We study nonchiral wave functions for systems with continuous spins obtained from the conformal field theory (CFT) of a free, massless boson. In contrast to the case of discrete spins, these can be treated as bosonic Gaussian states, which allows us to efficiently compute correlations and entanglement properties both in one (1D) and in two spatial dimensions (2D). In 1D, the computed entanglement entropy and spectra are in agreement with the underlying CFT. Furthermore, we construct a 1D parent Hamiltonian with a low-energy spectrum corresponding to that of a free, massless boson. In 2D, we find edge excitations in the entanglement spectrum, although the states do not have intrinsic topological order, as revealed by a determination of the topological entanglement entropy.
We study nonchiral wave functions for systems with continuous spins obtained from the conformal field theory (CFT) of a free, massless boson. In contrast to the case of discrete spins, these can be treated as bosonic Gaussian states, which allows us to efficiently compute correlations and entanglement properties both in one (1D) and in two spatial dimensions (2D). In 1D, the computed entanglement entropy and spectra are in agreement with the underlying CFT. Furthermore, we construct a 1D parent Hamiltonian with a low-energy spectrum corresponding to that of a free, massless boson. In 2D, we find edge excitations in the entanglement spectrum, although the states do not have intrinsic topological order, as revealed by a determination of the topological entanglement entropy.
We study states of one- and two-dimensional spin systems that are constructed as correlators within the conformal field theory of a massless, free boson. In one dimension, these are good variational wave functions for XXZ spin chains, and they are similar to lattice Laughlin states in two dimensions. We show that their zz correlations are determined by a modification of the original free-boson theory. An expansion to quadratic order leads to a solvable, effective theory for the correlations in these states. Compared to the massless boson, there is an additional term in this effective theory that explains the behavior of the correlations: a polynomial decay in one dimension and at the edge of a two-dimensional system and an exponential decay in the bulk of a two-dimensional system. We test the validity of our approximation by comparing it to Monte Carlo computations.
In this thesis, we study quantum many-body systems in one (1D) and in two spatial dimensions (2D).We adopt the approach established by Moore and Read, where model states are constructed using conformal field theory (CFT), a scale-invariant quantum field theory.The central themes of this thesis are the definition of states through CFT, their characterization, and the understanding of their properties in terms of the underlying CFT.The first part of this thesis considers a CFT with an additional SU(2) symmetry.We define a map from CFT states to those of spin-1 2 systems on lattices.In 1D, the CFT vacuum is mapped to the ground state of a Heisenberg-like spin chain with long-range inverse-square interactions.We show that the excited states of this spin chain can be constructed from excitations of the CFT.Thus, we establish a correspondence between the spectrum of the CFT and that of the spin chain.In a next step, we study these states in 2D, where the CFT ground state corresponds to a fractional quantum Hall (FQH) lattice state.Excited states of the CFT are mapped to wave functions describing edge excitations of the FQH system.Through Monte Carlo simulations, we provide numerical evidence for a central property of these edge modes, namely that their local bulk correlations coincide with those of the corresponding FQH state as the system size becomes large enough.These results confirm the bulk-edge correspondence stating that the CFT associated with the gapless edge can be used to describe wave functions of the bulk.We then consider a larger class of states constructed from CFT.In 1D, these are good descriptions of the ground state of the XXZ spin-1 2 chain, and they correspond to FQH lattice states in 2D.Through a path integral representation, we propose an approximation for their spin-spin correlations.The effective action determining this approximation differs from that of the underlying CFT through an additional mass-like