arXiv subjects

Boris L. Altshuler

Publications and source records attributed to Boris L. Altshuler.

At least 19 recordsLinked to original sources

arXiv2026

Semi-localized ground state in a 1D system with long-range hopping

We study the localization of a quantum particle in a one-dimensional disordered system with long-range hopping amplitudes $t(r)\propto r^{-a}$. In contrast to the standard one-dimensional Anderson model ($a\to\infty$), in which all states are localized and the localization length is minimal at the band edge, the long-range model with $1<a<3/2$ exhibits a disorder-driven transition at the band edge, while high-energy states remain localized at arbitrary disorder strength. We investigate this transition for the ground state in momentum space. In the weak-disorder regime, we derive perturbative expressions for the characteristic functions and moments of the momentum-space wave function, as well as for its fractal dimensions. Our results demonstrate that the ground state exhibits $\it semilocalization$ rather than conventional localization, thereby extending the class of models displaying the unusual $\it semifractality$ of wave functions.

cond-mat.dis-nn↗

arXiv2026

Semi-fractality and localization on a chiral Cayley tree

We study a quantum particle hopping on an infinite Cayley tree with nearest-neighbor hopping amplitudes drawn from a distribution singular as $|t|^{-a}$ near weak links and no on-site disorder. Because the graph is bipartite, the model has chiral symmetry, which strongly affects the statistics of eigenstates at the center of the spectrum. Using population dynamics to solve the cavity equations for the propagator, we analyze the distribution of the local density of states and show that it develops broad power-law tails. These tails imply an unusual form of wave-function statistics, which we call semi-fractality: the eigenstates occupy an extensive fraction of the system, but their higher moments behave as in a multifractal state. We find that the symmetry properties of the local-density-of-states distribution are not fixed only by the symmetry class, but vary continuously with the exponent controlling the power-law hopping distribution. As this exponent is changed, the system crosses from a semi-fractal regime to a localized one. At the transition, the wave functions realize an extreme intermediate form that we call semi-localized, simultaneously extended in their support but localized according to higher moments.

cond-mat.dis-nn↗

arXiv2026

Stability Analysis of Superconductivity in $\textit{P6/mmm}$-LaSc$_2$H$_{24}$ and its Experimental Reproducibility from La-Sc Alloys

In this work, we analyze the feasibility of room-temperature superconductivity in the lanthanum-scandium hydride $\textit{P6/mmm}$-LaSc$_2$H$_{24}$. We demonstrate that the electron-phonon coupling calculations performed using the $σ$-broadening of the double $δ$-function at the Fermi surface lead to a very strong dependence of $\textit{T$_c$}(σ)$ on the arbitrary $σ$, whereas the tetrahedral method for the electron-phonon interaction is free from this drawback and leads to $\textit{T$_c$}$ > 300 K at 300 GPa in agreement with previous predictions. By analyzing the stability of the metallic state of LaSc$_2$H$_{24}$ at 250-300 GPa, we show that this compound is at the edge of the stability region ($ξ$ = 0.54), similar to $\textit{fcc}$ LaH$_{10}$ at 140-150 GPa. Seven experimental attempts to synthesize LaSc$_2$H$_{24}$ at 250-280 GPa starting from the (La,Sc$_2$) alloy are unsuccessful and indicate the absence of even traces of superconductivity at 245-300 K in all the resulting La-Sc-H hydrides. The method for preparing the precursor by simultaneous deposition of La and Sc metals may be a key factor for the successful synthesis of LaSc$_2$H$_{24}$.

cond-mat.supr-con↗

arXiv2025

Effective delocalization in the one-dimensional Anderson model with stealthy disorder

