Self-dual Point Research Articles

One dimensional staggered bosons, clock models, and their noninvertible symmetries

We study systems of staggered boson Hamiltonians in a one dimensional lattice and in particular how the translation symmetry by one unit in these systems is in reality a noninvertible symmetry closely related to T-duality. We also study the simplest systems of clock models derived from these staggered boson Hamiltonians. We show that the noninvertible symmetries of these lattice models together with the discrete ZN symmetry predict that these are critical points with a U(1) current algebra at c=1 and radius 2N whenever N>4. We also present an independent computation of this value that arises directly from the staggered boson variables and does not use these additional symmetries. We also present a theoretical estimate of the values of critical coupling constants away from the self-dual symmetry point in these clock models. Published by the American Physical Society 2024

Strong/weak duality symmetries for Jacobi–Gordon field theory through elliptic functions

Abstract It is shown that the standard sin/sinh Gordon field theory with the strong/weak duality symmetry of its quantum S-matrix, can be formulated in terms of elliptic functions with their duality symmetries, which will correspond to the classical realization of that quantum symmetry. Specifically we show that the so called self-dual point that divides the strong and the weak coupling regimes, corresponds only to one point of a set of fixed points under the duality transformations for the elliptic functions. Furthermore, the equations of motion can be solved in exact form in terms of the inverse elliptic functions; in spontaneous symmetry breaking scenarios, these solutions show that 
kink-like solitons can decay to cusp-like solitons

Phase Diagram of the Ashkin–Teller Model

The Ashkin–Teller model is a pair of interacting Ising models and has two parameters: J is a coupling constant in the Ising models and U describes the strength of the interaction between them. In the ferromagnetic case J,U>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J,U>0$$\\end{document} on the square lattice, we establish a complete phase diagram conjectured in physics in 1970s (by Kadanoff and Wegner, Wu and Lin, Baxter and others): when J<U\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J<U$$\\end{document}, the transitions for the Ising spins and their products occur at two distinct curves that are dual to each other; when J≥U\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J\\ge U$$\\end{document}, both transitions occur at the self-dual curve. All transitions are shown to be sharp using the OSSS inequality. We use a finite-size criterion argument and continuity to extend the result of Glazman and Peled (Electron J Probab 28:1-53, 2023) from a self-dual point to its neighborhood. Our proofs go through the random-cluster representation of the Ashkin–Teller model introduced by Chayes–Machta and Pfister–Velenik and we rely on couplings to FK-percolation.

Competing phases and intertwined orders in coupled wires near the self-dual point

Competing phases and intertwined orders in coupled wires near the self-dual point

Electromagnetically unclonable functions generated by non-Hermitian absorber-emitter.

Physically unclonable functions (PUFs) are a class of hardware-specific security primitives based on secret keys extracted from integrated circuits, which can protect important information against cyberattacks and reverse engineering. Here, we put forward an emerging type of PUF in the electromagnetic domain by virtue of the self-dual absorber-emitter singularity that uniquely exists in the non-Hermitian parity-time (PT)-symmetric structures. At this self-dual singular point, the reconfigurable emissive and absorptive properties with order-of-magnitude differences in scattered power can respond sensitively to admittance or phase perturbations caused by, for example, manufacturing imperfectness. Consequently, the entropy sourced from inevitable manufacturing variations can be amplified, yielding excellent PUF security metrics in terms of randomness and uniqueness. We show that this electromagnetic PUF can be robust against machine learning-assisted attacks based on the Fourier regression and generative adversarial network. Moreover, the proposed PUF concept is wavelength-scalable in radio frequency, terahertz, infrared, and optical systems, paving a promising avenue toward applications of cryptography and encryption.

Universal Predictions of Modular Invariant Flavor Models near the Self-Dual Point.

We analyze a large set of modular invariant models of lepton masses and mixing angles, pointing out that many of them prefer to live close to the self-dual point τ=i. We show that in the vicinity of this point a universal behavior naturally emerges, independent from details of the theory, such as the finite modular group acting on the lepton multiplets, the weights of the matter multiplets and even the form of the kinetic terms, which are not required to be minimal nor flavor blind. The neutrino mass spectrum is normally ordered and universal relations describe the scaling of the physical observables in terms of the parameter |τ-i|.