term at the lattice positions.This explains the behavior of the correlations: While they decay polynomially in 1D and at the edge of 2D systems, bulk correlations in 2D decrease exponentially as a function of the distance.We test the accuracy of the approximation by comparing it to actual spin-spin correlations obtained through Monte Carlo simulations.Our approximation provides an analytical argument for the screening hypothesis, which states that FQH wave functions have exponentially decaying bulk correlations.The last part of this thesis uses CFT to define nonchiral states with continuous spins on lattices.Opposed to the case of discrete spins, these are Gaussian and thus their properties can be computed efficiently.Through an analysis of entanglement entropies and spectra, we identify signatures of the underlying CFT in 1D.In 2D, we find indications of edge states.Although the topological entanglement entropy vanishes, the entanglement spectrum contains modes that decay exponentially with the distance to the entanglement cut. ZusammenfassungDiese Dissertation untersucht Quanten-Vielteilchensysteme in einer (1D) und in zwei räumlichen Dimensionen (2D).Wir arbeiten im Rahmen der von Moore und Read etablierten Herangehensweise, wonach Modellzustände durch konforme Feldtheorie (CFT), einer skaleninvarianten Quantenfeldtheorie, konstruiert werden.Die Schwerpunktthemen dieser Arbeit sind: die Definition von Quantenzuständen vieler Teilchen mittels CFT, deren Charakterisierung und das Verständnis ihrer Eigenschaften in Bezug auf die zugrundeliegende CFT.Der erste Teil dieser Dissertation betrachtet eine CFT, welche eine zusätzliche SU(2)-Symmetrie aufweist.Wir definieren eine Abbildung von Zuständen der CFT zu solchen eines Spin-1 2 -Systems auf einem Gitter.In 1D wird das Vakuum der CFT auf den Grundzustand einer Spinkette abgebildet, welche ähnlich einem Heisenberg-Modell ist und langreichweitige Wechselwirkungen hat, die mit dem Quadrat der Distanz abfallen.Wir konstruieren Anregungszustände der Spinkette ausgehend von Anregungen der CFT.In einem weiteren Schritt untersuchen wir diese Zustände in 2D, wo der Grundzustand der CFT dem Zustand eines fraktionalen Quanten-Hall-Effekts (FQH) auf einem Gitter entspricht.Anregungen der CFT werden auf Wellenfunktionen abgebildet, welche Randzustände des FQH-Systems beschreiben.Mittels Monte-Carlo-Simulationen finden wir numerische Belege für eine zentrale Eigenschaft solcher Randzustände: Deren lokale Korrelationen im Inneren des Systems stimmen mit denen des zugrundeliegenden FQH-Zustands überein, sobald die Systemgröße groß genug wird.Dieses Ergebnis bestätigt das Konzept der Korrespondenz zwischen dem Inneren und dem Rand.Dieses besagt, dass die CFT des Randes Zustände des Inneren beschreiben kann.Danach betrachten wir eine größere Klasse von Zuständen, welche durch CFT definiert sind.Diese sind gute Näherungen für den Grundzustand der XXZ-Spin-1 2 -Kette in 1D und entsprechen FQH-Zuständen in 2D.Durch eine Pfandintegraldarstellung leiten wir eine Näherung für Spin-Spin-Korrelationen her.Die effektive Wirkung dieser Näherung unterscheidet sich von derjenigen der zugrundeliegenden CFT dadurch, dass sie einen zusätzlichen Term enthält, der einem Masseterm ähnlich ist.Dieser ist an den Gitterplätzen des Systems lokalisiert und erklärt somit das Verhalten der Korrelationen: Diese zerfallen polynomiell in 1D und am Rand eines 2D Systems und exponentiell im Inneren eines 2D Systems.Wir überprüfen die Genauigkeit der Näherung, indem wir sie mit den tatsächlichen Spin-Spin-Korrelationen vergleichen, welche wir mittels Monte-Carlo-Simulationen berechnen.Unsere Näherung stellt einen analytischen Beleg für die Abschirmhypothese dar.Diese besagt, dass Korrelationen von FQH-Zuständen im Inneren des Systems exponentiell zerfallen.Der letzte Teil dieser Dissertation definiert nicht-chirale Zustände mit kontinuierlichem Spin auf Gittern mittels CFT.Im Unterschied zum Fall diskreter Spins sind dies Gaußsche Zustände.Daher können deren Eigenschaften effizient berechnet werden.Mittels einer Analyse von Verschränkungsentropien und -spektren identifizieren wir Charakteristika der zugrundeliegenden CFT in 1D.In 2D finden wir Hinweise auf Randzustände.Obwohl die topologische Verschränkungsentropie verschwindet, enthält das Verschränkungsspektrum Moden, welche exponentiell am Rand lokalisiert sind.
