Integrable Quantum Research Articles

Volume-Law Entanglement Entropy of Typical Pure Quantum States

The entanglement entropy of subsystems of typical eigenstates of quantum many-body Hamiltonians has been recently conjectured to be a diagnostic of quantum chaos and integrability. In quantum chaotic systems it has been found to behave as in typical pure states, while in integrable systems it has been found to behave as in typical pure Gaussian states. In this tutorial, we provide a pedagogical introduction to known results about the entanglement entropy of subsystems of typical pure states and of typical pure Gaussian states. They both exhibit a leading term that scales with the volume of the subsystem, when smaller than one half of the volume of the system, but the prefactor of the volume law is fundamentally different. It is constant (and maximal) for typical pure states, and it depends on the ratio between the volume of the subsystem and of the entire system for typical pure Gaussian states. Since particle-number conservation plays an important role in many physical Hamiltonians, we discuss its effect on typical pure states and on typical pure Gaussian states. We prove that while the behavior of the leading volume-law terms does not change qualitatively, the nature of the subleading terms can change. In particular, subleading corrections can appear that depend on the square root of the volume of the subsystem. We unveil the origin of those corrections. Finally, we discuss the connection between the entanglement entropy of typical pure states and analytical results obtained in the context of random matrix theory, as well as numerical results obtained for physical Hamiltonians.

Statistical properties of the off-diagonal matrix elements of observables in eigenstates of integrable systems.

We study the statistical properties of the off-diagonal matrix elements of observables in the energy eigenstates of integrable quantum systems. They have been found to be dense in the spin-1/2 XXZ chain, while they are sparse in noninteracting systems. We focus on the quasimomentum occupation of hard-core bosons in one dimension and show that the distributions of the off-diagonal matrix elements are well described by generalized Gamma distributions, in both the presence and absence of translational invariance but not in the presence of localization. We also show that the results obtained for the off-diagonal matrix elements of observables in the spin-1/2 XXZ model are well described by a generalized Gamma distribution.

Classical and quantum integrability of the three-dimensional generalized trapped ion Hamiltonian

In this paper, we study the trapped ion Hamiltonian in three-dimensional (3D) with the generalized potential in the quadrupole field with superposition of the hexapole and octopole fields. We determine new integrable cases by using the Painlevé analysis and find the second and third classical invariants for each P-case. Moreover, we perturb this Hamiltonian by an inverse square potential and we prove that the 3D perturbed Hamiltonian is completely integrable in the sense of Liouville for the special conditions. Quantum invariants are obtained by adding deformation terms, computed using Moyal's bracket, to the corresponding classical counterparts. Furthermore, we use python programming language to plot the third classical invariant, the deformation and the third quantum invariant in phase space for each quantum integrable case in order to confirm the analytical results.

Onset of many-body quantum chaos due to breaking integrability

Integrable quantum systems of finite size are generically robust against weak enough integrability-breaking perturbations, but become quantum chaotic and thermalizing if the integrability-breaking is strong enough. We argue that the onset of quantum chaos can be described as a Fock-space delocalization process, with the eigenstates of the integrable system being taken as the "Fock states". The integrability-breaking perturbation introduces hopping in this Fock space, and chaos sets in when this hopping delocalizes the many-body eigenstates in this space. Depending on the range of the dominant Fock-space hopping, delocalization can occur either through a crossover, or via a transition that becomes sharp in the appropriate large-system dynamic limit. In either case, the perturbation strength at the onset of chaos scales to zero in the usual thermodynamic limit, with a size-dependence that we estimate analytically and compute numerically for a few specific models. We also identify two intermediate finite-size-dependent regimes: There is generally an intermediate nonchaotic regime in which integrability is broken strongly enough to produce some system-wide many-body resonances but not enough to thermalize the system. In spatially extended systems (but not in quantum dots) there is also a crossover or transition between chaotic regimes where the ratio of the system size to the mean free path of the quasiparticles of the integrable system is small versus large compared to unity.