We study analytically and numerically the Anderson model in one dimension with "stealthy" disorder, defined as having a power spectrum that vanishes in a continuous band of wave numbers. Motivated by recent studies on the optical transparency properties of stealthy hyperuniform layered media, we compute the localization length using a perturbative expansion of the self-energy. We find that, for fixed energy and small but finite disorder strength $W$, there exists for any finite length system a range of stealthiness $χ$ for which the localization length exceeds the system size. This kind of "effective delocalization" is the result of the novel kind of correlated disorder that spans a continuous range of length scales, a defining characteristic of stealthy systems. Unlike uncorrelated disorder, for which the localization length $ξ$ scales as $W^{-2}$ to leading order for small W, the leading order terms in the perturbation expansion of $ξ$ for stealthy disordered systems vanish identically for a progressively large number of terms as $χ$ increases such that $ξ$ scales as $W^{-2n}$ with arbitrarily large $n$. Moreover, we support our analytical results with numerical simulations. Our results introduce stealthy disorder into quantum tight-binding models and show that enforcing a low-$k$ spectral gap markedly alters the scattering landscape, enabling localization lengths that exceed the system size at fixed disorder strength. Since this mechanism relies only on the spectral properties of the disorder, it carries over directly to photonic and phononic wave systems.

cond-mat.dis-nn↗

arXiv2025

Trotter transition in BCS pairing dynamics

We study universal aspects of thermalization induced by Trotterization, a procedure routinely used in gate-based quantum computation. We use the reduced-BCS model -- quantum integrable with a classically integrable mean-field limit -- where the effects of Trotter chaos are expected to be particularly stark. The resulting Trotterized chaotic dynamics is characterized by its Lyapunov spectrum and rescaled Kolmogorov-Sinai entropy. The chaos quantifiers depend on the Trotterization time step $τ$. We observe a Trotter transition at a finite step value $τ_c \approx \sqrt{N}$. While the dynamics is weakly chaotic for time steps $τ\ll τ_c$, the regime of large Trotterization steps is characterized by short temporal correlations. We derive two different scaling laws for the two different regimes by numerically fitting the maximum Lyapunov exponent data. The scaling law of the large $τ$ limit agrees well with the one derived from the kicked top map. Beyond its relevance to current quantum computers, our work opens new directions -- such as probing observables like the Loschmidt echo, which lie beyond standard mean-field description -- across the Trotter transition we uncover.

quant-ph↗

arXiv2025

Kallen-Lehmann representation for Spinor-Scalar Loops in de Sitter Space-time. Spectral equations

Compact formulas (1), (28) presented in this paper permit to formulate the Kallen-Lehmann harmonic decompositions of the products of two spinor and of spinor and scalar harmonic functions on AdS and Wightman functions on dS spaces; the correct flat space limits of the corresponding Kallen-Lehmann densities is demonstrated, the connection between poles of the Kallen-Lehmann density and the late-time divergence of the spinor-scalar loop is traced in the Yukawa model with light scalar field. Also in Sec. 4 the relatively simple, suitable for calculations Kallen-Lehmann representations of the real parts of spinor and scalar one-loop self-energies on dS space-time are proposed, and corresponding spectral equations for conformal dimensions in the chain approximation are written down.

hep-th↗

arXiv2025

Fundamental limits on the electron-phonon coupling and superconducting $T_c$

Fundamental upper bounds on the electron-phonon interaction strength and superconducting transition temperature $T_c$ in metals are established based on the intrinsic instability of the equilibrium between electrons and the crystal lattice under strong interaction. This instability explains why observed electron-phonon coupling constants are limited to $λ\lesssim 4$. The theory also accounts for the mechanism of metastable superconductivity with enhanced $T_c$, which emerges near the instability threshold. Based on theoretical analysis and comparison with experimental data, room-temperature phonon-mediated superconductivity is found to be feasible exclusively in hydrogen compounds.

cond-mat.supr-con↗

arXiv2025

Bounds on $T_c$ in the Eliashberg theory of Superconductivity. II: Dispersive phonons

