Integrable Quantum Research Articles

Exact solution of the quantum integrable model associated with the Motzkin spin chain

The Motzkin spin chain is a spin-1 frustration-free model introduced by Shor & Movassagh. The ground state is constructed by mapping random walks on the upper half of the square lattice to spin configurations. It has unusually large entanglement entropy [quantum fluctuations]. The ground state of the Motzkin chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states. The model is not solvable. We simplify the model by removing one of the local equivalence moves of the Motzkin paths. The system becomes integrable [similar to the XXX spin chain]. We call it free Motzkin chain. From the point of view of quantum integrability, the model is special since its R-matrix does not have crossing unitarity. We solve the periodic free Motzkin chain by generalizing the functional Bethe Ansatz method. We construct a T − Q relation with an additional parameter to formulate the energy spectrum. This new parameter is related to the roots of unity and can be described by the Möbius function in number theory. We observe further patterns of number theory.

The full analytic trans-series in integrable field theories

We analyze a family of generalized energy densities in integrable quantum field theories in the presence of an external field coupled to a conserved charge. By using the Wiener-Hopf technique to solve the linear thermodynamic Bethe ansatz equations we derive the full analytic trans-series for these observables in terms of a perturbatively defined basis. We show how to calculate these basis elements to high orders analytically and reveal their complete resurgence structure. We demonstrate that the physical value of the energy density is obtained by the median resummation of the perturbative series.

Thermodynamic stability of cytosine tetramers mediated by partially reduced silver trimers: QM/MM free energy analysis

AbstractPartially reduced silver clusters growing in a polycytosine matrix could simultaneously possess stabilizing ability and remarkable optical features. The mechanisms of this phenomenon are not fully understood so far. We examined the thermodynamic stability of a cytosine–silver associate comprising four cytosines and three silver atoms that can be considered the smallest known luminescent silver marker stabilized on DNA. Quantum and molecular mechanical representations thermodynamic integration toward the dissociation pathway of the cytosine tetramer, time‐dependent density functional theory calculations of the lowest energy excited states and quantum theory of atoms in molecules analysis of bonding showed that the reduction of two ions within the silver trimer mediating cytosine tetramer leads to the appearance of the green‐emitting silver cluster with low stabilization ability of cytosines relative to a tetramer stabilized by two silver ions.

Integrable Digital Quantum Simulation: Generalized Gibbs Ensembles and Trotter Transitions.

The Trotter-Suzuki decomposition is a promising avenue for digital quantum simulation (DQS), approximating continuous-time dynamics by discrete Trotter steps of duration τ. Recent work suggested that DQS is typically characterized by a sharp Trotter transition: when τ is increased beyond a threshold value, approximation errors become uncontrolled at large times due to the onset of quantum chaos. Here, we contrast this picture with the case of integrable DQS. We focus on a simple quench from a spin-wave state in the prototypical XXZ Heisenberg spin chain, and study its integrable Trotterized evolution as a function of τ. Because of its exact local conservation laws, the system does not heat up to infinite temperature and the late-time properties of the dynamics are captured by a discrete generalized Gibbs ensemble (dGGE). By means of exact calculations we find that, for small τ, the dGGE depends analytically on the Trotter step, implying that discretization errors remain bounded even at infinite times. Conversely, the dGGE changes abruptly at a threshold value τ_{th}, signaling a novel type of Trotter transition. We show that the latter can be detected locally, as it is associated with the appearance of a nonzero staggered magnetization with a subtle dependence on τ. We highlight the differences between continuous and discrete GGEs, suggesting the latter as novel interesting nonequilibrium states exclusive to digital platforms.

Comment on the Vacuum Energy Density for λϕ4 Theory in d Spacetime Dimensions

In a recent article we showed that the vacuum energy density in two spacetime dimensions for a wide variety of integrable quantum field theories has the form ρvac=−m2/2g where m is a physical mass and g is a generalized coupling, where in the free field limit g→0, ρvac diverges. This vacuum energy density has the form ⟨Tμν⟩=−ρvacgμν, and has previously been considered as a contribution to the stress energy tensor in Einstein’s gravity as a “cosmological constant”. We speculated that in four spacetime dimensions ρvac takes a similar form ρvac=−m4/2g, but did not support this idea in any specific model. In this article, we study this problem for λϕ4 theory in d spacetime dimensions. We show how to obtain the exactρvac for the sinh–Gordon theory in the weak coupling limit by using a saddle point approximation. This calculation indicates that the vacuum energy can be well-defined, positive or negative, without spontaneous symmetry breaking. We also show that ρvac satisfies a Callan–Symanzik type of renormalization group equation. For the most interesting case physically, ρvac is positive and can arise from a marginally relevant negative coupling g and the vacuum energy flows to zero at low energies.

