This scheme thus gives a method for identifying Hamiltonian systems up to bi-rational maps, which is performed in this article for systems of quasi-Painlevé type having singularities that are either square-root type algebraic poles or ordinary poles.

The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation $\dot{x}(t)+\gamma x = \sqrt{2 D}\xi(t)$, where $\xi(t)$ is the fractional Gaussian noise with the Hurst exponent $0 \lt H \lt 1$. For $H\neq 1/2$ the fOU process is non-Markovian but Gaussian, and it has either vanishing (for $H \lt 1/2$), or divergent (for $H \gt 1/2$) spectral density at zero frequency. For $H \gt 1/2$, the fOU is long-correlated. Here we study dynamical large deviations of the fOU process and focus on the area $A_n = \int_{-T}^{T} x^n(t) \mathrm{d}t$, $n = 1,2,\ldots$ over a long time window 2T. Employing the optimal fluctuation method, we determine the optimal path of the conditioned process, which dominates the large-A_n tail of the probability distribution of the area, $\mathcal{P}(A_n,T)\sim \exp[-S(A_n,T)]$. We uncover a nontrivial phase diagram of scaling behaviors of the optimal paths and of the action $S(A_n\equiv 2 a_n T,T)\sim T^{\alpha(H,n)} a^{2/n}_n$ on the (H, n) plane. The phase diagram includes three distinct regions: (i) $H \gt 1-1/n$, where $\alpha(H,n) = 2-2H$, and the optimal paths are delocalized in time, (ii) n = 2 and $H\unicode{x2A7D} \frac{1}{2}$, where $\alpha(H,n) = 1$, and the optimal paths oscillate with an H-dependent frequency, and (iii) $H\unicode{x2A7D} 1-1/n$ and n > 2, where $\alpha(H,n) = 2/n$, and the optimal paths are strongly localized. We verify our theoretical predictions in large-deviation simulations of the fOU process. By combining the Wang-Landau Monte-Carlo algorithm with the circulant embedding method of generation of stationary Gaussian fields, we were able to measure probability densities as small as 10⁻¹⁷⁰. We also generalize our findings to other stationary Gaussian processes with either divergent, or vanishing spectral density at zero frequency.

For three-dimensional circular-quiver supersymmetric Chern–Simons theories, the questions, whether duality cascades always terminate and whether the endpoint is unique, were rephrased into the question whether a polytope defined in the parameter space of relative ranks for duality cascades is a parallelotope, filling the space by discrete translations. By regarding circular quivers as affine Dynkin diagrams, we generalize the arguments into other affine quivers. We find that, after rewriting properties into the group-theoretical language, most arguments work in the generalizations. Especially, we find that, instead of the original relation to parallelotopes, the corresponding polytope still fills the parameter space but with some gaps. This indicates that, under certain restrictions, duality cascades still terminate uniquely.

Real 3-manifold triangulations can be uniquely represented by isomorphism signatures. Databases of these isomorphism signatures are generated for a variety of 3-manifolds and knot complements, using SnapPy and Regina, then these language-like inputs are used to train various machine learning architectures to differentiate the manifolds, as well as their Dehn surgeries, via their triangulations. Gradient saliency analysis then extracts key parts of this language-like encoding scheme from the trained models. The isomorphism signature databases are taken from the 3-manifolds' Pachner graphs, which are also generated in bulk for some selected manifolds of focus and for the subset of the SnapPy orientable cusped census with $\lt 8$ initial tetrahedra. These Pachner graphs are further analysed through the lens of network science to identify new structure in the triangulation representation; in particular for the hyperbolic case, a relation between the length of the shortest geodesic (systole) and the size of the Pachner graph's ball is observed.