Time and band limiting operator and Bethe ansatz

The time and band limiting operator is introduced to optimize the reconstruction of a signal from only a partial part of its spectrum. In the discrete case, this operator commutes with the so-called algebraic Heun operator which appears in the context of the quantum integrable systems. The construction of both operators and the proof of their commutativity is recalled. A direct connection between their spectra is obtained. Then, the Bethe ansatz, a well-known method to diagonalize integrable quantum Hamiltonians, is used to diagonalize the Heun operator and to obtain insights on the spectrum of the time and band limiting operator.

$\mathrm{T}\overline{\mathrm{T}}$-deformed 1d Bose gas

\mathrm{T}\bar{\mathrm{T}}TT‾ deformation was originally proposed as an irrelevant solvable deformation for 2d relativistic quantum field theories (QFTs). The same family of deformations can also be defined for integrable quantum spin chains which was first studied in the context of integrability in AdS/CFT. In this paper, we construct such deformations for yet another type of models, which describe a collection of particles moving in 1d and interacting in an integrable manner. The prototype of such models is the Lieb-Liniger model. This shows that such deformations can be defined for a very wide range of systems. We study the finite volume spectrum and thermodynamics of the \mathrm{T}\overline{\mathrm{T}}TT¯-deformed Lieb-Liniger model. We find that for one sign of the deformation parameter (\lambda<0)(λ<0), the deformed spectrum becomes complex when the volume of the system is smaller than certain critical value, signifying the break down of UV physics. For the other sign (\lambda>0)(λ>0), there exists an upper bound for the temperature, similar to the Hagedorn behavior of the \mathrm{T}\overline{\mathrm{T}}TT¯ deformed QFTs. Both behaviors can be attributed to the fact that \mathrm{T}\overline{\mathrm{T}}TT¯ deformation changes the size the particles. We show that for \lambda>0λ>0, the deformation increases the spaces between particles which effectively increases the volume of the system. For \lambda<0λ<0, \mathrm{T}\overline{\mathrm{T}}TT¯ deformation fattens point particles to finite size hard rods. This is similar to the observation that the action of \mathrm{T}\overline{\mathrm{T}}TT¯-deformed free boson is the Nambu-Goto action, which describes bosonic strings — also an extended object with finite size.

Quantum integration of elementary particle processes

We apply quantum integration to elementary particle-physics processes. In particular, we look at scattering processes such as e+e−→qq¯ and e+e−→qq¯′W. The corresponding probability distributions can be first appropriately loaded on a quantum computer using either quantum Generative Adversarial Networks or an exact method. The distributions are then integrated using the method of Quantum Amplitude Estimation which shows a quadratic speed-up with respect to classical techniques. In simulations of noiseless quantum computers, we obtain per-cent accurate results for one- and two-dimensional integration with up to six qubits. This work paves the way towards taking advantage of quantum algorithms for the integration of high-energy processes.

Designing tomorrow's quantum internet

Principles of quantum mechanics promise a future quantum internet that connects a wide variety of quantum devices together in a coherent and secure fashion. It is well known that due to the size of this quantum internet, quantum repeaters will be a critical part in a similar fashion to the importance of repeaters in today's telecommunications internet. Given the inherent differences between classical and quantum physics, it is essential to establish how a quantum internet will function including how we route information as well as the functionality quantum repeaters will need to provide. Our considerations here go far beyond quantum key distribution and instead focus on a true network of connected quantum devices, including computers and sensors. We show how the efficient operation of such quantum networks relies on the seamless integration of both quantum and classical communication resources.

Renormalization in TT¯ -deformed nonintegrable theories

That the exact quantum S-matrix of $\text{T}\bar{\text{T}}$-deformed field theories is known has interesting consequences for their perturbative renormalisation. Recent investigations into the interplay between renormalisation and integrable deformations have focused on those cases where the (undeformed) seed theories are integrable. In this paper, we study the perturbative renormalisation of non-integrable $\text{T}\bar{\text{T}}$-deformed field theories, focusing on a theory of scalars with a quartic interaction. We show that in this theory, the 4-point 1-loop S-matrix exhibits features similar to its integrable cousins. We also carry out a similar analysis for the case of an integrable quantum field theory comprising a pair of chiral bosons. Our discussion of this latter theory supplies the first explicit example of a perturbative field theory computation in the formalism due to Sen.