Subclasses of p-Valent κ-Uniformly Convex and Starlike Functions Defined by the q-Derivative Operator

The potential for widespread applications of the geometric and mapping properties of functions of a complex variable has motivated this article. On the other hand, the basic or quantum (or q-) derivatives and the basic or quantum (or q-) integrals are extensively applied in many different areas of the mathematical, physical and engineering sciences. Here, in this article, we first apply the q-calculus in order to introduce the q-derivative operator Sη,p,qn,m. Secondly, by means of this q-derivative operator, we define an interesting subclass Tℵλ,pn,m(η,α,κ) of the class of normalized analytic and multivalent (or p-valent) functions in the open unit disk U. This p-valent analytic function class is associated with the class κ-UCV of κ-uniformly convex functions and the class κ-UST of κ-uniformly starlike functions in U. For functions belonging to the normalized analytic and multivalent (or p-valent) function class Tℵλ,pn,m(η,α,κ), we then investigate such properties as those involving (for example) the coefficient bounds, distortion results, convex linear combinations, and the radii of starlikeness, convexity and close-to-convexity. We also consider a number of corollaries and consequences of the main findings, which we derived herein.

The Poisson geometry of Plancherel formulas for triangular groups

In this paper we establish the existence of canonical coordinates for generic co-adjoint orbits on triangular groups. These orbits correspond to a set of full Plancherel measure on the associated dual groups. This generalizes a well-known coordinatization of co-adjoint orbits of a minimal (non-generic) type originally discovered by Flaschka. The latter had strong connections to the classical Toda lattice and its associated Poisson geometry. Our results develop connections with the Full Kostant–Toda lattice and its Poisson geometry. This leads to novel insights relating the details of Plancherel theorems for Borel Lie groups to the invariant theory for Borels and their subgroups. We also discuss some implications for the quantum integrability of the Full Kostant Toda lattice.

Mingling of the infrared and ultraviolet and the “cosmological constant” for interacting QFT in 2d

We propose a proper definition of the vacuum expectation value of the stress energy tensor 〈0| Tμν| 0〉 for integrable quantum field theories in two spacetime dimensions, which is the analog of the cosmological constant in 4d. For a wide variety of models, massive or massless, we show {rho}_{textrm{vac}}=-{m}^2/2mathfrak{g} exactly, where mathfrak{g} is a generalized coupling which we compute and m is a basic mass scale. The kinds of models we consider are the massive sinh-Gordon and sine-Gordon theories and perturbations of the Yang-Lee and 3-state Potts models, pure Toverline{T} perturbations of infra-red QFT’s, and UV completions of the latter which are massless flows between UV and IR fixed points. In the massive case m is the physical mass of the lightest particle and mathfrak{g} is related to parameters in the 2-body S-matrix. In some examples ρvac = 0 due to a fractional supersymmetry. For massless cases, m can be a scale of spontaneous symmetry breaking. The “cosmological constant problem” generically arises in the free field limit mathfrak{g} → 0, thus interactions can potentially resolve the problem at least for most cases considered in this paper. We speculate on extensions of these results to 4 spacetime dimensions and propose {rho}_{textrm{vac}}={m}^4/2mathfrak{g} , however without integrability we cannot yet propose a precise manner in which to calculate mathfrak{g} . Nevertheless, based on cosmological data on ρvac, if mathfrak{g} ~ 1 then it is worth pointing out that the lightest mass particle is on the order of experimental values of proposed neutrino masses.

Asymptotics of singular values for quantised derivatives on noncommutative tori

We obtain Weyl type asymptotics for pseudodifferential operators on quantum torus Tθd of the form TIθ−1. Here T stands for classical pseudodifferential operators on quantum torus of order 0, and Iθ−1 is the Riesz potential of order −1. As an application, we obtain Weyl type asymptotics for the quantised derivative ▪ of an operator x from the homogeneous Sobolev space W˙d1(Tθd). The asymptotic coefficient is equivalent to the norm of ▪ in the principal ideal Ld,∞, as well as the norm of x in W˙d1(Tθd). We precise and rectify earlier results in our previous paper (Comm. Math. Phys. 371 (2019), no. 3, 1231–1260) on quantum integration formula for ▪.

