Papers

Driven by the growing demand in the energy, medical, and industrial sectors, we investigate a hydrogen isotope separation technique that offers both a high separation factor and economic feasibility. Our findings reveal that filtering isotopes through two-dimensional graphene layers provides an exceptionally efficient quantum-mechanical method for isotope separation. Using a recently developed analytical pairwise potential between hydrogen isotopes and carbon atoms in graphene, we examine the classical trajectories of isotopes near the graphene layer, as well as the quantum-mechanical tunneling properties of isotopes through the graphene layer. Using various quantum-mechanical methods, we calculate both the isotope tunneling probabilities and the quantum-mechanical isotope sticking probabilities. Our study shows that quantum filtering through graphene layers can be an effective technique for enriching deuterium by separating it from protium.

Multiparticle collision dynamics (MPCD) is a mesoscale simulation technique that uses a simplified solvent to model hydrodynamic interactions. Rather than interact through pairwise forces, MPCD solvent particles undergo momentum-exchanging collisions within spatially localized cells according to prescribed rules. The conventional MPCD algorithm employs cubic collision cells, but this choice is not optimal for systems that are most naturally described using skewed simulation boxes. Here, we investigate the behavior of a modified MPCD scheme in which the collision cells are aligned with the vectors that define a triclinic (parallelepiped) simulation box. We find that skewing the collision cells has a small but statistically significant impact on the transport properties of the pure solvent. Similar, but more pronounced, effects are found for nearly hard spheres in solution, including a significant decrease in their nominal self-diffusion coefficient and unphysical anisotropy in their self-diffusion tensor. Thus, our analysis indicates that skewed MPCD collision cells may result in spurious behavior and should be used with caution. We posit that these artifacts may be mitigated by grid-free schemes for placing particles into collision cells.

Creation of charged spinless bosons from vacuum in the rigidly rotating frame is studied in presence of the external static electromagnetic fields, being formed either in the resting, or in the general rotating or in the local-flat frames. It is shown that the description remains the same in the resting and the local-flat frames. More specifically, the case of a solenoid (magnetic flux tube) embedded into the rotating empty cylinder (rotation frame) is studied in presence or absence of a static square electric potential well $eA_0$. A supervortex of a spinless-boson field can be created from vacuum when the rotation frequency exceeds a critical value $\Omega_c$. It is shown that $\Omega_c$ is smaller provided the solenoid rests in the resting frame. Then the case is considered when the rotating frame additionally moves with acceleration $\vec{w}(\vec{r})\neq 0$. A specific case $w(r)=G/r$ for $G=const$ is treated explicitly. It is shown that for $G<0$ the charged spinless-boson vortex can be created in rapidly rotating system even in the limit when the magnetic and electric fields tend to zero.

The contribution of magnetoelectric effect to Landau–Lifshitz-Gilbert equation is considered in case when medium polarization is caused by parallel component of neighboring spins. The result is presented for ferromagnetic and antiferromagnetic materials. A comparison is given with the contribution of magnetoelectric effect to Landau–Lifshitz-Gilbert equation when medium polarization is caused by perpendicular component of spins. The dispersion dependence of spin waves in antiferromagnets with cycloidal equilibrium spin order is derived. The electric susceptibility and permittivity of antiferromagnetic multiferroics in which magnetoelectric effect is caused by collinear component of spins is obtained analytically.

