Continuous Unitary Transformations Research Articles

The exact solution of the Wegner flow equation with the Mielke generator for 3 × 3 hermitian matrices

The Wegner flow equation is a differential equation for a family of Hamiltonians and can be considered as a continuous unitary transformation. In essence, the transformation continuously decouples degrees of freedom and gradually converts the transformed Hamiltonian into the near-diagonal form. The Wegner flow equation has so far been applied to many areas of physics, but mainly in perturbative calculations. However, the exact solutions are known only for the simplest of two-dimensional case and this certainly limits the practical application of the method. On the other hand, the knowledge of exact solutions is essential for many physical problems as well as it can help in constructing improved approximated approaches. In this paper we present the exact solution of the Wegner flow equation with the Mielke generator for 3 × 3 Hermitian matrices, the general solutions for N × N tridiagonal Hermitian matrices and partially for 4 × 4 real symmetric matrices.

Order-by-disorder and long-range interactions in the antiferromagnetic transverse-field Ising model on the triangular lattice—A perturbative point of view

We study the low-field ground-state (GS) properties of the antiferromagnetic transverse-field Ising model with long-range interactions (afLRTFIM) on the triangular lattice. We use the method of perturbative continuous unitary transformations (pCUT) to derive an effective model for the degenerate GS space of the antiferromagnetic nearest-neighbour (NN) Ising model on a finite system, by treating the transverse-field (TF) and the long-range interactions (LRI) as a perturbation. We determine a level-crossing between the gapped plain stripe phase at small TF and the gapped clock-ordered phase at intermediate TF at hc≅0.129 for α=6, N=36 spins in order three perturbation theory on the finite system. We discuss the qualitative layout of the quantum phase diagram of the afLRTFIM on the triangular lattice.

Monte Carlo Based Techniques for Quantum Magnets with Long-Range Interactions.

Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum-optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows extracting high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo integration enables calculations on large finite systems. Finite-size scaling can then be used to determine the physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations.

Continuous unitary transformation approach to the Kondo–Majorana interplay

We analyze a setup composed of a correlated quantum dot (QD) coupled to one metallic lead and one end of topological chain hosting a Majorana zero mode (MZM). In such a hybrid structure, a leakage of the MZM into the region of the QD competes with the Kondo resonance appearing as a consequence of the spin-exchange interactions between the dot and the lead. In the work, we use the nontrivial technique called the continuous unitary transformation (CUT) to analyze this competition. Using the CUT technique, we inspect the influence of the coupling between the QD and the chain on effective exchange and density–density interactions.

Continuously varying critical exponents in long-range quantum spin ladders

We investigate the quantum-critical behavior between the rung-singlet phase with hidden string order and the Néel phase with broken SU(2)-symmetry in quantum spin ladders with algebraically decaying unfrustrated long-range Heisenberg interactions. To this end, we determine high-order series expansions of energies and observables in the thermodynamic limit about the isolated rung-dimer limit. This is achieved by extending the method of perturbative continuous unitary transformations (pCUT) to long-range Heisenberg interactions and to the calculation of generic observables. The quantum-critical breakdown of the rung-singlet phase then allows us to determine the critical phase transition line and the entire set of critical exponents as a function of the decay exponent of the long-range interaction. We demonstrate long-range mean-field behavior as well as a non-trivial regime of continuously varying critical exponents implying the absence of deconfined criticality contrary to a recent suggestion in the literature.

Momentum-resolved spin-conserving two-triplon bound state and continuum in a cuprate ladder

Studying multi-particle elementary excitations has provided unique access to understand collective many-body phenomena in correlated electronic materials, paving the way towards constructing microscopic models. In this work, we perform O K-edge resonant inelastic X-ray scattering (RIXS) on the quasi-one-dimensional cuprate {{{{{{rm{Sr}}}}}}}_{14}{{{{{{rm{Cu}}}}}}}_{24}{{{{{{rm{O}}}}}}}_{41} with weakly-doped spin ladders. The RIXS signal is dominated by a dispersing sharp mode ~ 270 meV on top of a damped incoherent component ~ 400-500 meV. Comparing with model calculations using the perturbative continuous unitary transformations method, the two components resemble the spin-conserving ΔS = 0 two-triplon bound state and continuum excitations in the spin ladders. Such multi-spin response with long-lived ΔS = 0 excitons is central to several exotic magnetic properties featuring Majorana fermions, yet remains unexplored given the generally weak cross-section with other experimental techniques. By investigating a simple spin-ladder model system, our study provides valuable insight into low-dimensional quantum magnetism.