Finite-Size Scaling Theory at a Self-Dual Quantum Critical Point

The nondivergence of the generalized Grüneisen ratio (GR) at a quantum critical point (QCP) has been proposed to be a universal thermodynamic signature of self-duality. We study how the Kramers–Wannier-type self-duality manifests itself in the finite-size scaling behavior of thermodynamic quantities in the quantum critical regime. While the self-duality cannot be realized as a unitary transformation in the total Hilbert space for the Hamiltonian with the periodic boundary condition, it can be implemented in certain symmetry sectors with proper boundary conditions. Therefore, the GR and the transverse magnetization of the one-dimensional transverse-field Ising model exhibit different finite-size scaling behaviors in different sectors. This implies that the numerical diagnosis of self-dual QCP requires identification of the proper symmetry sectors.

Duality, Hidden Symmetry, and Dynamic Isomerism in 2D Hinge Structures.

Recently, a new type of duality was reported in some deformable mechanical networks that exhibit Kramers-like degeneracy in phononic spectrum at the self-dual point. In this work, we clarify the origin of this duality and propose a design principle of 2D self-dual structures with arbitrary complexity. We find that this duality originates from the partial central inversion (PCI) symmetry of the hinge, which belongs to a more general end-fixed scaling transformation. This symmetry gives the structure an extra degree of freedom without modifying its dynamics. This results in dynamic isomers, i.e., dissimilar 2D mechanical structures, either periodic or aperiodic, having identical dynamic modes, based on which we demonstrate a new type of wave guide without reflection or loss. Moreover, the PCI symmetry allows us to design various 2D periodic isostatic networks with hinge duality. At last, by further studying a 2D nonmechanical magnonic system, we show that the duality and the associated hidden symmetry should exist in a broad range of Hamiltonian systems.

Monte Carlo study of duality and the Berezinskii-Kosterlitz-Thouless phase transitions of the two-dimensional q-state clock model in flow representations.

The two-dimensional q-state clock model for q≥5 undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for q=5-9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are η_{1}=1/4 and η_{2}=4/q^{2} at the two BKT transitions. Further, the approximate self-dual points β_{sd}(L), defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size L and behave as β_{sd}≃q/2π asymptotically at the large-q limit. The exponent η at β_{sd} is consistent with 1/q within statistical error as long as q≥5. Based on this, we further conjecture that η(β_{sd})=1/q holds exactly and is universal for systems in the q-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.

Emergent quantum state designs and biunitarity in dual-unitary circuit dynamics

Recent works have investigated the emergence of a new kind of random matrix behaviour in unitary dynamics following a quantum quench. Starting from a time-evolved state, an ensemble of pure states supported on a small subsystem can be generated by performing projective measurements on the remainder of the system, leading to a projected ensemble. In chaotic quantum systems it was conjectured that such projected ensembles become indistinguishable from the uniform Haar-random ensemble and lead to a quantum state design. Exact results were recently presented by Ho and Choi [Phys. Rev. Lett. 128, 060601 (2022)] for the kicked Ising model at the self-dual point. We provide an alternative construction that can be extended to general chaotic dual-unitary circuits with solvable initial states and measurements, highlighting the role of the underlying dual-unitarity and further showing how dual-unitary circuit models exhibit both exact solvability and random matrix behaviour. Building on results from biunitary connections, we show how complex Hadamard matrices and unitary error bases both lead to solvable measurement schemes.

Non-invertible self-duality defects of Cardy-Rabinovici model and mixed gravitational anomaly

We study new symmetries of the Cardy-Rabinovici model and their dynamical applications. The Cardy-Rabinovici model is a 4d U(1) gauge theory with electric and magnetic matters, which is a good playground for studying the dynamics of the Yang-Mills theory with θ angle. In this model, the electromagnetic S L(2, ℤ) self-duality is not realized in a naive way. Still, the S L(2, ℤ) transformations become legitimate duality operations by appropriately gauging the ℤN 1-form symmetry. We construct new noninvertible symmetries from this duality at self-dual points and determine their non-group-like fusion rules. As an application, we can rule out the trivially gapped phase for some self-dual parameters due to a mixed gravitational anomaly of this new symmetry. We also show how the conjectured phase diagram of the Cardy-Rabinovici model is consistent with this anomaly matching condition.

Existence of corner modes in elastic twisted kagome lattices

This Letter investigates the emergence of corner modes in elastic twisted Kagome lattices at a critical twist angle, known as self-dual point. We show that a special type of corner modes exist and they are localized at a very specific type of corners independent of the overall shape of the finite lattice. Moreover, these modes appear even in the perturbed lattice at the corners with mirror planes. By exploring its counterpart in electronic system, we attribute such corner modes to charge accumulation at the boundary, which is confirmed by the plot of charge distribution in a finite lattice.