The standard Eliashberg theory of superconductivity is studied, in which the effective electron-electron interactions are mediated by generally dispersive phonons, with Eliashberg spectral function $α^2 F(ω)\geq 0$ that is $\proptoω^2$ for small $ω>0$ and vanishes for large $ω$. The Eliashberg function also defines the electron-phonon coupling strength $λ:= 2 \int_0^\infty\frac{α^2 F(ω)}ωdω$. Setting $\frac{2α^2 F(ω)}ωdω=: λP(dω)$, formally defining a probability measure $P(dω)$ with compact support, and assuming as usual that the phase transition between normal and superconductivity coincides with the linear stability boundary $\mathscr{S}_{\!c}$ of the normal region against perturbations toward the superconducting region, it is shown that $\mathscr{S}_{\!c}$ is a graph of a function $Λ(P,T)$ that is determined by a variational principle: if $(λ,P,T)\in\mathscr{S}_{\!c}$, then $λ= 1/\mathfrak{k}(P,T)$, where $\mathfrak{k}(P,T)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{K}(P,T)$ on $\ell^2$ sequences constructed in the paper. Given $P$, sufficient conditions on $T$ are stated under which the map $T\mapsto λ= Λ(P,T)$ is invertible. For sufficiently large $λ$ this yields: (i) the existence of a critical temperature $T_c$ as function of $λ$ and $P$; (ii) a sequence of lower bounds on $T_c(λ,P)$ that converges to $T_c(λ,P)$. Also obtained is an upper bound on $T_c(λ,P)$. It agrees with the asymptotic form $T_c(λ,P) \sim C \sqrt{\langle ω^2\rangle} \sqrtλ$ valid for $λ\sim\infty$, given $P$, though with a constant $C$ that is a factor $\approx 2.034$ larger than the sharp constant. Here, $\langleω^2\rangle := \int_0^\infty ω^2 P(dω)$.

math-ph↗

arXiv2025

Instability of metals with respect to strong electron-phonon interaction

We show that thermal equilibrium between conduction electrons and phonons becomes kinetically unstable when the renormalized electron-phonon coupling exceeds a certain threshold. We prove that negative electronic specific heat, $C_\mathrm{el} < 0$, is sufficient to trigger the instability. Specifically, the instability sets in as soon as the quasiparticle weight becomes negative over a range of energies, even before $C_\mathrm{el}$ turns negative. This is an inherently nonequilibrium phenomenon, occurring prior to the formation of any equilibrium phase. Depending on the system, it can proceed along different pathways, ultimately resulting in a structural transition to an insulating or metallic state.

cond-mat.str-el↗

arXiv2025

Bounds on $T_c$ in the Eliashberg theory of Superconductivity. I: The $γ$-model

Using the recent reformulation for the Eliashberg theory of superconductivity in terms of a classical interacting Bloch spin chain model, rigorous upper and lower bounds on the critical temperature $T_c$ are obtained for the $γ$ model -- a version of Eliashberg theory in which the effective electron-electron interaction is proportional to $(g/|ω_n-ω_m|)^γ$, where $ω_n-ω_m$ is the transferred Matsubara frequency, $g>0$ a reference energy, and $γ>0$ a parameter. The rigorous lower bounds are based on a variational principle that identifies $(T_c/g)^γ$ with the largest (positive) eigenvalue of an explicitly constructed compact, self-adjoint operator $\mathfrak{G}(γ)$. These lower bounds form an increasing sequence that converges to $T_c(g,γ)$. The upper bound on $T_c(g,γ)$ is based on fixed point theory, proving linear stability of the normal state for $T$ larger than the upper bound on $T_c(g,γ)$.

math-ph↗

arXiv2025

Bounds on $T_c$ in the Eliashberg theory of Superconductivity. III: Einstein phonons

The dispersionless limit of the standard Eliashberg theory of superconductivity is studied. The effective electron-electron interactions are mediated by Einstein phonons of frequency $Ω>0$, equipped with electron-phonon coupling strength $λ$. This allows for a detailed evaluation of the general results on $T_c$ for phonons with non-trivial dispersion relation, obtained in a previous paper, (II), by the authors. The variational principle for the linear stability boundary $\mathscr{S}_{\!c}$ of the normal state region against perturbations toward the superconducting region, obtained in (II), simplifies as follows: If $(λ,Ω,T)\in\mathscr{S}_{\!c}$, then $λ= 1/\mathfrak{h}(\varpi)$, where $\varpi:=Ω/2πT$, and where $\mathfrak{h}(\varpi)>0$ is the largest eigenvalue of a compact self-adjoint operator $\mathfrak{H}(\varpi)$ on $\ell^2$ sequences; $\mathfrak{H}(\varpi)$ is the dispersionless limit $P(dω)\toδ(ω-Ω)dω$ of the operator $\mathfrak{K}(P,T)$ of (II). It is shown that when $\varpi \leq \sqrt{2}$, then the map $\varpi\mapsto\mathfrak{h}(\varpi)$ is invertible. For $λ>0.77$ this yields: (i) the existence of a critical temperature $T_c(λ,Ω) = Ωf(λ)$; (ii) a sequence of lower bounds on $f(λ)$ that converges to $f(λ)$. Also obtained is an upper bound on $T_c(λ,Ω)$, which agrees with the asymptotic behavior $T_c(λ,Ω) \sim C Ω\sqrtλ$ for $λ\sim\infty$, given $Ω$, though with $C\approx 2.034 C_\infty$, where $C_\infty := \frac{1}{2π}\mathfrak{k}(2)^\frac12 =0.1827262477...$ is the optimal constant, and $\mathfrak{k}(γ)>0$ the largest eigenvalue of a compact self-adjoint operator for the $γ$ model, determined in the first paper, (I), on $T_c$ by the authors.