Utilizing the intrinsic history-dependence and nonlinearity of hardware, physical reservoir computing is a promising neuromorphic approach to encode time-series data for in-sensor computing. The accuracy of this encoding critically depends on the distinguishability of multi-state outputs, which is often limited by suboptimal and empirically chosen reservoir operation conditions. In this work, we demonstrate a machine learning approach, Bayesian optimization, to improve the encoding fidelity of solution-processed Al2O3/In2O3 thin-film transistors (TFTs). We show high-fidelity 6-bit temporal encoding by exploring five key pulse parameters and using the normalized degree of separation (nDoS) as the metric of output state separability. Additionally, we show that a model trained on simpler 4-bit data can effectively guide optimization of more complex 6-bit encoding tasks, reducing experimental cost. Specifically, for the encoding and reconstruction of binary-patterned images of a moving car across 6 sequential frames, we demonstrate that the encoding is more accurate when operating the TFT using optimized pulse parameters and the 4-bit optimized operating condition performs almost as well as the 6-bit optimized condition. Finally, interpretability analysis via Shapley Additive Explanations (SHAP) reveals that gate pulse amplitude and drain voltage are the most influential parameters in achieving higher state separation. This work presents the first systematic method to identify optimal operating conditions for reservoir devices, and the approach can be extended to other physical reservoir implementations across different material platforms.

We investigate Josephson arrays consisting of a dice-lattice network of superconducting weak links surrounding rhombic plaquettes of proximitized semiconductor. Josephson coupling of the weak links and electron density in the plaquettes are independently controlled by separate electrostatic gates. Applied magnetic flux results in an intricate pattern of switching currents associated with frustration, $f$. For depleted plaquettes, the switching current is nearly periodic in $f$, expected for a phase-only description, while occupied plaquettes yield a decreasing envelope of switching currents with increasing $f$. A model of flux dependence based on ballistic small-area junctions and diffusive large-area plaquettes yields excellent agreement with experiment.

This review is based on lectures given by the author at the Enrico Fermi Summer School in Varenna. It presents the basics of Density Functional Theory (DFT) for Fermi superfluids, with particular emphasis on nuclear systems. Special attention is given to the foundations of both DFT and time-dependent DFT (TDDFT). The review explores the advantages and challenges involved in the practical application of TDDFT to superfluid systems, as well as the typical approximations employed. Various applications of the TDDFT framework to the description of phenomena related to nonequilibrium superfluidity in atomic nuclei, neutron stars, and ultracold atoms are discussed.

These are lecture notes expanding upon a set of lectures given by G.M. at the TASI 2023 School. Part I is an introduction to topological field theory, including extended topological field theory. Part II is an introduction to generalized Abelian gauge theories and their relation to differential cohomology.

To observe Majorana bound states, and especially to use them as a qubit, requires careful optimization of competing quality metrics. We systematically compare Majorana quality in proximitized semiconductor nanowires and quantum dot chains. Using multi-objective optimization, we analyze the fundamental trade-offs between topological gap and localization length, two key metrics that determine MBS coherence and operational fidelity. We demonstrate that these quantities cannot be simultaneously optimized in realistic models, creating Pareto frontiers that define the achievable parameter space. Our results show that QD chains achieve both comparable quality as nanowires and a regime with a much shorter localization length, making them particularly promising for near-term quantum computing applications where device length and disorder are limiting factors.

The dynamics of hybrid systems – i.e. ones in which classical and quantum degrees of freedom co-exist and interact – feature both diffusion in the classical sector and decoherence in the quantum state. In this article, we will consider the simple setup of a classical damped oscillator interacting with its quantum counterpart and show that, for any initial state, the dynamics flows to a unique (non-equilibrium) steady state, which we compute explicitly. To do so, we make use of a useful mapping between hybrid and classical diffusive dynamics, which we characterise in detail in the master equation formalism.

Phase transitions mark qualitative reorganizations of collective behavior, yet identifying their boundaries remains challenging whenever analytic solutions are absent and conventional simulations fail. Here we introduce learnability as a universal criterion, defined as the ability of a transformer model containing attention mechanism to extract structure from microscopic states. Using self-supervised learning and Monte Carlo generated configurations of the two-dimensional Ising model, we show that ordered phases correspond to enhanced learnability, manifested in both reduced training loss and structured attention patterns, while disordered phases remain resistant to learning. Two unsupervised diagnostics, the sharp jump in training loss and the rise in attention entropy, recover the critical temperature in excellent agreement with the exact value. Our results establish learnability as a data-driven marker of phase transitions and highlight deep parallels between long-range order in condensed matter and the emergence of structure in modern language models.

