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1. Random anti-commuting Hermitian matrices

   https://doi.org/10.1142/S2010326324500199  

   McCarthy, John E; McCarthy, Hazel T (October 2024, Random Matrices: Theory and Applications)

   We consider pairs of anti-commuting [Formula: see text]-by-[Formula: see text] Hermitian matrices that are chosen randomly with respect to a Gaussian measure. Generically such a pair decomposes into the direct sum of [Formula: see text]-by-[Formula: see text] blocks on which the first matrix has eigenvalues [Formula: see text] and the second has eigenvalues [Formula: see text]. We call [Formula: see text] the skew spectrum of the pair. We derive a formula for the probability density of the skew spectrum, and show that the elements are repelling. 

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2. Preparing Hamiltonians for quantum simulation: A computational framework for Cartan decomposition via Lax dynamics

   https://doi.org/10.1090/mcom/4056  

   Chu, Moody (February 2025, Mathematics of Computation)

   Quantum algorithms usually are described via quantum circuits representable as unitary operators. Synthesizing the unitary operators described mathematically in terms of the unitary operators recognizable as quantum circuits is essential. One such a challenge lies in the Hamiltonian simulation problem, where the matrix exponential of a large-scale skew-Hermitian matrix is to be computed. Most current techniques are prone to approximation errors, whereas the parametrization of the underlying Hamiltonian via the Cartan decomposition is more promising. To prepare for such a simulation, this work proposes to tackle the Cartan decomposition by means of the Lax dynamics. The advantages include not only that it is numerically feasible with no matrices involved, but also that this approach offers a genuine unitary synthesis within the integration errors. This work contributes to the theoretic and algorithmic foundations in three aspects: exploiting the quaternary representation of Hamiltonian subalgebras; describing a common mechanism for deriving the Lax dynamics; and providing a mathematical theory of convergence. 

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3. Non‐Hermitian Selective Thermal Emitters using Metal–Semiconductor Hybrid Resonators

   https://doi.org/10.1002/adma.201904154  

   Doiron, Chloe_F; Naik, Gururaj_V (September 2019, Advanced Materials)

   Abstract All open systems that exchange energy with their environment are non‐Hermitian. Thermal emitters are open systems that can benefit from the rich set of physical phenomena enabled by their non‐Hermitian description. Using phase, symmetry, chirality, and topology, thermal radiation from hot surfaces can be unconventionally engineered to generate light with new states. Such thermal emitters are necessary for a wide variety of applications in sensing and energy conversion. Here, a non‐Hermitian selective thermal emitter is experimentally demonstrated, which exhibits passivePT‐symmetry in thermal emission at 700 °C. Furthermore, the effect of internal phase of the oscillator system on far‐field thermal radiation is experimentally demonstrated. The ability to tune the oscillator phase provides new pathways for both engineering and controlling selective thermal emitters for applications in sensing and energy conversion. 

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4. On the optimal arrangement of 2 d lines in ℂ d

   https://doi.org/10.1093/imaiai/iaaf008  

   Fallon, Kean; Iverson, Joseph W (March 2025, Information and Inference: A Journal of the IMA)

   Abstract We show the optimal coherence of $2d$ lines in $$\mathbb{C}^{d}$$ is given by the Welch bound whenever a skew Hadamard matrix of order $d+1$ exists. Our proof uses a variant of Hadamard matrix doubling that converts any equiangular tight frame of size $$\tfrac{d-1}{2} \times d$$ into another one of size $$d \times 2d$$. Among $$d \leq 160$$, this produces equiangular tight frames of new sizes when $d = 11$, $35$, $39$, $43$, $47$, $59$, $67$, $71$, $83$, $95$, $103$, $107$, $111$, $119$, $123$, $127$, $131$, $143$, $151$ and $155$. 

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5. Quantum electrodynamics of non-Hermitian Dirac fermions

   https://doi.org/10.1007/JHEP01(2024)143  

   Murshed, Sk Asrap; Roy, Bitan (January 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> We develop an effective quantum electrodynamics for non-Hermitian (NH) Dirac materials interacting with photons. These systems are described by nonspatial symmetry protected Lorentz invariant NH Dirac operators, featuring two velocity parametersυ_Handυ_NHassociated with the standard Hermitian and a masslike anti-Hermitian Dirac operators, respectively. They display linear energy-momentum relation, however, in terms of an effective Fermi velocity$$ {\upsilon}_{\textrm{F}}=\sqrt{\upsilon_{\textrm{H}}^2-{\upsilon}_{\textrm{NH}}^2} $$ ${\upsilon }_F=\sqrt{{\upsilon }_H^2-{\upsilon }_{\mathrm{NH}}^2}$of NH Dirac fermions. Interaction with the fluctuating electromagnetic radiation then gives birth to an emergent Lorentz symmetry in this family of NH Dirac materials in the deep infrared regime, where the system possesses a unique terminal velocityυ_F=c, withcbeing the speed of light. While in two dimensions such a terminal velocity is set by the speed of light in the free space, dynamic screening in three spatial dimensions permits its nonuniversal values. Manifestations of such an emergent spacetime symmetry on the scale dependence of various physical observables in correlated NH Dirac materials are discussed. 

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