Dynamic structure factor of the antiferromagnetic Kitaev model in large magnetic fields

We investigate the dynamic structure factor of the antiferromagnetic Kitaev honeycomb model in a magnetic field by applying perturbative continuous unitary transformations about the high-field limit. One- and two-quasiparticle properties of the dressed elementary spin-flip excitations of the high-field polarized phase are calculated which account for most of the spectral weight in the dynamic structure factor. We discuss the evolution of spectral features in these quasiparticle sectors in terms of one-quasiparticle dispersions, two-quasiparticle continua, the formation of antibound states, and quasiparticle decay. In particular, a comparably strong spectral feature above the upper edge of the upmost two-quasiparticle continuum represents three antibound states which form due to nearest-neighbor density-density interactions.

Two-triplon excitations of the Kitaev-Heisenberg bilayer

We study the spectrum of a bilayer of Kitaev magnets on the honeycomb lattice, coupled by Heisenberg exchange in the quantum-dimer phase at strong interlayer coupling. Using the perturbative Continuous Unitary transformation (pCUT) we perform series expansion starting from the fully dimerized limit, to evaluate the elementary excitations, reaching up to and focusing on the two-triplon sector. In stark contrast to conventional bilayer quantum magnets, and because of the broken $SU(2)$-invariance, encoded in the intralayer directional compass-exchange, the bilayer Kitaev magnet is shown to exhibit a rich structure of two-triplon scattering-state continua, as well as several collective two-triplon (anti)bound states. Direct physical pictures for the occurrence of the latter are provided and the (anti)bound states are studied versus the stacking type, the spin components, and the exchange parameters. In addition to the two-triplon spectra, we investigate a corresponding experimental probe by evaluating the magnetic Raman-scattering intensity. We find a very strong sensitivity of this intensity on the two-triplon interactions and the scattering geometry, however a signal from the (anti)bound states appears only in very close proximity to the continuum.

Quantum criticality and excitations of a long-range anisotropic XY chain in a transverse field

The critical breakdown of a one-dimensional quantum magnet with long-range interactions is studied by investigating the high-field polarized phase of the anisotropic XY model in a transverse field for the ferromagnetic and antiferromagnetic cases. While for the limiting case of the isotropic long-range XY model we can extract the elementary one-quasiparticle dispersion analytically and calculate two-quasiparticle excitation energies quantitatively in a numerical fashion, for the long-range Ising limit as well as in the intermediate regime we use perturbative continuous unitary transformations on white graphs in combination with classical Monte Carlo simulations for the graph embedding to extract high-order series expansions in the thermodynamic limit. This enables us to determine the quantum-critical breakdown of the high-field polarized phase by analyzing the gap closing including associated critical exponents and multiplicative logarithmic corrections. In addition, for the ferromagnetic isotropic XY model we determined the critical exponents $z$ and $\ensuremath{\nu}$ analytically by a bosonic quantum-field theory.

Generalized diagonalization scheme for many-particle systems

Despite the advances in the development of numerical methods analytical approaches still play the key role on the way towards a deeper understanding of many-particle systems. In this regards, diagonalization schemes for Hamiltonians represent an important direction in the field. Among these techniques the method, presented here, might be that approach with the widest range of possible applications: We demonstrate that both stepwise and continuous unitary transformations to diagonalize the many-particle Hamiltonian as well as perturbation theory and also non-perturbative treatments can be understood within the same theoretical framework. The new method is based on the introduction of generalized projection operators and allows to develop a renormalization scheme which is used to evaluate directly the physical quantities of a many-particle system. The applicability of this approach is shown for two important elementary many-particle problems.

Dynamic structure factor of Heisenberg bilayer dimer phases in the presence of quenched disorder and frustration

We investigate the influence of quenched disorder on the dynamic structure factor of Heisenberg bilayers on the square, triangular, and kagome lattice in the quantum paramagnetic phase. Perturbative continuous unitary transformations and white graphs are employed to calculate the one-triplon contribution up to high orders in perturbation about the dimer limit for bimodal and continuous disorder. For the square lattice we find that the lifetime of the gap mode is increased by stronger quantum correlations while stronger disorder effects are observed for the triangular lattice due to geometric frustration. For intra-dimer disorder, in-band energy gaps are observed for both lattices which can be understood in terms of a level repulsion on dimers with low and high intra-dimer exchange that are close in energy at the momentum where the in-band gap opens. For the highly frustrated kagome lattice disorder even allows to decrease the gap energy. In addition, the localization length of the low-energy flat band is increased up to order $7$ in perturbation theory. The interplay of quenched disorder, geometric frustration, and strong correlations leads therefore to rich structures in the dynamical structure factor of two-dimensional quantum magnets.

Fine Grained Tensor Network Methods.