We discuss intrinsic relaxation rates in superconductors, and how they may be measured using non-linear optical (terahertz) response. We consider both $s$ and $d$-wave superconductors, both with and without a phenomenological (energy dependent) damping. Intrinsic relaxation rates of interest include the Higgs mode decay rate, the quasiparticle redistribution rate ($1/T_1$) and the quasiparticle dephasing rate ($1/T_2$), where the latter two rates are zero in the pure BCS model, but non-zero in the presence of damping. Using the Anderson pseudospin formalism, we illustrate how these intrinsic relaxation rates are related to measurable quantities such as the time-dependent gap function and the non-linear current (a.k.a. third harmonic generation). Hence, we show how intrinsic relaxation rates may be experimentally extracted and discuss what one may thereby learn about the underlying damping. We also discuss the effects of polarization control (viz. non-linear response to light polarized in different directions), which offers a useful experimental knob, especially for $d$-wave superconductors, enabling selective excitation of modes in different irreducible representations (and readout of their corresponding relaxation rates).

A quantum error-correcting code with a nonzero error threshold undergoes a mixed-state phase transition when the error rate reaches that threshold. We explore this phase transition for Haar-random quantum codes, in which the logical information is encoded in a random subspace of the physical Hilbert space. We focus on the spectrum of the encoded system density matrix as a function of the rate of uncorrelated, single-qudit errors. For low error rates, this spectrum consists of well-separated bands, representing errors of different weights. As the error rate increases, the bands for high-weight errors merge. The evolution of these bands with increasing error rate is well described by a simple analytic ansatz. Using this ansatz, as well as an explicit calculation, we show that the threshold for Haar-random quantum codes saturates the hashing bound, and thus coincides with that for random $\textit{stabilizer}$ codes. For error rates that exceed the hashing bound, typical errors are uncorrectable, but postselected error correction remains possible until a much higher $\textit{detection}$ threshold. Postselection can in principle be implemented by projecting onto subspaces corresponding to low-weight errors, which remain correctable past the hashing bound.

The dynamics of extended many-body systems are generically chaotic. Classically, a hallmark of chaos is the exponential sensitivity to initial conditions captured by positive Lyapunov exponents. Supplementing chaotic dynamics with stochastic resetting drives a sharp dynamical phase transition: We show that the Lyapunov spectrum, i.e., the complete set of Lyapunov exponents, abruptly collapses to zero above a critical resetting rate. At criticality, we find a sudden loss of analyticity of the velocity-dependent Lyapunov exponent, which we relate to the transition from ballistic scrambling of information to an arrested regime where information becomes exponentially localized over a characteristic length diverging at criticality with an exponent $\nu = 1/2$ and a dynamical exponent $z=2$. We illustrate our analytical results on generic chaotic dynamics by numerical simulations of coupled map lattices.

Lattice gauge theories (LGTs) represent one of the most ambitious goals of quantum simulation. From a practical implementation perspective, non-Abelian theories present significantly tougher challenges than Abelian LGTs. However, it is unknown whether this is also reflected in increased values of quantum resources relating to the complexity of simulating quantum many-body models. Here, we compare three paradigmatic measures of quantum resources – stabilizer Rényi entropy, generalized geometric measure of entanglement, and fermionic antiflatness – for pure-gauge theories on a ladder with Abelian $\mathbb{Z}_N$ as well as non-Abelian $D_3$ and SU(2) gauge symmetries. We find that non-Abelian symmetries are not necessarily inherently harder to simulate than Abelian ones, but rather the required quantum resources depend nontrivially on the interplay between the group structure, superselection sector, and encoding of the gauge constraints. Our findings help indicate where quantum advantage could emerge in simulations of LGTs, both in NISQ and fault-tolerant eras.

