Continuous unitary transformations, or flow equations, provide a powerful non-perturbative framework for diagonalizing quantum many-body Hamiltonians. This work presents a first-principles proof that the complex, high-dimensional dynamics generated by a Wegner-type flow equation can, under specific and well-defined conditions, rigorously collapse to an exactly integrable second-order ordinary differential equation. We demonstrate that for a one-dimensional spinless fermion model in the non-interacting limit (Δ₀=0), the flow equations for the coupling constants can be derived exactly, revealing a hidden mathematical structure. We term this phenomenon "Emergent Integrability," where the diagonalization flow itself possesses an integrable submanifold, even if the initial Hamiltonian is not integrable. This discovery, established through rigorous and unabridged derivations, reveals that the process of diagonalization can be fundamentally simpler than the initial problem, shedding new light on the intrinsic mathematical structure of renormalization group-like transformations.
Continuous unitary transformations, or flow equations, provide a powerful non-perturbative framework for diagonalizing quantum many-body Hamiltonians. This work presents a first-principles proof that the complex, high-dimensional dynamics generated by a Wegner-type flow equation can, under specific and well-defined conditions, rigorously collapse to an exactly integrable second-order ordinary differential equation. We demonstrate that for a one-dimensional spinless fermion model in the non-interacting limit (Δ₀=0), the flow equations for the coupling constants can be derived exactly, revealing a hidden mathematical structure. We term this phenomenon "Emergent Integrability," where the diagonalization flow itself possesses an integrable submanifold, even if the initial Hamiltonian is not integrable. This discovery, established through rigorous and unabridged derivations, reveals that the process of diagonalization can be fundamentally simpler than the initial problem, shedding new light on the intrinsic mathematical structure of renormalization group-like transformations.
We use a novel parametrization of the flowing Hamiltonian to show that the flow equations based on continuous unitary transformations, as proposed by Wegner, can be implemented through a nonlinear partial differential equation involving one flow parameter and two system specific auxiliary variables. The implementation is non-perturbative as the partial differential equation involves a systematic expansion in fluctuations, controlled by the size of the system, rather than the coupling constant. The method is applied to the Lipkin model to construct a mapping which maps the non-interacting spectrum onto the interacting spectrum to a very high accuracy. This function is universal in the sense that the full spectrum for any (large) number of particles can be obtained from it. In a similar way expectation values for a large class of operators can be obtained, which also makes it possible to probe the structure of the eigenstates.
Abstract The iterative methods to diagonalize matrices and many-body Hamiltonians can be reformulated as flows of Hamiltonians towards diagonalization driven by unitary transformations that preserve the spectrum. After a comparative overview of the various types of discrete flows (Jacobi, QR-algorithm) and differential flows (Toda, Wegner, White) that have been introduced in the past, we focus on the random XXZ chain with random fields in order to determine the best closed flow within a given subspace of running Hamiltonians. For the special case of the free-fermion random XX chain with random fields, the flow coincides with the Toda differential flow for tridiagonal matrices which is related to the classical integrable Toda chain and which can be seen as the continuous analog of the discrete QR-algorithm. For the random XXZ chain with random fields that displays a many-body-localization transition, the present differential flow should be an interesting alternative to compare with the discrete flow that has been proposed recently to study the many-body-localization properties in a model of interacting fermions (Rademaker and Ortuno 2016 Phys. Rev. Lett. 116 , 010404 ).
Continuous unitary transformations are a powerful tool to extract valuable information out of quantum many-body Hamiltonians, in which the so-called flow equation transforms the Hamiltonian to a diagonal or block-diagonal form in second quantization. Yet, one of their main challenges is how to approximate the infinitely-many coupled differential equations that are produced throughout this flow. Here we show that tensor networks offer a natural and non-perturbative truncation scheme in terms of entanglement. The corresponding scheme is called “entanglement-CUT” or eCUT. It can be used to extract the low-energy physics of quantum many-body Hamiltonians, including quasiparticle energy gaps. We provide the general idea behind eCUT and explain its implementation for finite 1d systems using the formalism of matrix product operators. We also present proof-of-principle results for the spin-(1/2) 1d quantum Ising model and the 3-state quantum Potts model in a transverse field. Entanglement-CUTs can also be generalized to higher dimensions and to the thermodynamic limit.
