We study lattice wave functions obtained from the SU(2)${}_{1}$ Wess-Zumino-Witten conformal field theory. Following Moore and Read's construction, the Kalmeyer-Laughlin fractional quantum Hall state is defined as a correlation function of primary fields. By an additional insertion of Kac-Moody currents, we associate a wave function with each state of the conformal field theory. These wave functions span the complete Hilbert space of the lattice system. On the cylinder, we study global properties of the lattice states analytically and correlation functions numerically using a Metropolis Monte Carlo method. By comparing short-range bulk correlations, numerical evidence is provided that the states with one current operator represent edge states in the thermodynamic limit. We show that the edge states with one Kac-Moody current of lowest order have a good overlap with low-energy excited states of a local Hamiltonian, for which the Kalmeyer-Laughlin state approximates the ground state. For some states, exact parent Hamiltonians are derived on the cylinder. These Hamiltonians are SU(2) invariant and nonlocal with up to four-body interactions.
We study lattice wave functions obtained from the SU(2)1 Wess-Zumino-Witten conformal field theory. Following Moore and Read's construction, the Kalmeyer-Laughlin fractional quantum Hall state is defined as a correlation function of primary fields. By an additional insertion of Kac-Moody currents, we associate a wave function with each state of the conformal field theory. These wave functions span the complete Hilbert space of the lattice system. On the cylinder, we study global properties of the lattice states analytically and correlation functions numerically using a Metropolis Monte Carlo method. By comparing short-range bulk correlations, numerical evidence is provided that the states with one current operator represent edge states in the thermodynamic limit. We show that the edge states with one Kac-Moody current of lowest order have a good overlap with low-energy excited states of a local Hamiltonian, for which the Kalmeyer-Laughlin state approximates the ground state. For some states, exact parent Hamiltonians are derived on the cylinder. These Hamiltonians are SU(2) invariant and nonlocal with up to four-body interactions.
The Kalmeyer-Laughlin state, which is a lattice version of the bosonic Laughlin state at filling factor one half, has attracted much attention due to its topological and chiral spin liquid properties. Here we show that the Kalmeyer-Laughlin state on the torus can be expressed in terms of a correlator of conformal fields from the SU(2)1 Wess-Zumino-Witten model. This reveals an interesting underlying mathematical structure and provides a natural way to generalize the Kalmeyer-Laughlin state to arbitrary lattices on the torus. We find that the many-body Chern number of the states is unity for more different lattices, which suggests that the topological properties of the states are preserved when the lattice is changed. Finally, we analyze the symmetry properties of the states on square lattices.
We investigate lattice effects on wave functions that are lattice analogs of bosonic and fermionic Laughlin wave functions with number of particles per flux $\ensuremath{\nu}=1/q$ in the Landau levels. These wave functions are defined analytically on lattices with $\ensuremath{\mu}$ particles per lattice site, where $\ensuremath{\mu}$ may be different than $\ensuremath{\nu}$. We give numerical evidence that these states have the same topological properties as the corresponding continuum Laughlin states for different values of q and for different fillings $\ensuremath{\mu}$. These states define, in particular, particle-hole symmetric lattice fractional quantum Hall states when the lattice is half filled. On the square lattice it is observed that for $q\ensuremath{\le}4$ this particle-hole symmetric state displays the topological properties of the continuum Laughlin state at filling fraction $\ensuremath{\nu}=1/q$, while for larger q there is a transition towards long-range ordered antiferromagnets. This effect does not persist if the lattice is deformed from a square to a triangular lattice, or on the kagome lattice, in which case the topological properties of the state are recovered. We then show that changing the number of particles while keeping the expression of these wave functions identical gives rise to edge states that have the same correlations in the bulk as the reference lattice Laughlin states but a different density at the edge. We derive an exact parent Hamiltonian for which all these edge states are ground states with different number of particles. In addition this Hamiltonian admits the reference lattice Laughlin state as its unique ground state of filling factor 1/q. Parent Hamiltonians are also derived for the lattice Laughlin states at other fillings of the lattice, when $\ensuremath{\mu}\ensuremath{\le}1/q$ or $\ensuremath{\mu}\ensuremath{\ge}1\ensuremath{-}1/q$ and when q=4 also at half filling.
Conformal field theory has recently been applied to derive few-body Hamiltonians whose ground states are lattice versions of fractional quantum Hall states. The exact lattice models involve interactions over long distances, which is difficult to realize in experiments. It seems, however, that such long-range interactions should not be necessary, as the correlations decay exponentially in the bulk. This poses the question, whether the Hamiltonians can be truncated to contain only local interactions without changing the physics of the ground state. Previous studies have, in a couple of cases with particularly much symmetry, obtained such local Hamiltonians by keeping only a few local terms and numerically optimizing the coefficients. Here, we investigate a different strategy to construct truncated Hamiltonians, which does not rely on optimization, and which can be applied independently of the choice of lattice. We test the approach on two models with bosonic Laughlin-like ground states with filling factor 1/2 and 1/4, respectively. We first investigate how the coupling strengths in the exact Hamiltonians depend on distance, and then we study the truncated models. For the case of 1/2 filling, we find that the truncated model with truncation radius 2 lattice constants on the square lattice and 1 lattice constant on the triangular lattice has an approximate twofold ground-state degeneracy on the torus, and the overlap per site between these states and the states constructed from conformal field theory is higher than 0.99 for the lattices considered. For the model at 1/4 filling, our results give some hints that a truncation radius of 5 on the square lattice and 7 on the triangular lattice might be enough, but the finite-size effects are too large to judge whether the topology is, indeed, present in the thermodynamic limit. The states with high overlap also have the expected topological entanglement entropies.
We employ the SU(n)k Wess-Zumino-Witten (WZW) model in conformal field theory to construct lattice wave functions in both one and two dimensions. The spins on all lattice sites are chosen to transform under the SU(n) irreducible representation with a single row and k boxes in the Young tableau. It is demonstrated that the wave functions can be reinterpreted as parton states, which enables efficient conversion to matrix product states, such that many physical properties can be evaluated directly. In one dimension, these wave functions describe critical spin chains whose universality classes are in one-to-one correspondence with the WZW models used in the construction. In two dimensions, our constructions yield model wave functions for chiral spin liquids, and we show how to find all topological sectors of them in a systematic way. Using the null vectors of Kac-Moody algebras, parent Hamiltonians of the SU(3)k series are derived. The SU(3)k chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the k=2 member hosts Fibonacci anyons.
We introduce one-dimensional lattice models with exact matrix-product ground states describing the fractional quantum Hall (FQH) states in Laughlin series (given by filling factors $\ensuremath{\nu}=1/q$) on torus geometry. Surprisingly, the exactly solvable Hamiltonian has the same mathematical structure as that of the pseudopotential for the Laughlin wave function, and naturally derives the general properties of the Laughlin wave function such as the Z2 properties of the FQH states and the fermion-boson relation. The obtained exact ground states have high overlaps with the Laughlin states and well describe their properties. Using the matrix product method, density functions and correlation functions are calculated analytically. Especially, obtained entanglement spectra reflect gapless edge states as was discussed by Li and Haldane.
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.