The gap equations lie at the core of the Bardeen-Cooper-Schrieffer (BCS) theory, a standard tool in the description of superfluidity. As a set of non-linear integral equations, the gap equations' inherent difficulties oftentimes hinder even the crudest descriptions of superfluid states. Hard-core potentials, high-density superfluids, and coupled-channel pairing are all reasons that have historically required one to provide special treatment to the gap equations to get a solution. In this paper we present a new method for solving the gap equations that holds the promise of being an efficient universal solver that requires the minimum amount of a priori knowledge of the targeted solutions. With theoretical evidence of exotic nuclear superfluidity posing new questions to our understanding of this fundamental property of nuclear systems, the presented method can be a valuable tool when exploring new pairing states, finite-temperature properties, or the development of sophisticated descriptions of nuclear superfludity.

While experimental evidence for spacetime supersymmetry (SUSY) in particle physics remains elusive, condensed matter systems offer a promising arena for its emergence at quantum critical points (QCPs). Although there have been a variety of proposals for emergent SUSY at symmetry-breaking QCPs, the emergence of SUSY at fractionalized QCPs remains largely unexplored. Here, we demonstrate emergent space-time SUSY at a fractionalized QCP in the Kitaev honeycomb model with Su-Schrieffer-Heeger (SSH) spin-phonon coupling. Specifically, through numerical computations and analytical analysis, we show that the anisotropic SSH-Kitaev model hosts a fractionalized QCP between a Dirac spin liquid and an incommensurate/commensurate valence-bond-solid phase coexisting with $\mathbb{Z}_2$ topological order. A low-energy field theory incorporating phonon quantum fluctuations reveals that this fractionalized QCP features an emergent $\mathcal{N}=2$ spacetime SUSY. We further discuss their universal experimental signatures in thermal transport and viscosity, highlighting the concrete lattice realization of emergent SUSY at a fractionalized QCP in 2D.

From atomic crystals to macroscopic material structures, twisted bilayer systems have emerged as a promising route to control wave phenomena. In few-layer van der Waals (vdW) materials, however, the intrinsically weak interlayer coupling typically demands fine control of small twist angles to reach magic-angle conditions. Here, we show that one-dimensional photonic crystal bilayers can overcome this limitation by accessing a regime of strong interlayer coupling – comparable to intralayer coupling. This strong coupling enables flat-band formation over a broad angular range even at large twist angles. We experimentally realize this regime by stacking WS$_2$ gratings using a two-step lithography method, resulting in ultra-wide chiral flat-band cascades in magic-angle twisted bilayer gratings. Our work not only provides a platform for designing photonic applications with tunable localization but also explores a new regime of physics unattainable in conventional solid-state based moiré systems.

Predicting the superconducting transition temperature ($T_c$) from crystal structure and composition remains a central challenge in condensed-matter physics, reflecting the absence of a broadly predictive framework connecting microscopic bonding to macroscopic quantum behavior. Here, we introduce $\mathcal{GP}$-$T_c$, an interpretable, structure- and chemistry-aware Gaussian process model that enables uncertainty-quantified $T_c$ prediction from experimentally accessible inputs. By encoding local bonding environments as graphlet histograms, we find that the predictive space collapses to a compact set of descriptors: the distribution of electron-affinity (EA) differences between neighboring atoms, together with interatomic distances and simple elemental features, suffices to predict $T_c$ across disparate superconducting families—identifying an overlooked chemical control parameter that underscores the essential role of local structure beyond composition-only approaches. Our results demonstrate that the EA differences serves as an accessible window into electronic structure providing a mechanism-agnostic physical basis that captures $T_c$ across conventional and unconventional families, including doped charge transfer insulators. $\mathcal{GP}$-$T_c$ reproduces the experimentally reported $T_c$ range of the infinite-layer nickelate Nd$_{0.8}$Sr$_{0.2}$NiO$_2$, and we predict and experimentally confirm superconductivity in stoichiometric PtPb$_3$Bi ($T_c \approx 3$~K). To facilitate broad community use, $\mathcal{GP}$-$T_c$ is made available through a web interface for crystal-structure-based prediction, and the same framework identifies additional high-priority superconducting candidates—including SrNiO$_2$ and K(PRh)$_2$—that provide concrete targets for ongoing and future experimental exploration.