Quantum sensors are now universally acknowledged as one of the most promising near-term quantum technologies. The traditional formulation of quantum sensing introduces a concrete bound on ultimate precision through the so-called local sensing framework, in which a significant knowledge of prior information about the unknown parameter value is implicitly assumed. Moreover, the framework provides a systematic approach for optimizing the sensing protocol. In contrast, the paradigm of global sensing aims to find a precision bound for parameter estimation in the absence of such prior information. In recent years, vigorous research has been pursued to describe the contours of global quantum estimation. Here, we review some of these emerging developments. These developments are both in the realm of finding ultimate precision bounds with respect to appropriate figures of merit in the global sensing paradigm, as well as in the search for algorithms that achieve these bounds. We categorize these developments into two largely mutually exclusive camps; one employing Bayesian updating and the other seeking to generalize the frequentist picture of local sensing towards the global paradigm. In the first approach, in order to achieve the best performance, one has to optimize the measurement settings adaptively. In the second approach, the measurement setting is fixed, however the challenge is to identify this fixed measurement optimally.

More then 35 approaches to the Dirac equation derivation are presented. Various physical principles and mathematical methods are used. A review of well-known and not enough known contributions to the problem is given, unexpected and unconventional derivations are presented as well. Three original approaches to the problem suggested by the author are considered: (i) generalization of H. Sallhofer derivation, (ii) obtaining of massless Dirac equation from the Maxwell equations in a maximally symmetrical form, (iii) derivation of the Dirac equation with nonzero mass from relativistic canonical quantum mechanics of the fermion-antifermion spin s = 1/2 doublet. Today we are able to demonstrate new features of our derivations given in original papers. In some sense the important role of the Dirac equation in contemporary theoretical physics is demonstrated. A criterion for the usefulness of one or another derivation of the Dirac equation is established.

We review different exact approaches to string theory. In the context of the Green–Schwarz superstring, we discuss the action in curved backgrounds and its supercoset formulation, with particular attention to superstring backgrounds of the AdS₃ type supported by both Ramond–Ramond and Neveu–Schwarz–Neveu–Schwarz fluxes. This is the basis for the discussion of classical integrability, of worldsheet-scattering factorisation in the uniform lightcone gauge, and eventually of the string spectrum through the mirror thermodynamic Bethe ansatz, which for AdS₃ backgrounds was only derived and analysed very recently. We then illustrate some aspects of the Ramond–Neveu–Schwarz string, and introduce the formalism of Berkovits–Vafa–Witten, which has seen very recent applications to AdS₃ physics, which we also briefly review. Finally, we present the relation between M-theory in the discrete lightcone quantisation and decoupling limits of string theory that exhibit non-relativistic behaviors, highlighting the connection with integrable $T\bar{T}$ deformations, as well as the relation between spin-matrix theory and Landau–Lifshitz models.

This review is based on lectures given at the Young Researchers Integrability School and Workshop 2022 'Taming the string worldsheet' at NORDITA, Stockholm.

The quantum-mechanical transition amplitudes for atomic and molecular processes in strong laser fields are expressed in the form of multidimensional integrals of highly oscillatory functions. Such integrals are ideally suited for the evaluation by asymptotic methods for integrals. Furthermore, using these methods it is possible to identify, in the sense of Feynman's path-integral formalism, the partial contributions of quantum orbits, which are related to particular solutions of the saddle-point equations. This affords insight into the physics of the problem, which would not have been possible by only solving these integrals numerically. We apply the saddle-point method to various quantum processes that are important in strong-field physics and attoscience. The special case of coalescing or near-coalescing saddle points requires application of the uniform approximation. We also present two modifications of the saddle-point method, for the cases where a singular point of the subintegral function exactly overlaps with a saddle point or is located in its close vicinity. Particular emphasis is on the classification of the saddle-point solutions. This problem is solved for the one-dimensional integral over the ionization time, relevant for above-threshold ionization (ATI), while for two-dimensional integrals a classification by the multi-index $(\alpha,\beta,m)$ is introduced, which is particularly useful for the medium- and high-energy spectrum of high-order harmonic generation (HHG) and backward-scattered electrons (for high-order ATI). For the low-energy structures a classification using the multi-index $(\nu,\rho,\mu)$ is introduced for the forward-scattering quantum orbits. In addition to laser-induced processes such as ATI, HHG and high-order ATI, we consider laser-assisted scattering as an example of laser-assisted processes for which real solutions of the saddle-point equation exist. Particular attention is devoted to the quantum orbits that describe and visualize these processes. We also consider finite laser pulses, the semiclassical approximation, the role of the Coulomb field and the case of laser fields intense enough to lead into the relativistic regime.

