We develop a method of constructing excited states in one-dimensional spin chains which are derived from the $\text{SU}{(2)}_{1}$ Wess-Zumino-Witten conformal field theory (CFT) using a parent Hamiltonian approach. The resulting systems are equivalent to the Haldane-Shastry model. In our ansatz, correlation functions between primary fields correspond to the ground state of the spin system, whereas excited states are obtained by insertion of descendant fields. Our construction is based on the current algebra of the CFT and emphasizes the close relation between the spectrum of the spin system and the underlying CFT. This general structure might imply that the method could be applied to a wider range of model systems.
We develop a method of constructing excited states in one-dimensional spin chains which are derived from the SU(2)1 Wess-Zumino-Witten conformal field theory (CFT) using a parent Hamiltonian approach. The resulting systems are equivalent to the Haldane-Shastry model. In our ansatz, correlation functions between primary fields correspond to the ground state of the spin system, whereas excited states are obtained by insertion of descendant fields. Our construction is based on the current algebra of the CFT and emphasizes the close relation between the spectrum of the spin system and the underlying CFT. This general structure might imply that the method could be applied to a wider range of model systems.
We investigate a family of spin-S chain Hamiltonians recently introduced by one of us [Greiter, Mapping of Parent Hamiltonians, Springer Tracts in Modern Physics, Vol. 244 (Springer, Berlin, 2011)]. For S=1/2, it corresponds to the Haldane-Shastry model. For general spin S, we find indication that the low-energy theory of these spin chains is described by the SU(2)k Wess-Zumino-Witten model with coupling k=2S. In particular, we investigate the S=1 model whose ground state is given by a Pfaffian for even number of sites N. We reconcile aspects of the spectrum of the Hamiltonian for arbitrary N with trial states obtained by Schwinger projection of two Haldane-Shastry chains.
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 propose to use null vectors in conformal field theories to derive model Hamiltonians of quantum spin chains and the corresponding ground state wavefunction(s). The approach is quite general, and we illustrate it by constructing a family of Hamiltonians whose ground states are the chiral correlators of the SU (2) k WZW model for integer values of the level k . The simplest example corresponds to k = 1 and is essentially a nonuniform generalization of the Haldane–Shastry model with long-range exchange couplings. At level k = 2, we analyse the model for N spin 1 fields. We find that the Renyi entropy and the two-point spin correlator show, respectively, logarithmic growth and algebraic decay. Furthermore, we use the null vectors to derive a set of algebraic, linear equations relating spin correlators within each model. At level k = 1, these equations allow us to compute the two-point spin correlators analytically for the finite chain uniform Haldane–Shastry model and to obtain numerical results for the nonuniform case and for higher-point spin correlators in a very simple way and without resorting to Monte Carlo techniques.
Phase transitions of second order play an important role in solid state physics. It is in their nature that they occur only in systems that are composed from an infinite number of components since, only that way, the necessary conditions for this (for example, scale invariance) are granted. Such a system at a second order phase transition is called critical. The infinite number of particles poses a particular challenge, even for the description of classical systems. Quantum mechanical systems, however, are distinctly more difficult to treat in this context, since the dimension of their state space grows exponentially with the number of their particles, not to mention the fact that only a few – and usually only one-dimensional – quantum systems are exactly solvable. One of these exactly solvable one-dimensional quantum systems is the SU(N) Haldane-Shastry spin chain that may be regarded as the archetype of long-range spin chains. Moreover, it is critical in the continuum limit and its effective low-energy theory is the so-called SU(N)_1 Wess-Zumino-Witten model. It is a quantum field theory which is not only conformally invariant, so, in particular, scale invariant, but, furthermore, exhibits additional symmetry in the shape of an infinite extension of the Lie algebra su(N) associated with SU(N). Recent studies show that, from these very structures of the SU(N)_1 WZW model, one can, in turn, derive spin systems, whose arrangement is not necessarily the one of a spin chain, but even two-dimensional distributions of the spins in the plane are possible. These systems are again characterized by long-range interactions, comparable to those of the Haldane-Shastry spin chain, which is also obtained as a result of an appropriate choice of parameters. \n \nIn this thesis, we extend the construction already known for the SU(N) case to the supersymmetric case GL(m|n). Here, we construct explicitly both, a special quantum state as well as a Hamiltonian that projects this quantum state to zero. We also discuss the Hamiltonian in the special case of the GL(1|1) spin chain and compare it to the respective GL(1|1) Haldane-Shastry spin chain on a bipartite state space. Both are critical and we identify the corresponding conformal field theories. Subsequently, we describe a generalization of this system in terms of two parameters and explain how its spectrum was found. It is then analyzed and its continuum limit is determined. In doing so, it shows that the system displays criticality only for generic values of one of the two parameters.
We generalize the matrix product states method using the chiral vertex operators of conformal field theory and apply it to study the ground states of the XXZ spin chain, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model and random Heisenberg models. We compute the overlap with the exact wave functions, spin-spin correlators, and the Renyi entropy, showing that critical systems can be described by this method. For rotational invariant ansatzs we construct an inhomogenous extension of the Haldane-Shastry model with long-range exchange interactions.