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arXiv:1512.07234 · Journal of Physics A Mathematical and Theoretical

Spinon bases in supersymmetric CFTs

Thessa Fokkema, Kareljan Schoutens

Published Jun 6, 2016• 3 citations
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Abstract

Abstract We present a novel way to organise the finite size spectra of a class of conformal field theories (CFT) with  = 2 or (nonlinear)  = 4 superconformal symmetry. Generalising the spinon basis of the SU ( n ) 1 WZW theories, we introduce supersymmetric spinons ( ϕ − , ϕ + ) , which form a representation of the supersymmetry algebra. In each case, we show how to construct a multi-spinon basis of the chiral CFT spectra. The multi-spinon states are labelled by a collection { n j } of (discrete) momenta. The state-content for given choice of { n j } is determined through a generalised exclusion principle, similar to Haldane's ‘motif’ rules for the SU ( n ) 1 theories. In the simplest case, which is the  = 2 superconformal theory with central charge c = 1, we develop an algebraic framework similar to the Yangian symmetry of the SU ( 2 ) 1 theory. It includes an operator H 2 , akin to a CFT Haldane–Shastry Hamiltonian, which is diagonalised by multi-spinon states. In all cases studied, we obtain finite partition sums by capping the spinon-momenta to some finite value. For the  = 2 superconformal CFTs, this finitisation precisely leads to the so-called M k supersymmetric lattice models with characteristic order- k exclusion rules on the lattice. Finitising the c = 2 CFT with nonlinear  = 4 superconformal symmetry similarly gives lattice model partition sums for spin-full Fermions with on-site and nearest neighbour exclusion.

DOI
10.1088/1751-8113/49/28/285004

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Abstract We present a novel way to organise the finite size spectra of a class of conformal field theories (CFT) with  = 2 or (nonlinear)  = 4 superconformal symmetry. Generalising the spinon basis of the SU ( n ) 1 WZW theories, we introduce supersymmetric spinons ( ϕ − , ϕ + ) , which form a representation of the supersymmetry algebra. In each case, we show how to construct a multi-spinon basis of the chiral CFT spectra. The multi-spinon states are labelled by a collection { n j } of (discrete) momenta. The state-content for given choice of { n j } is determined through a generalised exclusion principle, similar to Haldane's ‘motif’ rules for the SU ( n ) 1 theories. In the simplest case, which is the  = 2 superconformal theory with central charge c = 1, we develop an algebraic framework similar to the Yangian symmetry of the SU ( 2 ) 1 theory. It includes an operator H 2 , akin to a CFT Haldane–Shastry Hamiltonian, which is diagonalised by multi-spinon states. In all cases studied, we obtain finite partition sums by capping the spinon-momenta to some finite value. For the  = 2 superconformal CFTs, this finitisation precisely leads to the so-called M k supersymmetric lattice models with characteristic order- k exclusion rules on the lattice. Finitising the c = 2 CFT with nonlinear  = 4 superconformal symmetry similarly gives lattice model partition sums for spin-full Fermions with on-site and nearest neighbour exclusion.
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.
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Mk_kk​ models: the field theory connection
The M _k k models for 1D lattice fermions are characterised by {\cal N}=2 supersymmetry and by an order- k k clustering property. This paper highlights connections with quantum field theories (QFTs) in various regimes. At criticality the QFTs are minimal models of {\cal N}=2 supersymmetric conformal field theory (CFT) - we analyse finite size spectra on open chains with a variety of supersymmetry preserving boundary conditions. Specific staggering perturbations lead to a gapped regime corresponding to massive {\cal N}=2 supersymmetric QFT with Chebyshev superpotentials. At ‘extreme staggering’ we uncover a simple physical picture with degenerate supersymmetric vacua and mobile kinks. We connect this kink-picture to the Chebyshev QFTs and use it to derive novel CFT character formulas. For clarity the focus in this paper is on the simplest models, M _1 1 , M _2 2 and M _3 3 .
We employ the SU(n)k\mathrm{SU}{(n)}_{k}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)\mathrm{SU}(n)SU(n) irreducible representation with a single row and kkk 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\mathrm{SU}{(3)}_{k}SU(3)k​ series are derived. The SU(3)k\mathrm{SU}{(3)}_{k}SU(3)k​ chiral spin liquids are lattice analogs of non-Abelian spin-singlet fractional quantum Hall states, and the k=2k=2k=2 member hosts Fibonacci anyons.