cond-mat.supr-con↗

arXiv2024

Renormalization group for Anderson localization on high-dimensional lattices

We discuss the dependence of the critical properties of the Anderson model on the dimension $d$ in the language of $β$-function and renormalization group recently introduced in Ref.[arXiv:2306.14965] in the context of Anderson transition on random regular graphs. We show how in the delocalized region, including the transition point, the one-parameter scaling part of the $β$-function for the fractal dimension $D_{1}$ evolves smoothly from its $d=2$ form, in which $β_2\leq 0$, to its $β_\infty\geq 0$ form, which is represented by the regular random graph (RRG) result. We show how the $ε=d-2$ expansion and the $1/d$ expansion around the RRG result can be reconciled and how the initial part of a renormalization group trajectory governed by the irrelevant exponent $y$ depends on dimensionality. We also show how the irrelevant exponent emerges out of the high-gradient terms of expansion in the nonlinear sigma-model and put forward a conjecture about a lower bound for the fractal dimension. The framework introduced here may serve as a basis for investigations of disordered many-body systems and of more general non-equilibrium quantum systems.

cond-mat.dis-nn↗

arXiv2024

Renormalization Group Analysis of the Anderson Model on Random Regular Graphs

We present a renormalization group analysis of the problem of Anderson localization on a Random Regular Graph (RRG) which generalizes the renormalization group of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The renormalization group equations necessarily involve two parameters (one being the changing connectivity of sub-trees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the non-monotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of Many-Body Localization.

cond-mat.dis-nn↗

arXiv2024

Dynamical chaos in the integrable Toda chain induced by time discretization

We use the Toda chain model to demonstrate that numerical simulation of integrable Hamiltonian dynamics using time discretization destroys integrability and induces dynamical chaos. Specifically, we integrate this model with various symplectic integrators parametrized by the time step $τ$ and measure the Lyapunov time $T_Λ$ (inverse of the largest Lyapunov exponent $Λ$). A key observation is that $T_Λ$ is finite whenever $τ$ is finite but diverges when $τ\rightarrow 0$. We compare the Toda chain results with the nonitegrable Fermi-Pasta-Ulam-Tsingou chain dynamics. In addition, we observe a breakdown of the simulations at times $T_B \gg T_Λ$ due to certain positions and momenta becoming extremely large (``Not a Number''). This phenomenon originates from the periodic driving introduced by symplectic integrators and we also identify the concrete mechanism of the breakdown in the case of the Toda chain.

nlin.CD↗

arXiv2022

Superconductivity near a quantum critical point in the extreme retardation regime

We study fermions at quantum criticality with extremely retarded interactions of the form $V(ω_l)=(g/|ω_l|)^γ$, where $ω_l$ is the transferred Matsubara frequency. This system undergoes a normal-superconductor phase transition at a critical temperature $T=T_c$. The order parameter is the frequency-dependent gap function $Δ(ω_n)$ as in the Eliashberg theory. In general, the interaction is extremely retarded for $γ\gg 1$, except at low temperatures $γ>3$ is sufficient. We evaluate the normal state specific heat, $T_c$, the jump in the specific heat, $Δ(ω_n)$ near $T_c$, and the Landau free energy. Our answers are asymptotically exact in the limit $γ\to\infty$. At low temperatures, we prove that the global minimum of the free energy is nondegenerate and determine the order parameter, the free energy, and the specific heat. These answers are exact for $T\to0$ and $γ>3$. We also uncover and investigate an instability of the $γ$ model: negative specific heat at $T\to0$ and just above $T_c$.