Heterogeneous integration of a III-V quantum dot laser on high thermal conductivity silicon carbide.

Heat accumulation prevents semiconductor lasers from operating at their full potential. This can be addressed through heterogeneous integration of a III-V laser stack onto non-native substrate materials with high thermal conductivity. Here, we demonstrate III-V quantum dot lasers heterogeneously integrated on silicon carbide (SiC) substrates with high temperature stability. A large T0 of 221 K with a relatively temperature-insensitive operation occurs near room temperature, while lasing is sustained up to 105°C. The SiC platform presents a unique and ideal candidate for realizing monolithic integration of optoelectronics, quantum, and nonlinear photonics.

Integrable Floquet systems related to logarithmic conformal field theory

We study an integrable Floquet quantum system related to lattice statistical systems in the universality class of dense polymers. These systems are described by a particular non-unitary representation of the Temperley-Lieb algebra. We find a simple Lie algebra structure for the elements of Temperley-Lieb algebra which are invariant under shift by two lattice sites, and show how the local Floquet conserved charges and the Floquet Hamiltonian are expressed in terms of this algebra. The system has a phase transition between local and non-local phases of the Floquet Hamiltonian. We provide a strong indication that in the scaling limit this non-equilibrium system is described by the logarithmic conformal field theory.

On convergence of form factor expansions in the infinite volume quantum Sinh-Gordon model in 1+1 dimensions

This paper develops a technique allowing one to prove the convergence of a class of series of multiple integrals which corresponds to the form factor expansion of space-like separated two-point functions in the 1+1 dimensional massive integrable Sinh-Gordon quantum field theory.

Asymptotics in an asymptotic CFT

In this work we illustrate the resurgent structure of the λ-deformation; a two-dimensional integrable quantum field theory that has an RG flow with an SU(N)k Wess-Zumino-Witten conformal fixed point in the UV. To do so we use modern matched asymptotic techniques applied to the thermodynamic Bethe ansatz formulation to compute the free energy to 38 perturbative orders in an expansion of large applied chemical potential. We find numerical evidence for factorial asymptotic behaviour with both alternating and non-alternating character which we match to an analytic expression. A curiosity of the system is that the leading non-alternating factorial growth vanishing when k divides N. The ambiguities associated to Borel resummation of this series are suggestive of non-perturbative contributions. This is verified with an analytic study of the TBA system demonstrating a cancellation between perturbative and non-perturbative ambiguities.

Embedding integrable superspin chain in string theory

Using results on topological line defects of 4D Chern-Simons theory and the algebra/cycle homology correspondence in complex surfaces S with ADE singularities, we study the graded properties of the sl(m|n) chain and its embedding in string theory. Because of the Z2-grading of sl(m|n), we show that the (m+n)!/m!n! varieties of superspin chains with underlying super geometries have different cycle homologies. We investigate the algebraic and homological features of these integrable quantum chains and give a link between graded 2-cycles and genus-g Rieman surfaces Σg. Moreover, using homology language, we yield the brane realisation of the sl(m|n) chain in type IIA string and its uplift to M-theory. Other aspects like graded complex surfaces with sl(m|n) singularity as well as super magnons are also described.

Robustness of Kardar-Parisi-Zhang scaling in a classical integrable spin chain with broken integrability

Recent investigations have observed superdiffusion in integrable classical and quantum spin chains. An intriguing connection between these spin chains and Kardar-Parisi-Zhang (KPZ) universality class has emerged. Theoretical developments (e.g. generalized hydrodynamics) have highlighted the role of integrability as well as spin-symmetry in KPZ behaviour. However understanding their precise role on superdiffusive transport still remains a challenging task. The widely used quantum spin chain platform comes with severe numerical limitations. To circumvent this barrier, we focus on a classical integrable spin chain which was shown to have deep analogy with the quantum spin-$\frac{1}{2}$ Heisenberg chain. Remarkably, we find that KPZ behaviour prevails even when one considers integrability-breaking but spin-symmetry preserving terms, strongly indicating that spin-symmetry plays a central role even in the non-perturbative regime. On the other hand, in the non-perturbative regime, we find that energy correlations exhibit clear diffusive behaviour. We also study the classical analog of out-of-time-ordered correlator (OTOC) and Lyapunov exponents. We find significant presence of chaos for the integrability-broken cases even though KPZ behaviour remains robust. The robustness of KPZ behaviour is demonstrated for a wide class of spin-symmetry preserving integrability-breaking terms.