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.

Describing systems with non-Hermitian (NH) operators remains a challenge in quantum theory due to singularities (e.g., exceptional points and decoherence) arising from interactions with the environment. We introduce a well-defined computational basis for representing the NH Hamiltonian eigenstates, wherein singularities are shifted from the basis states to the expansion coefficients, simplifying the mathematical treatment of open quantum systems. Furthermore, we introduce a local `space-time' transformation on the computational basis that defines a generic dual space mapping. Interestingly, this transformation reveals a static/global symmetry for real/imaginary energy values, unveiling inherent conserved quantities in open quantum systems. Our formalism provides new insights into key features such as exceptional points, dual space maps, and the origin of symmetry-enforced real eigenvalues. The applicability of our framework extends to various branches of physics where NH operators manifest as ladder operators, order parameters, self-energies, projectors, and other entities.

The stationary Dirac equation, confined to a two-dimensional (2D) region, supports states propagating along the boundary and decaying exponentially away from the boundary. These edge states appear on the 2D surface of a 3D topological insulator, where massless fermionic quasiparticles are governed by the Dirac equation and confined by a magnetic insulator. We show how the continuous system can be simulated on a 2D square lattice, without running into the fermion-doubling obstruction. For that purpose we adapt the existing tangent fermion discretization on an unbounded lattice to account for a lattice termination that simulates the magnetic insulator interface.

Atomic frequency comb (AFC) quantum memories are a promising technology for quantum repeater networks because they enable multi-mode, long-time, and high-fidelity storage of photons with on-demand retrieval. The optimization of the retrieval efficiency of an AFC memory is important because it strongly impacts the entanglement distribution rate in quantum networks. Despite initial theoretical analyses and recent experimental demonstrations, a rigorous proof of the universally optimal configuration for the highest AFC retrieval efficiency has not been presented. In this paper we present a simple analytical proof which shows that the optimized square tooth offers the highest retrieval efficiency among all tooth shapes, under the physical constraint of finite optical depth of an atomic ensemble. The optimality still holds when the non-zero background absorption and the finite optical linewidth of atoms are considered. We further compare square, Lorentzian and Gaussian tooth shapes to reinforce the practical advantage of the square-tooth AFC in retrieval efficiency. Our proof lays rigorous foundation for the recipe of creating optimal AFC under realistic experimental conditions.

In this work we present a computation of the averages of conserved charge densities and currents of (1+1)-dimensional Integrable Quantum Field Theories in Generalised Gibbs Ensembles. Our approach is based on the quasi-particle description provided by the Thermodynamic Bethe Ansatz combined with the principles of Generalised Hydrodynamics, and we focus on Non-Equilibrium Steady State averages. When considering the ultraviolet (i.e. high temperature) limit of such averages, we recover the famous result by Bernard and Doyon (2012) for the energy current and density in Conformal Field Theories, and we extend it to conserved quantities with spin $s > 1$. We show that their averages are proportional to $T_L^{s+1}\pm T_R^{s+1}$, with $T_L$, $T_R$ the temperatures of two asymptotic thermal reservoirs. The same power law is obtained when considering some non-thermal generalised Gibbs states. In Conformal Field Theory, the power law is a consequence of the transformation properties of conserved charge operators, while the proportionality coefficient depends on the spin of the operator and on the central charge of the theory. We present an exact analytic expression for this coefficient in the case of a massive free fermion. At equilibrium, proportionality of spin-$s$ density averages to $T^{s+1}$ can be thought of as a generalisation of Stefan-Boltzmann's law, which states that the energy per unit surface area radiated by a black body scales as $T^4$.