cond-mat.supr-con↗

arXiv2022

Migdal-Eliashberg theory as a classical spin chain

We formulate the Migdal-Eliashberg theory of electron-phonon interactions in terms of classical spins by mapping the free energy to a Heisenberg spin chain in a Zeeman magnetic field. Spin components are energy-integrated normal and anomalous Green's functions and sites of the chain are fermionic Matsubara frequencies. The Zeeman field grows linearly with the spin coordinate and competes with ferromagnetic spin-spin interaction that falls off as the square of the inverse distance. The spin-chain representation makes a range of previously unknown properties plain to see. In particular, infinitely many new solutions of the Eliashberg equations both in the normal and superconducting states emerge at strong coupling. These saddle points of the free-energy functional correspond to spin flips. We argue that they are also fixed points of kinetic equations and play an essential role in far from equilibrium dynamics of strongly coupled superconductors. Up to an overall phase, the frequency-dependent gap function that minimizes the free energy must be non-negative. There are strong parallels between our Eliashberg spins and Anderson pseudospins, though the two sets of spins never coincide.

cond-mat.supr-con↗

arXiv2022

Breakdown of the Migdal-Eliashberg theory and a theory of lattice-fermionic superfluidity

We show that the Migdal-Eliashberg theory loses validity at a finite value $λ_c$ of the electron-phonon coupling $λ$ regardless of the underlying model Hamiltonian. The value of $λ_c$ is approximately between 3.0 and 3.7. The new phase that emerges at $λ>λ_c$ breaks the lattice translational symmetry. Depending on the filling fraction and crystal symmetry, it is an insulator or a Fermi liquid. Its characteristic feature is a gap or a pronounced depression of the fermionic density of states near the Fermi level. We establish the breakdown from within the Migdal-Eliashberg theory by demonstrating that the normal state specific heat is negative for $λ\ge 3.7$ and the quasiparticle lifetime vanishes in the strong coupling limit. At fixed $λ>λ_c$, the transition to the new phase occurs at a critical temperature higher than the superconducting transition temperature. In addition, there is a first order phase transition between the new phase and the superconducting state as we vary $λ$ across $λ_c$ at fixed temperature. We put forward a new theory - lattice-fermionic theory of superfluidity - that bridges the gap between the Migdal-Eliashberg approach and the physics at stronger coupling. At small $λ$, our theory reduces to the Migdal-Eliashberg theory and, past $λ_c$, it describes the new phase and a range of other phenomena.

cond-mat.supr-con↗

arXiv2020

Driven-Dissipative Dynamics of Atomic Ensembles in a Resonant Cavity II: Quasiperiodic Route to Chaos and Chaotic Synchronization

We analyze the origin and properties of the chaotic dynamics of two atomic ensembles in a driven-dissipative experimental setup, where they are collectively damped by a bad cavity mode and incoherently pumped by a Raman laser. Starting from the mean-field equations, we explain the emergence of chaos by way of quasiperiodicity -- presence of two or more incommensurate frequencies. This is known as the Ruelle-Takens-Newhouse route to chaos. The equations of motion have a $\mathbb{Z}_{2}$-symmetry with respect to the interchange of the two ensembles. However, some of the attractors of these equations spontaneously break this symmetry. To understand the emergence and subsequent properties of various attractors, we concurrently study the mean-field trajectories, Poincaré sections, maximum and conditional Lyapunov exponents, and power spectra. Using Floquet analysis, we show that quasiperiodicity is born out of non $\mathbb{Z}_{2}$-symmetric oscillations via a supercritical Neimark-Sacker bifurcation. Changing the detuning between the level spacings in the two ensembles and the repump rate results in the synchronization of the two chaotic ensembles. In this regime, the chaotic intensity fluctuations of the light radiated by the two ensembles are identical. Identifying the synchronization manifold, we understand the origin of synchronized chaos as a tangent bifurcation intermittency of the $\mathbb{Z}_{2}$-symmetric oscillations. At its birth, synchronized chaos is unstable. The interaction of this attractor with other attractors causes on-off intermittency until the synchronization manifold becomes sufficiently attractive. We also show coexistence of different phases in small pockets near the boundaries.

nlin.CD↗