New Orthogonality Relations for Super-Jack Polynomials and an Associated Lassalle–Nekrasov Correspondence

AbstractThe super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in$$n+m$$n+mvariables, which reduce to the Jack polynomials when$$n=0$$n=0or$$m=0$$m=0and provide joint eigenfunctions of the quantum integrals of the deformed trigonometric Calogero–Moser–Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form$$(p,q)\mapsto (L_pq)(0)$$(p,q)↦(Lpq)(0), with$$L_p$$Lpquantum integrals of the deformed rational Calogero–Moser–Sutherland system. In addition, we provide a new proof of the Lassalle–Nekrasov correspondence between deformed trigonometric and rational harmonic Calogero–Moser–Sutherland systems and infer orthogonality of super-Hermite polynomials, which provide joint eigenfunctions of the latter system.

Deforming the ODE/IM correspondence with textrm{T}overline{textrm{T}}

The ODE/IM correspondence is an exact link between classical and quantum integrable models. The primary purpose of this work is to show that it remains valid after textrm{T}overline{textrm{T}} perturbation on both sides of the correspondence. In particular, we prove that the deformed Lax pair of the sinh-Gordon model, obtained from the unperturbed one through a dynamical change of coordinates, leads to the same Burgers-type equation governing the quantum spectral flow induced by textrm{T}overline{textrm{T}} . Our main conclusions have general validity, as the analysis may be easily adapted to all the known ODE/IM examples involving integrable quantum field theories.

Geometrical, topological, and dynamical description of N interacting spin- s particles in a long-range Ising model and their interplay with quantum entanglement

Comprehending the connections between the geometric, topological, and dynamical structures of integrable quantum systems with quantum phenomena exploitable in quantum information tasks, such as quantum entanglement, is a major problem in geometric information science. In this work we investigate these issues in a physical system of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model. We discover the relevant dynamics, identify the corresponding quantum phase space and we derive the associated Fubini-Study metric. Through the application of the Gauss-Bonnet theorem and the derivation of the Gaussian curvature, we have proved that the dynamics occurs on a spherical topology manifold. Afterwards, we analyze the gained geometrical phase under the arbitrary and cyclic evolution processes and solve the quantum brachistochrone problem by establishing the time-optimal evolution. Moreover, by narrowing the system to a two spin-$\mathtt{s}$ system, we explore the relevant entanglement from two different perspectives; The first is geometrical in nature and involves the investigation of the interplay between the entanglement degree and the geometrical structures, such as the Fubini-Study metric, the Gaussian curvature and the geometrical phase. The second is dynamical in nature and tackles the entanglement effect on the evolution speed and geodesic distance. Additionally, we resolve the quantum brachistochrone problem based on the entanglement degree.

Quantum Phase Transitions for an Integrable Quantum Rabi-like Model with Two Interacting Qubits.

A two-interacting-qubit quantum Rabi-like model with vanishing transverse fields on the qubit pair is studied. Independently of the coupling regime, this model can be exactly and unitarily reduced to two independent single-spin quantum Rabi models, where the spin-spin coupling plays the role of the transverse field. This transformation and the analytical treatment of the single-spin quantum Rabi model provide the key to prove the integrability of our model. The existence of different first-order quantum phase transitions, characterized by discontinuous two-spin magnetization, mean photon number, and concurrence, is brought to light.

Topological adiabatic dynamics in classical mass-spring chains with clamps

The path dependence of adiabatic evolution in classical harmonic chains with clamps is examined. It is shown that cutting and joining a chain may braid adiabatic normal mode frequencies. Accordingly, different adiabatic paths with the same endpoints may transport a normal mode to a different one, and an adiabatic cycle pumps action variables, i.e., the adiabatic invariants of integrable classical systems. Another adiabatic pump for artificial edge modes induced by clamps is shown as an application. Extensions to completely integrable systems and quantum systems are outlined.