We show how Wegner's flow equations can be reformulated as ordinary differential equations through the use of the Moyal bracket. In finite-dimensional Hilbert spaces the introduction of the Moyal bracket leads naturally to the identification of a small expansion parameter, namely the inverse of the dimensionality of the space. This expansion corresponds to a non-perturbative treatment of the coupling constant. In the case of infinite-dimensional spaces ℏ plays the role of the small parameter and the Moyal formulation then allows for a semi-classical treatment of the flow equation. We demonstrate these statements for the Lipkin and Dicke models as well as for the symmetric x 4 and double-well potentials.
The emergence of complex behavior out of simple elementary building blocks is a typical property of condensed matter systems. While the individual constituents may be well-studied, strongly interacting many-particle systems can give rise to novel collective behavior that can not be understood by considering the individual properties alone. Effective models are a powerful tool to handle this complexity: Instead of aiming at a full description a complex many-particle system on the high-energy scale, an effective model covers only those degrees of freedom which are essential for the low-energy physics. While the high-energy degrees of freedom are integrated out, they influence the low-energy degrees of freedom due to a renormalization of the effective coupling constants. In particular, the concept of the quasi-particle allows for the description of a strongly-interacting many-particle excitations in terms of a simple objects with similar degrees of freedom as a dispersive elementary particle, which gives rise to phenomena like quasi-particle bound states or quasi-particle decay. This thesis aims at the application and methodological improvement of a systematic and well-controlled framework to derive effective low-energy models, denoted as Continuous Unitary Transformations (CUT). In this method, a Hamiltonian is mapped continuously to an effective basis, in which the low-energy degrees of freedom can be interpreted as quasi-particles. Mathematically, the transformation is parametrized by a flow parameter and is represented as a differential equation, denoted as flow equation, while the direction of the transformation is controlled by the generator scheme. As foundation of our studies, we use the sCUT method, in which the types of possible interaction terms are fixed by a truncation scheme. In a common project, we established a new flavor of CUT, called epCUT. It allows us to derive the effective Hamiltonian in a perturbation series. In contrast to the already existent pCUT method, epCUT can handle system with a non-equidistant spectrum as well. Moreover, we discovered the deepCUT method, the non-perturbative twin of epCUT. While it bears large similarities to sCUT, it replaces the cumbersome definition of an appropriate truncation scheme by the order of the flow equation system. Compared to sCUT, deepCUT offers a larger robustness while the ambiguities of the truncation scheme are eliminated. My contribution comprises the computational, parallelized construction of the flow equation system as well. In another project, we investigated the breakdown of the conventional, particle-sorting generator schemes in system with strongly overlapping quasi-particle sub-spaces and identified the emergence of a spectrum unbounded from below due to truncation errors as origin of the divergence. As a testing ground, we used a system of two coupled bosonic degrees of freedom. To overcome these problems, we developed a family of three novel variational generator schemes, denoted as scalar, vectorial and tensorial optimization. After this, we investigated the dimerized, two-dimensional S=1/2 Heisenberg model with an emphasis on the quantum phase transition between a magnetically disordered and a magnetically ordered phase using sCUT. Parametrizing the starting point by a variational parameter, we were able to describe the spontaneous breaking of the SU(2) spin symmetry by CUT for the first time. We were able to calculate the ground state energy and the dispersion of triplons, and magnons, respectively, in both phases. However, we were not able to provide quantitative results for the spontaneous magnetization. Furthermore, we found a way to derive an optimal starting point for the integration of the flow equations that is independent on the particularities of the model and does not require a parametrization by a variational parameter. As a further application, we analyzed the S=1 Heisenberg chain by means of deepCUT. To this end, we mapped the system to an extended S=1/2 Heisenberg ladder with the help of a variational parameter. Using the variational generator schemes developed before, we were able to decouple the low-energy spectrum and to calculate spectral densities. In this investigation, we found an indication for a S=0 bound state. Furthermore, we were able to find some signs of spontaneous quasi-particle decay, but a quantitatively correct description is pending.
Linear dynamics restricted to invariant submanifolds generally gives rise to nonlinear dynamics. Submanifolds in the quantum framework may emerge for several reasons: one could be interested in specific properties possessed by a given family of states, either as a consequence of experimental constraints or inside an approximation scheme. In this work we investigate such issues in connection with a one parameter group ϕt of transformations on a Hilbert space, H, defining the unitary evolutions of a chosen quantum system. Two procedures will be presented: the first one consists in the restriction of the vector field associated with the Schr\"{o}dinger equation to a submanifold invariant under the flow ϕt. The second one makes use of the Lagrangian formalism and can be extended also to non-invariant submanifolds, even if in such a case the resulting dynamics is only an approximation of the flow ϕt. Such a result, therefore, should be conceived as a generalization of the variational method already employed for stationary problems.