Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes

Searches for possible new quantum phases and classifications of quantum phases have been central problems in physics. Yet, they are indeed challenging problems due to the computational difficulties in analyzing quantum many-body systems and the lack of a general framework for classifications. While frustration-free Hamiltonians, which appear as fixed point Hamiltonians of renormalization group transformations, may serve as representatives of quantum phases, it is still difficult to analyze and classify quantum phases of arbitrary frustration-free Hamiltonians exhaustively. Here, we address these problems by sharpening our considerations to a certain subclass of frustration-free Hamiltonians, called stabilizer Hamiltonians, which have been actively studied in quantum information science. We propose a model of frustration-free Hamiltonians which covers a large class of physically realistic stabilizer Hamiltonians, constrained to only three physical conditions; the locality of interaction terms, translation symmetries and scale symmetries, meaning that the number of ground states does not grow with the system size. We show that quantum phases arising in two-dimensional models can be classified exactly through certain quantum coding theoretical operators, called logical operators, by proving that two models with topologically distinct shapes of logical operators are always separated by quantum phase transitions.     

Finite-Dimensional PT-Symmetric Hamiltonians

This paper investigates finite-dimensional PT-symmetric Hamiltonians. It is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) The usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex PT-symmetric Hamiltonians. In the first approach the spectrum remains real, while in the second approach the spectrum remains real if the PT symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D>2, a D-dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional PT-symmetric matrix Hamiltonian.     

Non-linear supersymmetry for non-Hermitian,non-diagonalizable Hamiltonians: I. General properties

We study complex potentials and related non-diagonalizable Hamiltonians with special emphasis on formal definitions of associated functions and Jordan cells. The non-linear SUSY for complex potentials is considered and the theorems characterizing its structure are presented. We define the class of complex potentials invariant under SUSY transformations for (non-)diagonalizable Hamiltonians and formulate several results concerning the properties of associated functions. We comment on the applicability of these results for softly non-Hermitian PT-symmetric Hamiltonians. The role of SUSY (Darboux) transformations in increasing/decreasing of Jordan cells in SUSY partner Hamiltonians is thoroughly analyzed and summarized in the Index Theorem. The properties of non-diagonalizable Hamiltonians as well as the Index Theorem are illustrated in the solvable examples of non-Hermitian reflectionless Hamiltonians. The rigorous proofs are relegated to part II of this paper. At last, some peculiarities in resolution of identity for discrete and continuous spectra with a zero-energy bound state at threshold are discussed.     

Quaternionic Formulation of Supersymmetric Quantum Mechanics

Quaternionic formulation of supersymmetric quantum mechanics has been developed consistently in terms of Hamiltonians, super partner Hamiltonians, and supercharges for free particle and interacting field in one and three dimensions. Supercharges, super partner Hamiltonians and energy eigenvalues are discussed and it has been shown that the results are consistent with the results of quantum mechanics.     

New family of models for incompressible quantum liquids in d>/=2

Through Haldane's construction, the fractional quantum Hall states on a two-sphere were shown to be the ground states of one-dimensional SU(2) spin Hamiltonians. In this Letter we generalize this construction to obtain a new class of SU(N) spin Hamiltonians. These Hamiltonians describe center-of-mass-position conserving pair hopping fermions in space dimension d>/=2.     

A Class of Asymmetric Gapped Hamiltonians on Quantum Spin Chains and its Characterization II

We give a characterization of the class of gapped Hamiltonians introduced in Part I (Ogata, A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I, 2015). The Hamiltonians in this class are given as MPS (Matrix product state) Hamiltonians. In Ogata (A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I, 2015), we list up properties of ground state structures of Hamiltonians in this class. In this Part II, we show the converse. Namely, if a (not necessarily MPS) Hamiltonian H satisfies five of the listed properties, there is a Hamiltonian H′ from the class by Ogata (A class of asymmetric gapped Hamiltonians on quantum spin chains and its classification I, 2015), satisfying the following: The ground state spaces of the two Hamiltonians on the infinite interval coincide. The spectral projections onto the ground state space of H on each finite intervals are approximated by that of H′ exponentially well, with respect to the interval size. The latter property has an application to the classification problem with open boundary conditions.     

