Solvable Quantum Research Articles

Multi-Unitary Complex Hadamard Matrices

We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of [Formula: see text] subsystems with [Formula: see text] levels each to the set of complex Hadamard matrices of order [Formula: see text]. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ([Formula: see text] or [Formula: see text]), two-unitary ([Formula: see text] and [Formula: see text] are unitary), or [Formula: see text]-unitary. Here [Formula: see text] denotes reshuffling of a matrix [Formula: see text] describing a bipartite system, and [Formula: see text] its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in [Formula: see text] dimensions.

Analytically solvable many-body Rosen-Zener quantum battery

Analytically solvable many-body Rosen-Zener quantum battery

Summation formulas generated by Hilbert space eigenproblem

We demonstrate that certain classes of Schlömilch-like infinite series and series that include generalized hypergeometric functions can be calculated in closed form starting from a simple quantum model of a particle trapped inside an infinite potential well and using principles of quantum mechanics. We provide a general framework based on the Hilbert space eigenproblem that can be applied to different exactly solvable quantum models. Obtaining series from normalization conditions in well-defined quantum problems secures their convergence.

Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions

A family of exactly solvable quantum square wells with discrete coordinates and with certain non-stationary Hermiticity-violating Robin boundary conditions is proposed and studied. Manifest non-Hermiticity of the model in conventional Hilbert space Hfriendly is required to coexist with the unitarity of system in another, ad hoc Hilbert space Hphysical . Thus, quantum mechanics in its non-Hermitian interaction picture (NIP) representation is to be used. We must construct the time-dependent states (say, ψ(t)) as well as the time-dependent observables (say, Λ(t)). Their evolution in time is generated by the operators denoted, here, by the respective symbols G(t) (a Schrödinger-equation generator) and Σ(t) (a Heisenberg-equation generator, a.k.a. quantum Coriolis force). The unitarity of evolution in Hphysical is then guaranteed by the reality of spectrum of the energy observable alias Hamiltonian H(t) = G(t) + Σ(t). The applicability of these ideas is illustrated via an N by N matrix model. At N = 2, closed formulae are presented not only for the measurable instantaneous energy spectrum but also for all of the eligible time-dependent physical inner-product metrics Θ(N=2)(t), for the related Dyson maps Ω(N=2)(t), for the Coriolis force Σ(N=2)(t) as well as, in the very ultimate step of the construction, for the truly nontrivial Schrödinger-equation generator G (N=2)(t).

Kink solutions in generalized 2D dilaton gravity

We study static kink solutions in a generalized two-dimensional dilaton gravity model, where the kinetic term of the dilaton is generalized to be an arbitrary function of the canonical one X=−12(∇φ)2, say F(X), and the kink is generated by a canonical scalar matter field ϕ. It is found that for arbitrary F(X), the background field equations have a simple first-order formalism, and the linear perturbation equation can always be written as a Schrödinger-like equation with factorizable Hamiltonian operator. After choosing appropriate F(X) and superpotential, we obtain a sine-Gordon type kink solution with pure AdS2 metric. The linear perturbation issue of this solution becomes an exactly solvable conformal quantum mechanics problem, if one of the model parameter takes a critical value.

The Dunkl oscillator on a space of nonconstant curvature: An exactly solvable quantum model with reflections

We introduce the Dunkl-Darboux III oscillator Hamiltonian in N dimensions as a λ−deformation of the Dunkl oscillator. This deformation is interpreted as the introduction of a non-constant curvature on the underlying space or, equivalently, as a quadratic position-dependent mass for the Dunkl oscillator. This new ND quantum model is shown to be exactly solvable, and its eigenvalues and eigenfunctions are explicitly presented. It is shown that in the 2D case both Darboux III and Dunkl oscillators can be coupled with a constant magnetic field, thus giving rise to two new exactly solvable quantum systems in which the effect of the λ−deformation and of the Dunkl derivatives on the Landau levels can be studied. Finally, the full 2D Dunkl-Darboux III oscillator is coupled with the magnetic field and shown to define an exactly solvable Hamiltonian, where the interplay between the λ−deformation and the magnetic field is explicitly illustrated.