On the basis of the extended classical elasticity theory, we propose universal semi-empirical analytical expressions for the energy and the equation of state for poly-crystalline solids. The validation of the relations has been made by means of first principle density functional theory simulations with the use of pseudo-potential approach and generalized gradient approximation for the exchange-correlation energy. The calculations performed for a large number of inorganic crystalline compounds with metal, covalent and ionic bonding (including diamond, Mg, sphalerite, B, magnesium carboboride, topaz, rocksalt, etc.) within the pressure range up to 300 GPa demonstrated an excellent agreement with the predictions of the analytical theory comparable in accuracy with Birch-Murnaghan approach.

We study the dynamics of a mean-field Ising model whose coupling depends on the magnetization via a linear feedback function. A key feature of this linear feedback Ising model (FIM) is the possibility of temperature-induced bistability, where a temperature increase can favor bistability between two phases. We show that the linear FIM provides a minimal model for a transcritical bifurcation as the temperature varies. Moreover, there can be two or three critical temperatures when the external magnetic field is non-negative. In the bistable region, we identify a Maxwell temperature where the two phases are equally probable, and we find that increasing the temperature favors the lower phase. We show that the probability distribution becomes non-Gaussian on certain time intervals when the magnetization converges algebraically at either zero temperature or critical temperatures. Near critical points in the parameter space, we derive a Fokker-Planck equation, construct the families of equilibrium distributions, and formulate scaling laws for transition rates between two stable equilibria. The linear FIM offers considerable flexibility in controlling steady-state bifurcations and their associated equilibrium distributions, which can be desirable for modeling feedback systems across various disciplines.

We present a tangent-space Krylov method for efficient computation of zero-temperature real-frequency spectral functions on top of ground-state matrix product states obtained from the density matrix renormalization group. It relies on projecting the Hamiltonian to the tangent space of the ground-state matrix product states, where resolvents can be efficiently represented using Krylov space techniques. This enables direct computations of excitation energies and the corresponding spectral weights. We demonstrate the accuracy and efficiency of the tangent-space Krylov approach by showcasing spectral data for various models. These include the 1D Haldane-Shastry and Heisenberg models as benchmarks. As an interesting application, we study the Hubbard model on a cylinder at half filling, augmented by a density-assisted hopping (DAH) term. We find that DAH leads to particle-hole asymmetric single-particle mobilities and lifetimes in the resulting Mott insulator, and identify the responsible scattering processes. Furthermore, we find that DAH influences the dispersion of Green's function zeros beyond its range, which has a frustrating effect on the Mott insulator studied here.

The spectral properties, momentum dispersion, and broadening of bulk plasmonic excitations of 26 elemental metals are studied from first principles calculations in the random-phase approximation. Spectral band structures are constructed from the resulting momentum- and frequency-dependent inverse dielectric function. We develop an effective analytical representation of the main collective excitations in the dielectric response, extending our earlier model based on multipole-Padé approximants (MPAs) to incorporate both momentum and frequency dependence [MPA($\q$)]. With this representation, we identify plasmonic quasiparticle dispersions exhibiting complex features, including non-parabolic energy and intensity dispersions, discontinuities due to anisotropy, and overlapping effects that lead to band crossings and anti-crossings. Comparing with available experimental data, mainly in the optical limit, we find good agreement with the computed spectra. The results for elemental metals and their effective MPA($\q$) representation establish a reference point that can guide both fundamental studies and practical applications in plasmonics and spectroscopy.

We introduce a family of hybrid quantum circuits involving unitary gates and projective measurements that display a measurement-induced phase transition. Remarkably, the volume-law phase featuring logarithmic entanglement growth for certain initial states. We attribute this slow entanglement growth to the similarly slow growth of the participation entropy, which bounds the entanglement. Furthermore, the quantum circuit can be mapped to a classical spin model with real positive Boltzmann weights which involves local multi-spin interactions and displays glassy dynamics at finite temperature. We trace the origin of both the slow quantum dynamics and the classical glassiness to the presence of large, non-local symmetry operators. Our work establishes a novel connection between quantum entanglement dynamics and classical glassy behavior, offering a new geometric perspective on entanglement phase transitions.