Entanglement dynamics of thermofield double states in integrable models

We study the entanglement dynamics of thermofield double (TFD) states in integrable spin chains and quantum field theories. We show that, for a natural choice of the Hamiltonian eigenbasis, the TFD evolution may be interpreted as a quantum quench from an initial state which is low-entangled in the real-space representation and displays a simple quasiparticle structure. Based on a semiclassical picture analogous to the one developed for standard quantum quenches, we conjecture a formula for the entanglement dynamics, which is valid for both discrete and continuous integrable field theories, and expected to be exact in the scaling limit of large space and time scales. We test our conjecture in two prototypical examples of integrable spin chains, where numerical tests are possible. First, in the XY-model, we compare our predictions with exact results obtained by mapping the system to free fermions, finding excellent agreement. Second, we test our conjecture in the interacting XXZ Heisenberg model, against numerical iTEBD calculations. For the latter, we generally find good agreement, although, for some range of the system parameters and within the accessible simulation times, some small discrepancies are visible, which we attribute to finite-time effects.

Eigenstate entanglement in integrable collective spin models

The average entanglement entropy (EE) of the energy eigenstates in non-vanishing partitions has been recently proposed as a diagnostic of integrability in quantum many-body systems. For it to be a faithful characterization of quantum integrability, it should distinguish quantum systems with a well-defined classical limit in the same way as the unequivocal classical integrability criteria. We examine the proposed diagnostic in the class of collective spin models characterized by permutation symmetry in the spins. The well-known Lipkin-Meshov-Glick (LMG) model is a paradigmatic integrable system in this class with a well-defined classical limit. Thus, this model is an excellent testbed for examining quantum integrability diagnostics. First, we calculate analytically the average EE of the Dicke basis{|j,m⟩}m=−jjin any non-vanishing bipartition, and show that in the thermodynamic limit, it converges to1/2of the maximal EE in the corresponding bipartition. Using finite-size scaling, we numerically demonstrate that the aforementioned average EE in the thermodynamic limit is universal for all parameter values of the LMG model. Our analysis illustrates how a value of the average EE far away from the maximal in the thermodynamic limit could be a signature of integrability.

Higher Order Deformed Elliptic Ruijsenaars Operators

We present four infinite families of mutually commuting difference operators which include the deformed elliptic Ruijsenaars operators. The trigonometric limit of this kind of operators was previously introduced by Feigin and Silantyev. They provide a quantum mechanical description of two kinds of relativistic quantum mechanical particles which can be identified with particles and anti-particles in an underlying quantum field theory. We give direct proofs of the commutativity of our operators and of some other fundamental properties such as kernel function identities. In particular, we give a rigorous proof of the quantum integrability of the deformed Ruijsenaars model.

Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models

Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs’ classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, mathcal{N} = 2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the “Dotsenko-Fateev matrix model action”. The latter is the “DF action” evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. For instance, analogous quantities as the “DF on-shell action” appear in 2d CFT realization of the so-called Richardson’s solution of the reduced BCS model describing physics of ultra-small superconducting grains. Precisely, the Richardson solution and its particular limit characterizing the rational Gaudin model have known implementation in 2d CFT in terms of a free field representation of certain (perturbed and unperturbed) WZW blocks. The WZW analogue of the “DF on-shell action” appears here and plays a crucial role, it is a generating function for eigenvalues of quantum integrals of motion. An exploration of relationships between power expansions and Coulomb gas representations of the aforementioned WZW blocks could pave a way for new analytical tools useful in the study of models of the Richardson-Gaudin type.