A solvable quantum field theory with asymptotic freedom in (3+1) dimensions

Recently, Ai, Bender and Sarkar gave a prescription on how to obtain [Formula: see text]-symmetric field theory results from an analytic continuation of Hermitian field theories. We perform this analytic continuation for the massless (critical) [Formula: see text] model with quartic interaction in (3+1) dimensions. In the large-[Formula: see text] limit, this theory is exactly solvable, and has negative [Formula: see text]-function in the ultraviolet, and a stable bound state in the infrared. The coupling diverges at a scale [Formula: see text], but can be continued into the far infrared. At finite temperature, the theory exhibits two phases separated by a second-order phase transition near [Formula: see text].

Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models

In quantum mechanics of unitary systems using non-Hermitian (or, more precisely, Θ-quasi-Hermitian) Hamiltonians H such that , the exactly solvable M-level bound-state models with arbitrary are rare. A new class of such models is proposed here, therefore. Its exact algebraic solvability (involving not only the closed formulae for wave functions but also the explicit description of all of the eligible metrics Θ) was achieved due to an extremely sparse (viz., just -parametric) but still nontrivial ‘zig-zag-matrix’ choice of the form of H.

Exactly solvable piecewise analytic double well potential VD(x) = min[(x + d)2, (x − d)2] and its dual single well potential VS(x) = max[(x + d)2, (x − d)2

By putting two harmonic oscillator potentials x2 side by side with a separation 2d, two exactly solvable piecewise analytic quantum systems with a free parameter d > 0 are obtained. Due to the mirror symmetry, their eigenvalues {E} for the even and odd parity sectors are determined exactly as the zeros of certain combinations of the confluent hypergeometric function F11 of d and E, which are common to VD and VS but in two different branches. The eigenfunctions are the piecewise square integrable combinations of F11, the so-called U functions. By comparing the eigenvalues and eigenfunctions for various values of the separation d, vivid pictures unfold showing the tunneling effects between the two wells.

Dual unitary circuits in random geometries

Recently introduced dual unitary brickwork circuits have been recognised as paradigmatic exactly solvable quantum chaotic many-body systems with tunable degree of ergodicity and mixing. Here we show that regularity of the circuit lattice is not crucial for exact solvability. We consider a circuit where random 2-qubit dual unitary gates sit at intersections of random arrangements of straight lines in two dimensions (mikado) and analytically compute the variance of the spatio-temporal correlation function of local operators. Note that the average correlator vanishes due to local Haar randomness of the gates. The result can be physically motivated for two random mikado settings. The first corresponds to the thermal state of free particles carrying internal qubit degrees of freedom which experience interaction at kinematic crossings, while the second represents rotationally symmetric (random euclidean) space-time.

Fermionic matrix models and bosonization

We explore different limits of exactly solvable vector and matrix fermionic quantum mechanical models with quartic interactions at finite temperature. The models preserve a $U(1)\times SU(N)\times SU(L)$ symmetry at the classical level and we analyze them through bosonization techniques introducing scalar (singlet) and matrix (non-singlet) bosonic fields. The bosonic path integral representations in the vector limits $(N,1)$ and $(1,L)$ are matched to fermionic Fock space Hamiltonians expressed in terms of quadratic Casimirs and some additional terms involving the Cartan subalgebra, which makes explicit the symmetries preserved by scalar and matrix bosonizations at the quantum level. For the case of non-singlet bosonization we find an equivalence between the vector model and the Polychronakos+Frahm spin model. Using this relation we compute the free energy. Finally, we compute the eigenvalue distribution in the large $N,L$-limit with $ \alpha = \frac{L}{N}$ fixed. The model displays a third order phase transition as we vary the temperature which, in the $\alpha\gg1$ limit, can be characterized analytically. We conclude finding the critical curve in the parameter space were the eigenvalue distribution transitions from single to double cut.