We develop a strategy for tensor network algorithms that allows to deal very efficiently with lattices of high connectivity. The basic idea is to fine grain the physical degrees of freedom, i.e., decompose them into more fundamental units which, after a suitable coarse graining, provide the original ones. Thanks to this procedure, the original lattice with high connectivity is transformed by an isometry into a simpler structure, which is easier to simulate via usual tensor network methods. In particular this enables the use of standard schemes to contract infinite 2D tensor networks-such as corner transfer matrix renormalization schemes-which are more involved on complex lattice structures. We prove the validity of our approach by numerically computing the ground-state properties of the ferromagnetic spin-1 transverse-field Ising model on the 2D triangular and 3D stacked triangular lattice, as well as of the hardcore and softcore Bose-Hubbard models on the triangular lattice. Our results are benchmarked against those obtained with other techniques, such as perturbative continuous unitary transformations and graph projected entangled pair states, showing excellent agreement and also improved performance in several regimes.

Multicarrier continuous-variable quantum key distribution

The multicarrier continuous-variable quantum key distribution (CVQKD) protocol is defined. In a CVQKD protocol, the information is conveyed by coherent quantum states. The quantum continuous variables are sent through a noisy quantum channel. For a quantum channel with additive-multiplicative noise both additive and multiplicative disturbances are present in the transmission. The multiplicative disturbance is an inherent attribute of diverse physical environments. Physical links with additive and multiplicative disturbances can represent a more general approach than purely additive noise links in several practical scenarios. In a standard CVQKD setting, the noise is modeled as an additive white Gaussian noise caused by an eavesdropper (Gaussian quantum link). As a corollary, standard CVQKD protocols are not optimal for arbitrary Gaussian quantum channels if multiplicative disturbances are also present in the physical link. Here, we define the adaptive multicarrier quadrature division (AMQD) modulation technique for CVQKD. The AMQD method is optimal for arbitrary Gaussian quantum channels with arbitrary multiplicative disturbances. The protocol granulates the Gaussian random input into Gaussian subcarrier continuous variables in the encoding phase, which are then decoded by a continuous unitary transformation. The subcarrier coherent variables formulate sub-channels from the physical link which leads to improved transmission efficiency, higher tolerable loss, and excess noise in comparison to standard CVQKD protocols. We also derive the security proof of multicarrier CVQKD at optimal Gaussian attacks in the finite-size and asymptotic regimes.

Quantum Criticality of Two-Dimensional Quantum Magnets with Long-Range Interactions.

We study the critical breakdown of two-dimensional quantum magnets in the presence of algebraically decaying long-range interactions by investigating the transverse-field Ising model on the square and triangular lattice. This is achieved technically by combining perturbative continuous unitary transformations with classical MonteCarlo simulations to extract high-order series for the one-particle excitations in the high-field quantum paramagnet. We find that the unfrustrated systems change from mean-field to nearest-neighbor universality with continuously varying critical exponents. In the frustrated case on the square lattice the system remains in the universality class of the nearest-neighbor model independent of the long-range nature of the interaction, while we argue that the quantum criticality for the triangular lattice is terminated by a first-order phase transition line.

Classical mean-field, spin-wave, and perturbative continuous unitary transformation analyses of the squaric acid system H2SQ

We study an organic Hydrogen bonded material $\rm{H_2SQ}$ analytically and map out the phase diagram as well as low energy excitations in the relevant parameter space. At zeroth order the dynamics is governed by plaquette interaction (product of $\sigma_z$ over a plaquette) which defines a $Z_2$ gauge theory and a deconfinement phase satisfying "ice rules". The system is studied under additional interactions such as an external Zeeman field (with strength $K$) in $x$-direction, a inter-molecular interaction (with strength $J_1$) and a dipole-dipole interaction with strength $J_2$ such that $K>J_1>J_2$. The effect of dipole-dipole interaction removes the local $Z_2$ symmetry and gives rise to four global degenerate states with Ferroeletric order. Using meanfield analysis we chart out the phase diagram for classical version of the model and find $K_c$ which defines the transition from disordered phase to ordered phase in $J_1, K$ plane for various values of $J_2$. We find that presence of $J_1$ and $J_2$ tends to stabilize the deconfined phases. Over the classical ground states we perform spin-wave analysis and surprisingly find that quantum fluctuations does not remove the classical degeneracy at all at quadratic level. The spin-wave spectrum is found for the four global degenerate ground states which shows both gapped spectrum and gapless spectrum (near the vicinity of CDT transition) for $J_2=0$. However for $J_2$ finite, the spectrum is always gapped. We perform PCUT analysis to improve the results of spin-wave analysis and calculate ground state energy and one particle dispersion and gap at high symmetry point. Using this we draw the phase boundary between confined and deconfined phase in the $K-J_1$ plane. The effect of $J_2$ is also discussed in the resulting phase boundary.