Minimal unitary representation of $SO(d,2)$ and its deformations describe all the conformally massless fields in $d$ dimensional Minkowskian spacetimes. In critical dimensions these spacetimes admit extensions with twistorial coordinates plus a dilatonic coordinate to causal spacetimes coordinatized by Jordan algebras $J_3^{\mathbb{A}}$ of degree three over the four division algebras $\mathbb{A}=\mathbb{R} , \mathbb{C} , \mathbb{H} , \mathbb{O} $. We study the minimal unitary representation (minrep) of the conformal group $E_{7(-25)}$ of the spacetime coordinatized by the exceptional Jordan algebra $J_3^{\mathbb{O}}$ defined over octonions $\mathbb{O}$. 
We show that the minrep of $E_{7(-25)}$ decomposes into infinitely many massless representations of the ten dimensional conformal group $SO(10,2)$. Corresponding conformal fields transform as symmetric tensors in spinor indices of $SO(9,1)$ subject to certain constraints. Even and odd tensorial fields describe bosonic and fermionic conformal fields , respectively. Each irrep of $SO(10,2)$ comes with infinite multiplicity and fall into a unitary representation of an $SU(1,1)$ subgroup that commutes with $SO(10,2)$. 
The noncompact generators transforming in spinor representation $16 $ of $SO(10)$ interpolate between the bosonic and fermionic representations and hence act like "bosonic supersymmetry" generators. These results truncate to similar results in the lower critical space-time dimensions. We also give the decomposition of the minrep of $E_{7(-25)}$ with respect to the subgroup $SO^*(12)\times SU(2)$ with $SO^*(12) $ acting as the conformal group of the spacetime coordinatized by the $J_3^{\mathbb{H}}$ defined over quaternions $\mathbb{H}$. Group $E_{7(-25)}$ is also the U-duality group of the exceptional $N=2$ Maxwell-Einstein supergravity in four dimensions. We discuss the relevance of our results to the composite scenario that was proposed for the exceptional supergravity so as to accommodate the families of quarks and leptons of the standard model as well as to the proposal that $E_{7(-25)}$ acts as spectrum generating symmetry group of the $5d$ exceptional supergravity.

Some new Hamiltonian systems of quasi-Painlevé type are presented and the analogue of Okamoto's space of initial conditions computed. Using the geometric approach that was introduced originally for the identification problem of Painlevé equations, comparing the irreducible components of the inaccessible divisors arising in the blow-up process, we find bi-rational coordinate changes between some of these systems that give rise to the same global Hamiltonian structure. This scheme thus gives a method for identifying Hamiltonian systems up to bi-rational maps, which is performed in this article for systems of quasi-Painlevé type having singularities that are either square-root type algebraic poles or ordinary poles.

The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation $\dot{x}(t)+\gamma x = \sqrt{2 D}\xi(t)$, where $\xi(t)$ is the fractional Gaussian noise with the Hurst exponent $0 \lt H \lt 1$. For $H\neq 1/2$ the fOU process is non-Markovian but Gaussian, and it has either vanishing (for $H \lt 1/2$), or divergent (for $H \gt 1/2$) spectral density at zero frequency. For $H \gt 1/2$, the fOU is long-correlated. Here we study dynamical large deviations of the fOU process and focus on the area $A_n = \int_{-T}^{T} x^n(t) \mathrm{d}t$, $n = 1,2,\ldots$ over a long time window 2T. Employing the optimal fluctuation method, we determine the optimal path of the conditioned process, which dominates the large-A_n tail of the probability distribution of the area, $\mathcal{P}(A_n,T)\sim \exp[-S(A_n,T)]$. We uncover a nontrivial phase diagram of scaling behaviors of the optimal paths and of the action $S(A_n\equiv 2 a_n T,T)\sim T^{\alpha(H,n)} a^{2/n}_n$ on the (H, n) plane. The phase diagram includes three distinct regions: (i) $H \gt 1-1/n$, where $\alpha(H,n) = 2-2H$, and the optimal paths are delocalized in time, (ii) n = 2 and $H\unicode{x2A7D} \frac{1}{2}$, where $\alpha(H,n) = 1$, and the optimal paths oscillate with an H-dependent frequency, and (iii) $H\unicode{x2A7D} 1-1/n$ and n > 2, where $\alpha(H,n) = 2/n$, and the optimal paths are strongly localized. We verify our theoretical predictions in large-deviation simulations of the fOU process. By combining the Wang-Landau Monte-Carlo algorithm with the circulant embedding method of generation of stationary Gaussian fields, we were able to measure probability densities as small as 10⁻¹⁷⁰. We also generalize our findings to other stationary Gaussian processes with either divergent, or vanishing spectral density at zero frequency.