No-go rules for multitime Landau–Zener models

Multitime Landau–Zener (MTLZ) model is a class of exactly solvable quantum many-body models which is multistate and multitime generalization of the two-state Landau–Zener model. Currently discovered MTLZ models include the “hypercubes”, the “fans” and their direct product models. In this work, we prove two no-go rules, named the “no \(K_{3,3}\)” rule and the “no 1221” rule, which forbid the existence of exact solutions for models with certain structures of interactions. We further apply these rules to show that for models with no more than 9 states, besides the models mentioned above there are no other MTLZ models. We also propose a scheme to systematically classify cases that could possibly host MTLZ models. Our work could serve as a guideline to search for new exactly solvable models within the MTLZ class.

Phase diagrams of Kitaev models for arbitrary magnetic field orientations

The Kitaev model is an exactly solvable quantum spin model within the language of constrained real fermions. In spite of numerous studies for magnetic fields along special orientations, there is a limited amount of knowledge on the complete field-angle characterization, which can provide valuable information on the existence of fractionalized excitations. For this purpose, we first study the pure ferromagnetic and antiferromagnetic Kitaev models for arbitrary external magnetic field directions via a mean-field theory, showing that there are many topological phases with different (or zero) Chern numbers, depending on the magnetic field strength and orientations. However, a realistic description of the candidate Kitaev materials, within the edge-sharing octahedra paradigm, requires additional coupling terms, including a large off-diagonal term $\mathrm{\ensuremath{\Gamma}}$ along with possible anisotropic corrections ${\mathrm{\ensuremath{\Gamma}}}_{p}$. It is therefore not sufficient to rely on the topological properties of the bare Kitaev model as the basis for the observed thermal Hall-conductivity signals, and an understanding of these extended Kitaev models with a complete field response is demanded. Starting from the zero-field phase diagram of $\mathrm{K}\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}_{p}$ models, we identify, besides the Kitaev spin liquid phase, antiferromagnetic zigzag, ferromagnetic phases, as well as an unusual Kitaev(-$\mathrm{\ensuremath{\Gamma}}$) spin liquid phase. The magnetic field response of these phases for arbitrary field orientations provides a remarkably rich phase diagram. For an extended parameter range and just above the critical field where the zigzag phase is suppressed, there is an intermediate phase region with suppressed energy gaps and substantial spin fractionalization. To comply our findings with experiments, we also reproduce a large asymmetry in the extent of this intermediate phases specifically for the two different field directions $\ensuremath{\theta}=\ifmmode\pm\else\textpm\fi{}{60}^{o}$ with respect to the normal to the plane of the honeycomb lattice.

Exactly solvable one-dimensional quantum models with gamma matrices.

In this paper we write exactly solvable generalizations of one-dimensional quantum XY and Ising-like models by using 2^{d}-dimensional gamma matrices as the degrees of freedom on each site. We show that these models result in quadratic Fermionic Hamiltonians with Jordan-Wigner-like transformations. We illustrate the techniques using a specific case of four-dimensional gamma matrices and explore the quantum phase transitions present in the model.

Infraparticle quantum fields and the formation of photon clouds

A non-perturbative and exactly solvable quantum field theoretical model for a “dressed Dirac field” is presented, that exhibits all the kinematical features of QED: an appropriate delocalization of the charged field as a prerequisite for the global Gauss Law, superselected photon clouds (asymptotic expectation values of the Maxwell field), infraparticle nature of charged particles that cannot be separated from their photon clouds, broken Lorentz symmetry. The model serves as an intermediate leg on a new roadmap towards full QED, formulated as an off-shell theory, i.e., including a perturbative construction of its interacting charged fields. It also fills a gap in recent discussions of the “Infrared Triangle”, and points the way towards a new scattering theory for theories with massless particles of helicity ≥ 1, in which infraparticles can be described at the level of charged fields, rather than just states.

Using quantum mechanics for calculation of different infinite sums

We demonstrate that a certain class of infinite sums can be calculated analytically starting from a specific quantum mechanical problem and using principles of quantum mechanics. For simplicity we illustrate the method by exploring the problem of a particle in a box. Twofold calculation of the mean value of energy for the polynomial wave function inside the well yields even argument p (p > 2) of Riemann zeta and related functions. This method can be applied to a wide class of exactly solvable quantum mechanical problems which may lead to different infinite sums. Besides, the analysis performed here provides deeper understanding of superposition principle and presents useful exercise for physics students.