We study the electrical and thermal transport properties and the violation of the Wiedemann-Franz (WF) law of two-carrier semimetals using exact treatments of the Boltzmann equation with the impurity and electron-electron scatterings in a magnetic field. For comparison, we also study those in the case of Baber scattering: a single-carrier system with an impurity scattering and phenomenological momentum-dissipative electron-electron scattering. In both systems, the longitudinal and transverse WF laws, $L = L_{\text{H}} = L_{0}= \pi^2k_B^2/3e^2$, hold at zero temperature, where the Lorenz ratio $L$ and the Hall Lorenz ratio $L_{\text{H}}$ are ratios of thermal conductivity $\kappa_{\mu\nu}$ to electrical conductivity $\sigma_{\mu\nu}$ divided by temperature. However, the electron-electron scattering makes Lorenz ratios deviate from $L_{0}$ with increasing temperature. To describe the WF law in a magnetic field, we introduce another set of Lorenz ratios, $\widetilde{L}$ and $\widetilde{L}_{\text{H}}$, defined as the ratios of the resistivity and the Hall coefficient to their thermal counterparts. The WF laws for them, $\widetilde{L} = \widetilde{L}_{\text{H}} = L_{0}$, and their violation are helpful for the discussion of $L$ and $L_{\text{H}}$. For Baber scattering, our exact result shows $L_{\text{H}}/L_{0} \sim (L/L_{0})^2$ in a weak magnetic field. In semimetals, the violations of the WF laws are significant, reflecting the different temperature dependence between the electrical and thermal resistivities in a magnetic field. This is because the momentum conservation of the electron-electron scattering has a completely different effect on electrical and thermal magnetotransport. We sort out these behaviors using $\widetilde{L}$ and $\widetilde{L}_{\text{H}}$. We also provide a relaxation time approximation, which is useful for comparing theory and experiment.

The generalized Brillouin zones (GBZs) are integral in the analysis of non-Hermitian band structures. Conventional wisdom suggests that the GBZ should be connected, where each point can be indexed by the real part of the wavevector, similar to the Brillouin zone. Here we demonstrate rich topological features of the GBZs in generic non-Hermitian one-dimensional models. We prove and discuss a set of sufficient conditions for the model to ensure the connectivity of its GBZ. In addition, we show that the GBZ can become disconnected and have more connected components than the number of bands, which results from the point-gap features of the band structure. This novel GBZ topology is applied to further demonstrate a counterintuitive effect, where the line gap of an open-boundary spectrum with sublattice symmetry may be closed without changing its point-gap topology. Our results challenge the current understanding of bands and gaps in non-Hermitian systems and highlight the need to further investigate the topological effects associated with the GBZ including topological invariants and open-boundary braiding.

We study $N$ run-and-tumble particles (RTPs) in one dimension interacting via a double-well potential $W(r)=-k_0 \, r^2/2+g \, r^4/4$, which is repulsive at short interparticle distance $r$ and attractive at large distance. At large time, the system forms a bound state where the density of particles has a finite support. We focus on the determination of the total density of particles in the stationary state $\rho_s(x)$, in the limit $N\to+\infty$. We obtain an explicit expression for $\rho_s(x)$ as a function of the ”renormalized" interaction parameter $k=k_0-3m_2$ where $m_2$ is the second moment of $\rho_s(x)$. Interestingly, this stationary solution exhibits a transition between a connected and a disconnected support for a certain value of $k$, which has no equivalent in the case of Brownian particles. Analyzing in detail the expression of the stationary density in the two cases, we find a variety of regimes characterized by different behaviors near the edges of the support and around $x=0$. Furthermore, we find that the mapping $k_0\to k$ becomes multi-valued below a certain value of the tumbling rate $\gamma$ of the RTPs for some range of values of $k_0$ near the transition, implying the existence of two stable solutions. Finally, we show that in the case of a disconnected support, it is possible to observe steady states where the density $\rho_s(x)$ is not symmetric. All our analytical predictions are in good agreement with numerical simulations already for systems of $N = 100$ particles. The non-uniqueness of the stationary state is a particular feature of this model in the presence of active (RTP) noise, which contrasts with the uniqueness of the Gibbs equilibrium for Brownian particles. We argue that these results are also relevant for a class of more realistic interactions with both an attractive and a repulsive part, but which decay at infinity.