Real 3-manifold triangulations can be uniquely represented by isomorphism signatures. Databases of these isomorphism signatures are generated for a variety of 3-manifolds and knot complements, using SnapPy and Regina, then these language-like inputs are used to train various machine learning architectures to differentiate the manifolds, as well as their Dehn surgeries, via their triangulations. Gradient saliency analysis then extracts key parts of this language-like encoding scheme from the trained models. The isomorphism signature databases are taken from the 3-manifolds' Pachner graphs, which are also generated in bulk for some selected manifolds of focus and for the subset of the SnapPy orientable cusped census with $\lt 8$ initial tetrahedra. These Pachner graphs are further analysed through the lens of network science to identify new structure in the triangulation representation; in particular for the hyperbolic case, a relation between the length of the shortest geodesic (systole) and the size of the Pachner graph's ball is observed.

We construct a relativistic quantum communication channel between two localized qubit systems, mediated by a relativistic quantum field, that can achieve the theoretical maximum for the quantum capacity in arbitrary curved spacetimes using the Unruh–DeWitt detector formalism. Using techniques from algebraic quantum field theory, we express the quantum capacity of the quantum communication channel purely in terms of the correlation functions of the field and the causal propagator for the wave equation. Consequently, the resulting quantum channel, and hence the quantum capacity, are by construction manifestly covariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and the choice of Hilbert space (quasifree) representations of the field.

The stationary Dirac equation, confined to a two-dimensional (2D) region, supports states propagating along the boundary and decaying exponentially away from the boundary. These edge states appear on the 2D surface of a 3D topological insulator, where massless fermionic quasiparticles are governed by the Dirac equation and confined by a magnetic insulator. We show how the continuous system can be simulated on a 2D square lattice, without running into the fermion-doubling obstruction. For that purpose we adapt the existing tangent fermion discretization on an unbounded lattice to account for a lattice termination that simulates the magnetic insulator interface.

In the article (Bosboom et al 2023 J. Phys. A: Math. Theor.56 345206) we presented a metriplectic formulation for the radiative transfer equation with polarization. An important ingredient in this process was the construction of a suitable Lie bracket for matrix-valued functions. The bracket presented in equation (20) in Bosboom et al (2023 J. Phys. A: Math. Theor.56 345206) is however not a Lie bracket since it does not satisfy the Jacobi identity. Here, we replace this bracket with a correct Lie bracket for matrix-valued functions and adjust the resulting proofs of the Jacobi identity of this bracket and the Casimir property of the entropy functional. All other results in the paper remain unaffected.

Atomic frequency comb (AFC) quantum memories are a promising technology for quantum repeater networks because they enable multi-mode, long-time, and high-fidelity storage of photons with on-demand retrieval. The optimization of the retrieval efficiency of an AFC memory is important because it strongly impacts the entanglement distribution rate in quantum networks. Despite initial theoretical analyses and recent experimental demonstrations, a rigorous proof of the universally optimal configuration for the highest AFC retrieval efficiency has not been presented. In this paper we present a simple analytical proof which shows that the optimized square tooth offers the highest retrieval efficiency among all tooth shapes, under the physical constraint of finite optical depth of an atomic ensemble. The optimality still holds when the non-zero background absorption and the finite optical linewidth of atoms are considered. We further compare square, Lorentzian and Gaussian tooth shapes to reinforce the practical advantage of the square-tooth AFC in retrieval efficiency. Our proof lays rigorous foundation for the recipe of creating optimal AFC under realistic experimental conditions.