Thermodynamics of perturbations of some single particle field theories

We study the thermodynamic Bethe ansatz (TBA) equations for pure perturbations of some simple integrable quantum field theories with a single bosonic or fermionic particle, in particular the massive sinh-Gordon model and its ultraviolet (UV) limit which is a deformation of the conformally invariant free massless boson. Whereas the TBA equations for deformations of massive theories are in principle known, the TBA equations we propose for the deformations of conformal field theories are relatively new and require a special factorization in rapidity variables of the CDD factor for the scattering of the massless particles. The latter TBA equations can be solved exactly and reproduce the known results for the ground state energy on a cylinder of circumference R which were previously obtained using different methods based for instance on the Burgers differential equation. Special attention is paid to the c-theorem in this context which is discussed in some detail. For positive infra-red central charge c IR, for flows consistent with the c-theorem the ground state energy develops a (previously known) square-root singularity toward the UV, which strongly suggests the theories are UV incomplete in these physically important cases. We suggest that the singularity indicates a tachyonic vacuum instability. Other cases with c IR < 0 do not have this singularity are interpreted as being UV complete with c UV = 0. We extend our results to a continuously variable c IR by introducing a chemical potential and suggest this as a possible toy model for the perturbed Liouville theory.

Toverline{T} deformations and the width of fundamental particles

We provide a simple geometric meaning for deformations of so-called Toverline{T} type in relativistic and non-relativistic systems. Deformations by the cross products of energy and momentum currents in integrable quantum field theories are known to modify the thermodynamic Bethe ansatz equations by a “CDD factor”. In turn, CDD factors may be interpreted as additional, fixed shifts incurred in scattering processes: a finite width added to the fundamental particles (or, if negative, to the free space between them). We suggest that this physical effect is a universal way of understanding Toverline{T} deformations, both in classical and quantum systems. We first show this in non-relativistic systems, with particle conservation and translation invariance, using the deformation formed out of the densities and currents of particles and momentum. This holds at the level of the equations of motion, and for any interaction potential, integrable or not. We then argue, and show by similar techniques in free relativistic particle systems, that Toverline{T} deformations of relativistic systems produce the equivalent phenomenon, accounting for length contractions. We also show that, in both the relativistic and non-relativistic cases, the width of particles is equivalent to a state-dependent change of metric, where the distance function discounts the particles’ widths, or counts the additional free space. This generalises and explains the known field-dependent coordinate change describing Toverline{T} deformations. The results connect such deformations with generalised hydrodynamics, where the relations between scattering shifts, widths of particles and state-dependent changes of metric have been established.

Branch Point Twist Field Form Factors in the sine-Gordon Model II: Composite Twist Fields and Symmetry Resolved Entanglement

In this paper we continue the program initiated in Part I, that is the study of entanglement measures in the sine-Gordon model. In both parts, we have focussed on one specific technique, that is the well-known connection between branch point twist field correlators and measures of entanglement in 1+1D integrable quantum field theory. Our papers apply this technique for the first time to a non-diagonal theory with an involved particle spectrum, the sine-Gordon model. In this Part II we focus on a different entanglement measure, the symmetry resolved entanglement, and develop its associated twist field description, exploiting the underlying U(1) symmetry of the theory. In this context, conventional branch point twist fields are no longer the fields required, but instead we must work with one of their composite generalisations, which can be understood as the field resulting from the fusion of a standard branch point twist field and the sine-Gordon exponential field associated with U(1) symmetry. The resulting composite twist field has correlators which as usual admit a form factor expansion. In this paper we write the associated form factor equations and solve them for various examples in the breather sector by using the method of angular quantisation. We show that, in the attractive regime, this is the sector which provides the leading contribution to the symmetry resolved entropies, both R\'enyi and von Neumann. We compute the latter in the limit of a large region size and show that they satisfy the property of equipartition, that is the leading contribution to the symmetry resolved entanglement is independent of the symmetry sector.