Nonperturbative gadget for topological quantum codes

Many-body entangled systems, in particular topologically ordered spin systems proposed as resources for quantum information processing tasks, often involve highly nonlocal interaction terms. While one may approximate such systems through two-body interactions perturbatively, these approaches have a number of drawbacks in practice. In this Letter, we propose a scheme to simulate many-body spin Hamiltonians with two-body Hamiltonians nonperturbatively. Unlike previous approaches, our Hamiltonians are not only exactly solvable with exact ground state degeneracy, but also support completely localized quasiparticle excitations, which are ideal for quantum information processing tasks. Our construction is limited to simulating the toric code and quantum double models, but generalizations to other nonlocal spin Hamiltonians may be possible.     

A family of affine quantum group invariant integrable extensions of the Hubbard Hamiltonian

We construct a family of spin chain Hamiltonians, which have the affine quantum group symmetry . Their eigenvalues coincide with the eigenvalues of the usual spin chain Hamiltonians, but have the degeneracy of levels, corresponding to the affine . The space of states of these spin chains is formed by the tensor product of the fully reducible representations of the quantum group.

The fermionic representations of the constructed spin chain Hamiltonians show that we have obtained new extensions of the Hubbard Hamiltonians. All of them are integrable and have the affine quantum group symmetry. The exact ground state of such type of model is presented, exhibiting superconducting behavior via the η-pairing mechanism.     

Efficient solvability of Hamiltonians and limits on the power of some quantum computational models

One way to specify a model of quantum computing is to give a set of control Hamiltonians acting on a quantum state space whose initial state and final measurement are specified in terms of the Hamiltonians. We formalize such models and show that they can be simulated classically in a time polynomial in the dimension of the Lie algebra generated by the Hamiltonians and logarithmic in the dimension of the state space. This leads to a definition of Lie-algebraic "generalized mean-field Hamiltonians." We show that they are efficiently (exactly) solvable. Our results generalize the known weakness of fermionic linear optics computation and give conditions on control needed to exploit the full power of quantum computing.     

On the concept of spectral singularities

In this paper, we discuss the concept of spectral singularities for non-Hermitian Hamiltonians. We exihibit spectral singularities of some well-known concrete Hamiltonians with complex-valued coefficients.     

Flux Hamiltonians, Lie algebras, and root lattices with minuscule decorations

We study a family of Hamiltonians of fermions hopping on a set of lattices in the presence of a background gauge field. The lattices are constructed by decorating the root lattices of various Lie algebras with their minuscule representations. The Hamiltonians are, in momentum space, themselves elements of the Lie algebras in these same representations. We describe various interesting aspects of the spectra, which exhibit a family resemblance to the Dirac spectrum, and in many cases are able to relate them to known facts about the relevant Lie algebras. Interestingly, various realizable lattices such as the kagomé and pyrochlore can be given this Lie algebraic interpretation, and the particular flux Hamiltonians arise as mean-field Hamiltonians for spin-1/2 Heisenberg models on these lattices.     

Exact solutions of graded Temperley-Lieb Hamiltonians

Orthosymplectic Hamiltonians derived from representations of the graded Temperley-Lieb algebra are presented and solved via the coordinate Bethe ansatz. The spectra of these Hamiltonians are obtained using open and closed boundary conditions.     

Construction of model hamiltonians in framework of Rayleigh-Schrödinger perturbation theory

A theory of the model Hamiltonians within the framework of Rayleigh-Schrödinger perturbation theory is elaborated. The approach of a model Hamiltonian is based on the assumption that if it is diagonalized in a chosen model space it will yield eigenvalues of the original Hamiltonian in the entire Hilbert space. The theory of the model Hamiltonians may be fruitful as a theoretical background for the study of effective Hamiltonians and as natural extension of the standard Rayleigh-Schrödinger perturbation theory.     