Hagedorn functions are carefully constructed generalizations of Hermite functions to the setting of many-dimensional squeezed and coupled harmonic systems. Wavepackets formed by superpositions of Hagedorn functions have been successfully used to solve the time-dependent Schrödinger equation exactly in harmonic systems and variationally in anharmonic systems. To evaluate typical observables, such as position or kinetic energy, it is sufficient to consider orthonormal Hagedorn functions with a single Gaussian center. Instead, we derive various relations between Hagedorn bases associated with different Gaussians, including their overlaps, which are necessary for evaluating quantities nonlocal in time, such as the time correlation functions needed for computing spectra. First, we use the Bogoliubov transformation to obtain the commutation relations between the ladder operators associated with different Gaussians. Then, instead of using numerical quadrature, we employ these commutation relations to derive exact recurrence relations for the overlap integrals between Hagedorn functions with different Gaussian centers. Finally, we present numerical experiments that demonstrate the accuracy and efficiency of our algebraic method as well as its suitability for treating problems in spectroscopy and chemical dynamics.

Minimal unitary representation of $SO(d,2)$ and its deformations describe all the conformally massless fields in $d$ dimensional Minkowskian spacetimes. In critical dimensions these spacetimes admit extensions with twistorial coordinates plus a dilatonic coordinate to causal spacetimes coordinatized by Jordan algebras $J_3^{\mathbb{A}}$ of degree three over the four division algebras $\mathbb{A}=\mathbb{R} , \mathbb{C} , \mathbb{H} , \mathbb{O} $. We study the minimal unitary representation (minrep) of the conformal group $E_{7(-25)}$ of the spacetime coordinatized by the exceptional Jordan algebra $J_3^{\mathbb{O}}$ defined over octonions $\mathbb{O}$. 
We show that the minrep of $E_{7(-25)}$ decomposes into infinitely many massless representations of the ten dimensional conformal group $SO(10,2)$. Corresponding conformal fields transform as symmetric tensors in spinor indices of $SO(9,1)$ subject to certain constraints. Even and odd tensorial fields describe bosonic and fermionic conformal fields , respectively. Each irrep of $SO(10,2)$ comes with infinite multiplicity and fall into a unitary representation of an $SU(1,1)$ subgroup that commutes with $SO(10,2)$. 
The noncompact generators transforming in spinor representation $16 $ of $SO(10)$ interpolate between the bosonic and fermionic representations and hence act like "bosonic supersymmetry" generators. These results truncate to similar results in the lower critical space-time dimensions. We also give the decomposition of the minrep of $E_{7(-25)}$ with respect to the subgroup $SO^*(12)\times SU(2)$ with $SO^*(12) $ acting as the conformal group of the spacetime coordinatized by the $J_3^{\mathbb{H}}$ defined over quaternions $\mathbb{H}$. Group $E_{7(-25)}$ is also the U-duality group of the exceptional $N=2$ Maxwell-Einstein supergravity in four dimensions. We discuss the relevance of our results to the composite scenario that was proposed for the exceptional supergravity so as to accommodate the families of quarks and leptons of the standard model as well as to the proposal that $E_{7(-25)}$ acts as spectrum generating symmetry group of the $5d$ exceptional supergravity.

We extend the standard semiclassical theory of excited-state quantum phase transitions (ESQPTs), based on a classification of stationary points in the classical Hamiltonian, to constrained systems. We adopt the method of Lagrange multipliers to find all stationary points and their properties directly from the Hamiltonian constrained by an arbitrary number of integrals of motion, and demonstrate the procedure on an algebraic $\mathfrak{u}(3)$ boson model with two independent constraints. We also elaborate the Holstein–Primakoff (HP) mapping, used to eliminate one degree of freedom in bosonic systems constrained by a conserved number of excitations, and address the fact that this mapping leads, in the classical limit, to a compact phase space with singular behaviour that conceals some stationary points at the phase space boundary. It is shown that the HP method reveals all ESQPTs only after constructing a complete atlas of different HP mappings.