The entanglement spectra for a subsystem in a spin chain fine-tuned to a quantum-critical point contains signatures of the underlying quantum field theory that governs its low-energy properties. For an open chain with given boundary conditions described by a 2D conformal field theory~(CFT), the entanglement spectrum of the left/right half of the system coincides with a boundary CFT spectrum, where one of the boundary conditions arise due to the `entanglement cut'. The latter has been argued to be conformal and has been numerically found to be the `free' boundary condition for Ising, Potts and free boson theories. For these models, the `free' boundary condition for the lattice degree of freedom has a counterpart in the continuum theory. However, this is not true in general. Here, this question is analyzed for the unitary minimal models of 2D CFTs using the density matrix renormalization group technique. The entanglement spectra are computed for blocks of spins in open chains of A-type restricted solid-on-solid models with identical boundary conditions at the ends. The imposed boundary conditions are realized exactly for these lattice models due to their integrable nature. The obtained entanglement spectra are in good agreement with certain boundary CFT spectra. The boundary condition for the entanglement cut is found to be conformal and to coincide with the one with the highest boundary entropy. This identification enables determination of the exponents governing the unusual corrections to the entanglement entropy from the CFT partition functions. These are compared with numerical results.
Abstract Entanglement or modular Hamiltonians play a crucial role in the investigation of correlations in quantum field theories. In particular, in 1 + 1 space–time dimensions, the spectra of entanglement Hamiltonians of conformal field theories (CFTs) for certain geometries are related to the spectra of the physical Hamiltonians of corresponding boundary CFTs. As a result, conformal invariance allows exact computation of the spectra of the entanglement Hamiltonians for these models. In this work, we perform this computation of the spectrum of the entanglement Hamiltonian for the free compactified boson CFT over a finite spatial interval. We compare the analytical results obtained for the continuum theory with numerical simulations of a lattice-regularized model for the CFT using density matrix renormalization group technique. To that end, we use a lattice regularization provided by superconducting quantum electronic circuits, built out of Josephson junctions and capacitors. Up to non-universal effects arising due to the lattice regularization, the numerical results are compatible with the predictions of the exact computations.
In this paper we discuss the topics concerning the local excitation and coherent state in 2D CFT. It is shown that the local excitation of primary operator can be taken as a coherent state of the global conformal group. We also discuss the entanglement property of such state. For rational CFT the entanglement entropy between the holomorphic and anti-holomorphic sector of the local excitation of some primary operator is related to the quantum dimension of the operator, consistent with previous approach, but by a differentmethod. We comment on the possible application of so-defined group coherent state in the holographic view. We also study the coherent state in the free massless boson field, their time evolution and entanglement property. We introduce the deformed local excitation and the entangled state constructed by them. It is shown the violation of Bell inequality for such entangled state.
Given a quantum critical wavefunction in any dimension, we propose a reconstructed Hamiltonian, analogous to the ones previously found for 1+1d CFT and for 2+1d bosonic liquid topologically-ordered states. We test numerically that, for known regularized approximate CFT groundstates (on the icosahedron and the fuzzy sphere), (1) they are close to the groundstate of their reconstructed Hamiltonian, and (2) the spectrum of their reconstructed Hamiltonian on the unit sphere has CFT properties (integer spacing of descendants) and matches known low-lying energies. We show that this provides an automated method to improve the finite-size effects in a fixed Hilbert space.