On the correspondence between fermion and boson states and operators

Mapping of shell-model (fermion) Hamiltonians onto boson Hamiltonians which underly the interaction boson model ^1–5) is investigated. A simple correspondence is defined and a sufficient condition given for shell-model Hamiltonians to simply correspond to finite hermitian boson Hamiltonians. A special case is discussed where diagonalization of a shell-model Hamiltonian for valence protons and neutrons can be exactly carried out in an equivalent (but different) boson space. If, however, the proton Hamiltonian and neutron Hamiltonian are diagonal in the seniority scheme, mapping of fermion states onto orthogonal boson states cannot be a simple correspondence. In that case the boson quadrupole operators equivalent to fermion guadrupole operators cannot be single-boson operators but must be more complicated, ones.     

Differential realizations of the two-mode bosonic and fermionic Hamiltonians: A unified approach

A method is developed to determine the eigenvalues and eigenfunction of two-boson 2 × 2 matrix Hamiltonians that include a wide class of quantum optical models. The quantum Hamiltonians are transformed in the form of the one variable differential equation and the conditions for their solvability are discussed. We present two different transformation procedures and we show our approach unify various approaches based on Lie algebraic technique. As an application, solutions of the modified Jaynes-Cummings and two-level Jahn-Teller Hamiltonians are studied.     

Exciton-polaritons in lattices: A non-linear photonic simulator

Microcavity polaritons are mixed light–matter quasiparticles with extraordinary nonlinear properties, which can be easily accessed in photoluminescence experiments. Thanks to the possibility of designing the potential landscape of polaritons, this system provides a versatile photonic platform to emulate 1D and 2D Hamiltonians. Polaritons allow transposing to the photonic world some of the properties of electrons in solid-state systems, and to engineer Hamiltonians for photons with novel transport properties. Here we review some experimental implementations of polariton Hamiltonians using lattice geometries.     

Energy-Level of Some Singular Harmonic Oscillators and Parametric Amplifiers with Singular Potential Derived via IEO Method

We employ the invariant eigen-operator （lEO） method to find the invariant eigen-operators of N-body singular oscillators＇ Hamiltonians and then derive their energy gaps. The Hamiltonians of parametric amplifiers with singular potential are also discussed in this way.     

Analysis of Adiabatic Approximation Using Stable Hamiltonian Method

In this paper, we deal with the adiabatic approximation of general Hamiltonians by splitting it into two parts, with one part a Hamiltonian that has at least one time-independent eigenstate up to a phase factor. We first develop the method of finding this kind of Hamiltonians. Then the relationship between adiabatic approximation and these Hamiltonians is discussed. Applying this to a general case, we give both a necessary condition and a sufficient condition for adiabatic approximation, followed by a spin-half example to illustrate.     

Exactly-solvable models derived from a generalized Gaudin algebra

We introduce a generalized Gaudin Lie algebra and a complete set of mutually commuting quantum invariants allowing the derivation of several families of exactly solvable Hamiltonians. Different Hamiltonians correspond to different representations of the generators of the algebra. The derived exactly-solvable generalized Gaudin models include the Hamiltonians of Bardeen–Cooper–Schrieffer, Suhl–Matthias–Walker, Lipkin–Meshkov–Glick, the generalized Dicke and atom–molecule, the nuclear interacting boson model, a new exactly-solvable Kondo-like impurity model, and many more that have not been exploited in the physics literature yet.     

Computational difficulty of computing the density of states

We study the computational difficulty of computing the ground state degeneracy and the density of states for local Hamiltonians. We show that the difficulty of both problems is exactly captured by a class which we call #BQP, which is the counting version of the quantum complexity class quantum Merlin Arthur. We show that #BQP is not harder than its classical counting counterpart #P, which in turn implies that computing the ground state degeneracy or the density of states for classical Hamiltonians is just as hard as it is for quantum Hamiltonians.