Geometric thermodynamic uncertainty relation in a periodically driven thermoelectric heat engine

Thermodynamic uncertainty relation, quantifying a trade-off among average current, the associated fluctuation (precision), and entropy production (cost), has been formulated in nonequilibrium steady state and various stochastic systems. Herein, we study the thermodynamic uncertainty relation in generic thermoelectric heat engines under a periodic control protocol, by uncovering the underlying Berry-phase-like contribution. We show that our thermodynamic uncertainty relation breaks the seminal steady-state results, originating from the non-vanishing geometric effect. Furthermore, by deriving the consequent trade-off relation binding efficiency, power, and constancy, we prove that the periodically driven thermoelectric heat engines can generally outperform the steady-state analogies. The general bounds are illustrated by an analytically solvable two-terminal single quantum dot heat engine under the periodic modulation. Our work provides a geometric framework in bounding and optimizing a wide range of periodically driven thermoelectric thermal machines.

Spectrum of the tight-binding model on Cayley trees and comparison with Bethe lattices

There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the spectrum of the tight-binding model (equivalently, the spectrum of the adjacency matrix) on Cayley trees. Recent approaches to the problem have relied on the similarity between the Cayley tree and the Bethe lattice. Here, we avoid to make any ansatz related to the Bethe lattice due to fundamental differences between the two lattices that persist even when taking the thermodynamic limit. Instead, we show that one can use a recursive procedure that starts from the boundary and then use the canonical basis to derive the complete spectrum of the tight-binding model on Cayley trees. Our resulting algorithm is extremely efficient, as witnessed with remarkable large trees having hundreds of shells. We also show that, in the thermodynamic limit, the density of states is dramatically different from that of the Bethe lattice.

Analytical results for the unusual Grüneisen ratio in the quantum Ising model with Dzyaloshinskii-Moriya interaction

The Gr\"uneisen ratio (GR) has emerged as a superb tool for the diagnosis of quantum phase transitions, which diverges algebraically upon approaching critical points of continuous phase transitions. However, this paradigm has been challenged recently by observations of a finite GR for self-dual criticality and divergent GR at symmetry-enhanced first-order transitions. To unveil the fascinating GR further, we exemplify the idea by studying an exactly solvable quantum Ising model with Dzyaloshinskii-Moriya interaction, which harbors a ferromagnetic phase, a paramagnetic phase, and a chiral Luttinger liquid. Although the self-dual criticality of the ferromagnetic--paramagnetic transition is undermined by the Dzyaloshinskii-Moriya interaction, we find that the GR at the transition is still finite albeit with an increasing value, signifying a proximate self-dual relation. By contrast, the GR at the transition between the gapped ferromagnetic phase and the gapless Luttinger liquid diverges and changes its sign when crossing the first-order transition. This implies that the GR could also probe the first-order transition between the gapped and gapless phases.

Intrinsic suppression of the topological thermal Hall effect in an exactly solvable quantum magnet

In contrast to electron (fermion) systems, topological phases of charge neutral bosons have been poorly understood despite recent extensive research on insulating magnets. The most important unresolved issue is how the inevitable inter-bosonic interactions influence the topological properties. It has been proposed that the quantum magnet SrCu$_2$(BO$_3$)$_2$ with an exact ground state serves as an ideal platform for this investigation, as the system is expected to be a magnetic analogue of a Chern insulator with electrons replaced by bosonic magnetic excitations (triplons). Here, in order to examine topologically protected triplon chiral edge modes in SrCu$_2$(BO$_3$)$_2$, we measured and calculated the thermal Hall conductivity $\kappa_{xy}$. Our calculations show that the sign of $\kappa_{xy}$ is negative, which is opposite to the previous calculations, and its magnitude is 2$\pi$ times smaller. No discernible $\kappa_{xy}$ was observed, which is at most 20-30% of our calculations if present. This implies that even relatively weak inter-particle interactions seriously influence the topological transport properties at finite temperatures. These demonstrate that, in contrast to fermionic cases, the picture of non-interacting topological quasi-particles cannot be naively applied to bosonic systems, calling special attention to the interpretation of the topological bosonic excitations reported for various insulating magnets.