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.
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.
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.
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.
Starting from the finite volume form factors of local operators, we show how and under which hypothesis the c = 1 free boson conformal field theory in two-dimensions emerges as an effective theory governing the large-distance regime of multi-point correlation functions for a large class of one-dimensional massless quantum Hamiltonians. In our approach, in the large-distance critical regime, the local operators of the initial model are represented by well suited vertex operators associated to the free boson model. This provides an effective field theoretic description of the large distance behaviour of correlation functions in 1D quantum critical models. We develop this description starting from the first principles and directly at the microscopic level, namely in terms of the properties of the finite volume matrix elements of local operators.
In this thesis the quantum Monte Carlo loop algorithm is applied to study the properties of correlation functions in low-dimensional quantum antiferromagnets. Two different types of lattice models are investigated. The first system is the quantum critical XXZ chain. The second class of models emerges when coupling the square lattice Heisenberg model to phonons. In case of the quantum critical XXZ chain, the loop algorithm is used to compute the two-point spin and dimer correlation functions at low temperatures. Starting from the well-known asymptotic expressions from conformal field theory for the decay of the correlations, the crossover from the finite-temperature to the ground state behavior is investigated. In addtion, the Monte Carlo data are used to extract numerical estimates for the correlation amplitudes. The deviations from asymptotic scaling at low finite temperatures and intermediate distances are examined. The results are also used to study the algebraic and logarithmic corrections in the ground state. The method is further applied to compute the low-temperature spin correlations of the square lattice Heisenberg model coupled to dispersionless Einstein phonons. Also, statically dimerized Heisenberg models in two dimensions are investigated. Here the quantum Monte Carlo technique is used to determine the specific pattern of dimerized bonds which has the largest gain in the magnetic energy. A direct comparison of ground state energies is only possible for a small number of dimerized systems. For this reason an alternative approach is developed, which allows to study a significantly larger class of models. Starting from a Landau expansion of the free energy, this amounts to a computation of specific structure factors of the dimer correlations of the uniform Heisenberg model. As a consequence, a Landau theory for the two-dimensional counterpart of the spin-Peierls transition is established. This adiabatic approach also applies to the Heisenberg model coupled to dispersive phonons. In this case the data for the dimer correlations are used to study the impact of the spin-phonon coupling on the phonon dispersion relations. In particular, the softening of phonon modes is investigated, which is relevant for experimental studies on substances with a quasi two-dimensional magnetic structure.
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.
Effective field theory is a very useful technique for understanding quantum many body systems. We use this approach to study a certain class of critical quantum spin-1 chains and symmetry protected topological phases of matter in two spatial dimensions. In the first context we consider the SU(3) symmetric spin-1 chain with additional interactions. We use bosonization to demonstrate that the effective field theories describing the critical behaviour of these spin chains can be mapped to free compact boson conformal field theories (CFTs) with central charge c=2. We also describe how some predictions from field theory can be verified in numerical calculations using exact diagonalization (ED) and the density matrix renormalization group (DMRG) algorithm. In particular, the bosonization method gives a formula for the evolution of four Tomonaga-Luttinger liquid (TLL) parameters as a function of the lattice parameters. Using the analytic formulae for the scaling dimensions in terms of the TLL parameters and matching of lowest scaling dimensions, we numerically calculate these field-theoretic parameters and confirm that their evolution agrees with the prediction using bosonization. We also tackle aspects of the challenging problem of understanding interacting topological phases. In this context, we aim to understand the effects if interactions in certain classes of symmetry protected topological (SPT) phases of matter. We consider non-chiral SPT phases in two spatial dimensions protected by a discrete symmetry such as Z_K or Z_K x Z_K symmetry. We argue that modular invariance/noninvariance of the partition function of the one-dimensional edge theory can be used to diagnose whether, by adding a suitable potential, the edge theory can be gapped or not without breaking the symmetry. By taking bosonic phases described by Chern-Simons K-matrix theories and fermionic phases relevant to topological superconductors as examples, we demonstrate explicitly that when modular invariance is achieved, we can construct an interaction potential that is consistent with the symmetry and can completely gap out the edge. We also briefly discuss preliminary results of a numerical approach for simulating the 3 dimensional Landau Level problem. This includes a brief description of a potential application of variational Monte Carlo to spin-orbit coupled ab initio systems.