We present a tensor-network-based method for simulating a weakly-measured quantum circuit. In particular, we use a Markov chain to efficiently sample measurements and contract the tensor network, propagating their effect forward along the spatial direction. Applications of our algorithm include validating quantum computers (capable of mid-circuit measurements) in regimes of easy classical simulability, and studying generative-machine-learning applications, where sampling from complex stochastic processes is the main task. As a demonstration of our algorithm, we consider a (1+1)-dimensional brickwall circuit of Haar-random unitaries, interspersed with generalized single-qubit measurements of variable strength. We simulate the dynamics for tens to hundreds of qubits if the circuit exhibits area-law entanglement (under strong measurements), and tens of qubits if it exhibits volume-law entanglement (under weak measurements). We observe signatures of a measurement-induced phase transition between the two regimes as a function of measurement strength.

Random graphs offer a useful mathematical representation of a variety of real world complex networks. Exponential random graphs, for example, are particularly suited towards generating random graphs constrained to have specified statistical moments. In this investigation, we elaborate on a generalization of the former where link probabilities are conditioned on the appearance of other links, corresponding to the introduction of interactions in an effective generalized statistical mechanical formalism. When restricted to the simplest non-trivial case of pairwise interactions, one can derive a closed form renormalization group transformation for maximum coordination number two on the corresponding line graph. Higher coordination numbers do not admit exact closed form renormalization group transformations, a feature that paraphrases the usual absence of exact transformations in two or more dimensional lattice systems. We introduce disorder and study the induced renormalization group flow on its probability assignments, highlighting its formal equivalence to time reversed anisotropic drift-diffusion on the statistical manifold associated with the effective Hamiltonian. We discuss the implications of our findings, stressing the long wavelength irrelevance of certain classes of pair-wise conditioning on random graphs, and conclude with possible applications. These include modeling the scaling behavior of preferential effects on social networks, opinion dynamics, and reinforcement effects on neural networks, as well as how our findings offer a systematic framework to deal with data limitations in inference and reconstruction problems.

Optimal constructions of classical LDPC codes can be obtained by choosing the Tanner graph uniformly at random among biregular graphs. We introduce a class of codes that we call “diffusion codes”, defined by placing each edge connecting bits and checks on some graph, and acting on that graph with a random SWAP network. By tuning the depth of the SWAP network, we can tune a tradeoff between the amount of randomness – and hence the optimality of code parameters – and locality with respect to the underlying graph. For diffusion codes defined on the cycle graph, if the SWAP network has depth $\sim Tn$ with $T> n^{2\beta}$ for arbitrary $\beta>0$, then we prove that almost surely the Tanner graph is a lossless “smaller set” vertex expander for small sets up size $\delta \sim \sqrt T \sim n^{\beta}$, with bounded bit and check degree. At the same time, the geometric size of the largest stabilizer is bounded by $\sqrt T$ in graph distance. We argue, based on physical intuition, that this result should hold more generally on arbitrary graphs. By taking hypergraph products of these classical codes we obtain quantum LDPC codes defined on the torus with smaller-set boundary and co-boundary expansion and the same expansion/locality tradeoffs as for the classical codes. These codes are self-correcting and admit single-shot decoding, while having the geometric size of the stabilizer growing as an arbitrarily small power law. Our proof technique establishes mixing of a random SWAP network on small subsystems at times scaling with only the subsystem size, which may be of independent interest.