Thermoelectric properties and Wiedemann-Franz-like relations in mixed-dimensional QEDs from particle-vortex dualities

We consider the thermoelectric properties of the mixed-dimensional quantum electrodynamics of the relativistic Dirac fermion and Wilson-Fisher boson. These models are self-dual, and can form nontrivial many-body phases depending on the values of chemical potential, background magnetic field and the electromagnetic fine-structure constant. Using particle-vortex duality, we derive a variety of thermoelectric relations for strongly interacting phases with classic paradigms such as the Wiedemann-Franz law and the Mott's relation in the dual weakly interacting regimes. Besides, at the self-dual point, for the fermionic theory we find the ratio of thermal conductivity of electrical conductivity depends on the determinant of the Seebeck tensor and the phenomenological parameter Hall angle ${\ensuremath{\theta}}_{H}$. As for the bosonic theory, the dual fermion description explains how its Seebeck tensor varies depending on the dynamic regime characterized by ${\ensuremath{\theta}}_{H}$.

Topological Order and Criticality in (2+1)D Monitored Random Quantum Circuits.

It has recently been discovered that random quantum circuits provide an avenue to realize rich entanglement phase diagrams, which are hidden to standard expectation values of operators. Here we study (2+1)D random circuits with random Clifford unitary gates and measurements designed to stabilize trivial area law and topologically ordered phases. With competing single qubit Pauli-Z and toric code stabilizer measurements, in addition to random Clifford unitaries, we find a phase diagram involving a tricritical point that maps to (2+1)D percolation, a possibly stable critical phase, topologically ordered, trivial, and volume law phases, and lines of critical points separating them. With Pauli-Y single qubit measurements instead, we find an anisotropic self-dual tricritical point, with dynamical exponent z≈1.46, exhibiting logarithmic violation of the area law and an anomalous exponent for the topological entanglement entropy, which thus appears distinct from any known percolation fixed point. The phase diagram also hosts a measurement-induced volume law entangled phase in the absence of unitary dynamics.

Self-duality of one-dimensional quasicrystals with spin-orbit interaction

Noninteracting spinless electrons in one-dimensional quasicrystals, described by the Aubry-Andr\'e-Harper (AAH) Hamiltonian with nearest-neighbor hopping, undergo a metal-to-insulator transition at a critical strength of the quasiperiodic potential. The AAH Hamiltonian is also known to be self-dual. Interestingly, the critical point and the self-dual point are identical in this case. In this work, we have studied the one-dimensional quasiperiodic AAH Hamiltonian in the presence of spin-orbit coupling of Rashba-type, which introduces an additional spin-conserving complex hopping and a spin-flip hopping. We have found that the AAH Hamiltonian remains self-dual in the presence of Rashba spin-orbit coupling, and the self-dual point follows a simple rescaled relationship among the parameters of the Hamiltonian. This system also undergoes a metal-to-insulator transition, and the nature of this transition has been found to be identical with the original AAH Hamiltonian. However, the critical points do not follow the same relationship as the self-dual points in general. In fact, the metal-to-insulator transition happens earlier than the self-dual point, except in some special cases in which they are observed to coincide.

Self-Assembly of Isostatic Self-Dual Colloidal Crystals.

Self-dual structures whose dual counterparts are themselves possess unique hidden symmetry, beyond the description of classical spatial symmetry groups. Here we propose a strategy based on a nematic monolayer of attractive half-cylindrical colloids to self-assemble these exotic structures. This system can be seen as a 2D system of semidisks. By using MonteCarlo simulations, we discover two isostatic self-dual crystals, i.e., an unreported crystal with pmg space-group symmetry and the twisted kagome crystal. For the pmg crystal approaching the critical point, we find the double degeneracy of the full phononic spectrum at the self-dual point and the merging of two tilted Weyl nodes into one critically tilted Dirac node. The latter is "accidentally" located on the high-symmetry line. The formation of this unconventional Dirac node is due to the emergence of the critical flatbands at the self-dual point, which are linear combinations of "finite-frequency" floppy modes. These modes can be understood as mechanically coupled self-dual rhombus chains vibrating in some unique uncoupled ways. Our work paves the way for designing and fabricating self-dual materials with exotic mechanical or phononic properties.