Traveling Waves for the Generalized Sinh-Gordon Equation with Variable Coefficients

The sinh-Gordon equation is simply the classical wave equation with a nonlinear sinh source term. It arises in diverse scientific applications including differential geometry theory, integrable quantum field theory, fluid dynamics, kink dynamics, and statistical mechanics. It can be used to describe generic properties of string dynamics for strings and multi-strings in constant curvature space. In the present paper, we study a generalized sinh-Gordon equation with variable coefficients with the goal of obtaining analytical traveling wave solutions. Our results show that the traveling waves of the variable coefficient sinh-Gordon equation can be derived from the known solutions of the standard sinh-Gordon equation under a specific selection of a choice of the variable coefficients. These solutions include some real single and multi-solitons, periodic waves, breaking kink waves, singular waves, periodic singular waves, and compactons. These solutions might be valuable when scientists model some real-life phenomena using the sinh-Gordon equation where the balance between dispersion and nonlinearity is perturbed.

Dynamical relaxation of correlators in periodically driven integrable quantum systems

We show that the correlation functions of a class of periodically driven integrable closed quantum systems approach their steady-state value as ${n}^{\ensuremath{-}(\ensuremath{\alpha}+1)/\ensuremath{\beta}}$, where $n$ is the number of drive cycles and $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ denote positive integers. We find that, generically, $\ensuremath{\beta}=2$ within a dynamical phase characterized by a fixed $\ensuremath{\alpha}$; however, its value can change to $\ensuremath{\beta}=3$ or $\ensuremath{\beta}=4$ either at critical drive frequencies separating two dynamical phases or at special points within a phase. We show that such decays are realized in both driven Su-Schrieffer-Heeger (SSH) and one-dimensional (1D) transverse field Ising models, discuss the role of symmetries of the Floquet spectrum in determining $\ensuremath{\beta}$, and chart out the values of $\ensuremath{\alpha}$ and $\ensuremath{\beta}$ realized in these models. We analyze the SSH model for a continuous drive protocol using a Floquet perturbation theory which provides analytical insight into the behavior of the correlation functions in terms of its Floquet Hamiltonian. This is supplemented by an exact numerical study of a similar behavior for the 1D Ising model driven by a square pulse protocol. For both models, we find a crossover timescale ${n}_{c}$ which diverges at the transition. We also unravel a long-time oscillatory behavior of the correlators when the critical drive frequency, ${\ensuremath{\omega}}_{c}$, is approached from below $(\ensuremath{\omega}&lt;{\ensuremath{\omega}}_{c})$. We tie such behavior to the presence of multiple stationary points in the Floquet spectrum of these models and provide an analytic expression for the time period of these oscillations.

Dynamic Reference Point-Oriented Consensus Mechanism in Linguistic Distribution Group Decision Making Restricted by Quantum Integration of Information

We present a consensus improvement mechanism based on prospect theory and quantum probability theory (QPT) that enables the manifestation of irrational and uncertain behaviors of decision makers (DMs) in linguistic distribution group decision making. In this framework, the DMs pursue the possibility of working with different partial agreements on prospect values. Considering that the reference information should be comprehensive and accurate as it guides information modification and affects consensus efficiency, objective and subjective information is integrated to obtain the information. Several studies have verified that the interference effect will occur when the brain beliefs flow towards the different decision classification paths. To address this problem, QPT is introduced into the information integration and the optimized value of the interference term can be acquired by the designed multi-objective programming model based on the maximum individual utility. Finally, as the reference point changes during the preference adjustment process, a dynamic reference point-oriented consensus model is constructed to obtain the optimized modification. A case study is performed on the emergency plan for the selection of designated hospitals, and comparative analyses are performed to demonstrate the feasibility and advantages of the proposed model. Several important insights are offered to simulate the most likely possibility of consciousness flowing into different decision classifications for DMs and moderators.

Some Generalizations of Different Types of Quantum Integral Inequalities for Differentiable Convex Functions with Applications

In this paper, we prove a new quantum integral equality involving a parameter, left and right quantum derivatives. Then, we use the newly established equality and prove some new estimates of quantum Ostrowski, quantum midpoint, quantum trapezoidal and quantum Simpson’s type inequalities for q-differentiable convex functions. It is also shown that the newly established inequalities are the refinements of the existing inequalities inside the literature. Finally, some examples and applications are given to illustrate the investigated results.