Nuclear Structure from the In-Medium Similarity Renormalization Group

Efforts to describe nuclear structure and dynamics from first principles have advanced significantly in recent years. Exact methods for light nuclei are now able to include continuum degrees of freedom and treat structure and reactions on the same footing, and multiple approximate, computationally efficient many-body methods have been developed that can be routinely applied for medium-mass nuclei. This has made it possible to confront modern nuclear interactions from Chiral Effective Field Theory, that are rooted in Quantum Chromodynamics with a wealth of experimental data.Here, we discuss one of these efficient new many-body methods, the In-Medium Similarity Renormalization Group (IMSRG), and its applications in modern nuclear structure theory. The IMSRG evolves the nuclear many-body Hamiltonian in second-quantized form through continuous unitary transformations that can be implemented with polynomial computational effort. Through suitably chosen generators, we drive the matrix representation of the Hamiltonian in configuration space to specific shapes, e.g., to implement a decoupling of low-and high-energy scales, or to extract energy eigenvalues for a given nucleus.We present selected results from Multireference IMSRG (MR-IMSRG) calculations of open-shell nuclei, as well as proof-of-principle applications for intrinsically deformed medium-mass nuclei. We discuss the successes and prospects of merging the (MR-)IMSRG with many-body methods ranging from Configuration Interaction to the Density Matrix Renormalization Group, with the goal of achieving an efficient simultaneous description of dynamic and static correlationsin atomic nuclei.

Qubit transport model for unitary black hole evaporation without firewalls

We give an explicit toy qubit transport model for transferring information from the gravitational field of a black hole to the Hawking radiation by a continuous unitary transformation of the outgoing radiation and the black hole gravitational field. The model has no firewalls or other drama at the event horizon, and it avoids a counterargument that has been raised for subsystem transfer models as resolutions of the firewall paradox. Furthermore, it fits the set of six physical constraints that Giddings has proposed for models of black hole evaporation. It does utilize nonlocal qubits for the gravitational field but assumes that the radiation interacts locally with these nonlocal qubits, so in some sense the nonlocality is confined to the gravitational sector. Although the qubit model is too crude to be quantitatively correct for the detailed spectrum of Hawking radiation, it fits qualitatively with what is expected.

Quantum phase transitions to topological Haldane phases in spin-one chains studied by linked-cluster expansions

We use linked-cluster expansions to analyze the quantum phase transitions between symmetry unbroken trivial and topological Haldane phases in two different spin-one chains. The first model is the spin-one Heisenberg chain in the presence of a single-ion anisotropy while the second one is the dimerized spin-one Heisenberg chain. For both models we determine the ground-state energy and the one-particle gap inside the non-topological phase as a high-order series using perturbative continuous unitary transformations. Extrapolations of the gap series are applied to locate the quantum critical point and to extract the associated critical exponent. We find that this approach works unsatisfactory for the anisotropic chain, since the quality of the extrapolation appears insufficient due to the large correlation length exponent. In contrast, extrapolation schemes display very good convergence for the gap closing in the case of the dimerized spin-one Heisenberg chain.

Stable unitary integrators for the numerical implementation of continuous unitary transformations

The technique of continuous unitary transformations has recently been used to provide physical insight into a diverse array of quantum mechanical systems. However, the question of how to best numerically implement the flow equations has received little attention. The most immediately apparent approach, using standard Runge-Kutta numerical integration algorithms, suffers from both severe inefficiency due to stiffness and the loss of unitarity. After reviewing the formalism of continuous unitary transformations and Wegner's original choice for the infinitesimal generator of the flow, we present a number of approaches to resolving these issues including a choice of generator which induces what we call the "uniform tangent decay flow" and three numerical integrators specifically designed to perform continuous unitary transformations efficiently while preserving the unitarity of flow. We conclude by applying one of the flow algorithms to a simple calculation that visually demonstrates the many-body localization transition.

Topological phase transition in the field induced Kitaev model on the Kagome lattice

We investigate the low-energy spectral properties of the Kitaev model on the Kagome lattice, which is a quantum spin model for the aim of fault-tolerant quantum computation, in the presence of a uniform magnetic field in the x- and z-directions. We explore the low-energy physics of the Kitaev model in the x- and z-magnetic fields, separately and establish a quasi-particle picture for anyonic excitations. Our study is based on the high-order series expansion of the low- and high field limits of the problem by means of perturbative continuous unitary transformations. We further show that the Kitaev model in the x-field is mapped to the Ising transverse field (ITF) model on the triangular lattice while, the system is mapped to another ITF model on the honeycomb lattice in the presence of the z-magnetic field. Additionally, we investigate the phase transitions of the model for the two cases and find that the topological phase of the Kitaev model breaks down to the polarized phase in either x- or z-directions by a second-order quantum phase transition in the 3D Ising universality class. We further detect dispersive bound states in high-field limits of the model for both cases of the magnetic field. The overall results further indicate that the Kitaev model on the Kagome lattice has the different stability as the toric code on the square lattice, while perturbed by magnetic fields.