Modular invariant dynamics and fermion mass hierarchies around \u03c4 = i

We discuss fermion mass hierarchies within modular invariant flavour models. We analyse the neighbourhood of the self-dual point τ = i, where modular invariant theories possess a residual Z4 invariance. In this region the breaking of Z4 can be fully described by the spurion ϵ ≈ τ − i, that flips its sign under Z4. Degeneracies or vanishing eigenvalues of fermion mass matrices, forced by the Z4 symmetry at τ = i, are removed by slightly deviating from the self-dual point. Relevant mass ratios are controlled by powers of |ϵ|. We present examples where this mechanism is a key ingredient to successfully implement an hierarchical spectrum in the lepton sector, even in the presence of a non-minimal Kähler potential.

Exact non-Hermitian mobility edges in one-dimensional quasicrystal lattice with exponentially decaying hopping and its dual lattice

We analytically determine the non-Hermitian mobility edges of a one-dimensional quasiperiodic lattice model with exponential decaying hopping and complex potentials as well as its dual model, which is just a non-Hermitian generalization of the Ganeshan-Pixley-Das Sarma model with nonreciprocal nearest-neighboring hopping. The presence of non-Hermitian term destroys the self-duality symmetry and thus prevents us exploring the localization-delocalization point through looking for self-dual points. Nevertheless, by applying Avila's global theory, the Lyapunov exponent of the Ganeshan-Pixley-Das Sarma model can be exactly derived, which enables us to get an analytical expression of mobility edge of the non-Hermitian dual model. Consequently, the mobility edge of the original model is obtained by using the dual transformation, which creates exact mappings between the spectra and wavefunctions of these two models.

Entanglement Phase Transitions in Measurement-Only Dynamics

Unitary circuits subject to repeated projective measurements can undergo an entanglement phase transition (EPT) as a function of the measurement rate. This transition is generally understood in terms of a competition between the scrambling effects of unitary dynamics and the disentangling effects of measurements. We find that, surprisingly, EPTs are possible even in the absence of scrambling unitary dynamics, where they are best understood as arising from measurements alone. This motivates us to introduce \emph{measurement-only models}, in which the "scrambling" and "un-scrambling" effects driving the EPT are fundamentally intertwined and cannot be attributed to physically distinct processes. This represents a novel form of an EPT, conceptually distinct from that in hybrid unitary-projective circuits. We explore the entanglement phase diagrams, critical points, and quantum code properties of some of these measurement-only models. We find that the principle driving the EPTs in these models is \emph{frustration}, or mutual incompatibility, of the measurements. Suprisingly, an entangling (volume-law) phase is the generic outcome when measuring sufficiently long but still local ($\gtrsim 3$-body) operators. We identify a class of exceptions to this behavior ("bipartite ensembles") which cannot sustain an entangling phase, but display dual area-law phases, possibly with different kinds of quantum order, separated by self-dual critical points. Finally, we introduce a measure of information spreading in dynamics with measurements and use it to demonstrate the emergence of a statistical light-cone, despite the non-locality inherent to quantum measurements.

Approaching the self-dual point of the sinh-Gordon model

One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the b → 1/b self-duality of its S-matrix, of which there is no trace in its Lagrangian formulation. Here b is the coupling appearing in the model’s eponymous hyperbolic cosine present in its Lagrangian, cosh(bϕ). In this paper we develop truncated spectrum methods (TSMs) for studying the sinh-Gordon model at a finite volume as we vary the coupling constant. We obtain the expected results for b ≪ 1 and intermediate values of b, but as the self-dual point b = 1 is approached, the basic application of the TSM to the ShG breaks down. We find that the TSM gives results with a strong cutoff Ec dependence, which disappears according only to a very slow power law in Ec. Standard renormalization group strategies — whether they be numerical or analytic — also fail to improve upon matters here. We thus explore three strategies to address the basic limitations of the TSM in the vicinity of b = 1. In the first, we focus on the small-volume spectrum. We attempt to understand how much of the physics of the ShG is encoded in the zero mode part of its Hamiltonian, in essence how ‘quantum mechanical’ vs ‘quantum field theoretic’ the problem is. In the second, we identify the divergencies present in perturbation theory and perform their resummation using a supra-Borel approximate. In the third approach, we use the exact form factors of the model to treat the ShG at one value of b as a perturbation of a ShG at a different coupling. In the light of this work, we argue that the strong coupling phase b > 1 of the Lagrangian formulation of model may be different from what is naïvely inferred from its S-matrix. In particular, we present an argument that the theory is massless for b > 1.