Spectral flow for an integrable staggered superspin chain

TL;DR

This paper investigates how the low energy states of a superspin chain with alternating representations evolve as boundary conditions change, revealing a flow between continuous and discrete spectrum levels.

Contribution

It introduces a detailed analysis of spectral flow in an integrable superspin chain with mixed representations, highlighting the impact of boundary condition variations.

Findings

Levels from the continuous spectrum flow into discrete levels when boundary conditions change.

Finite size data extrapolation reveals spectrum restructuring under boundary condition variation.

The study characterizes the spectrum using scaling dimensions and quasi momenta for two transfer matrix families.

Abstract

The flow of the low energy eigenstates of a $U_q[sl(2|1)]$ superspin chain with alternating fundamental ($3$) and dual ($\bar{3}$) representations is studied as function of a twist angle determining the boundary conditions. The finite size spectrum is characterized in terms of scaling dimensions and quasi momenta representing the two families of commuting transfer matrices for the model which are even and odd under the interchange $3\leftrightarrow \bar{3}$, respectively. Based on the extrapolation of our finite size data we find that under a variation of the boundary conditions from periodic to antiperiodic for the fermionic degrees of freedom levels from the continuous part of the spectrum flow into discrete levels and vice versa.

Equations

$${\mathcal{R}}_{12}^{({\omega }_1,{\omega }_2)}(\lambda ){\mathcal{R}}_{13}^{({\omega }_1,{\omega }_3)}(\lambda +\mu ){\mathcal{R}}_{23}^{({\omega }_2,{\omega }_3)}(\mu )={\mathcal{R}}_{23}^{({\omega }_2,{\omega }_3)}(\mu ){\mathcal{R}}_{13}^{({\omega }_1,{\omega }_3)}(\lambda +\mu ){\mathcal{R}}_{12}^{({\omega }_1,{\omega }_2)}(\mu ),$$

$${\tau }^{(3)}(\lambda )$$

$${\tau }^{(\bar{3})}(\lambda )$$

$$\tau (\lambda )={\tau }^{(3)}(\lambda ){\tau }^{(\bar{3})}(\lambda ).$$

$$\mathcal{H}=i\frac{\partial}{\partial\lambda}\ln\tau(\lambda)\Big{|}_{\lambda=0}\,.$$

$$\mathcal{K}$$

$$\begin{array}{rl}{\left[\frac{\sinh ({\lambda }_j^{(1)}+i\gamma )}{\sinh ({\lambda }_j^{(1)}-i\gamma )}\right]}^L& ={\mathrm{e}}^{i\alpha }\prod _{k=1}^{N_2}\frac{\sinh ({\lambda }_j^{(1)}-{\lambda }_k^{(2)}+i\gamma )}{\sinh ({\lambda }_j^{(1)}-{\lambda }_k^{(2)}-i\gamma )},j=1,\dots ,N_1,\\ {\left[\frac{\sinh ({\lambda }_j^{(2)}+i\gamma )}{\sinh ({\lambda }_j^{(2)}-i\gamma )}\right]}^L& ={\mathrm{e}}^{i\alpha }\prod _{k=1}^{N_1}\frac{\sinh ({\lambda }_j^{(2)}-{\lambda }_k^{(1)}+i\gamma )}{\sinh ({\lambda }_j^{(2)}-{\lambda }_k^{(1)}-i\gamma )},j=1,\dots ,N_2.\end{array}$$

$$E(\{{\lambda }_j^{(1)}\},\{{\lambda }_j^{(2)}\})$$

$$K(\{{\lambda }_j^{(1)}\},\{{\lambda }_j^{(2)}\})$$

$$X_{(m,w)}^{\mathrm{eff}}(\phi =\alpha +\pi )$$

$$X_{(0,0)}^{\ast }(\phi )$$

$$X_{(0,0)}^{\ast }(\pi )=\frac{1}{4}\frac{\pi -2\gamma }{\pi +2\gamma }=\frac{1}{4}\frac{1}{2k-1}$$

$${\lambda }_j^{(1)}=-{\lambda }_j^{(2)}\in \left\{{\mu }_k^{\pm }:\mathrm{Im}({\mu }_k^{\pm })\simeq \pm \frac{\gamma }{2},k=1\dots \frac{L-1}{2}\right\}\cup \left\{\infty \right\}.$$

$$X_{(0,-2)}^{\ast }=X_{(0,-2)}^{\mathrm{eff}}(\phi )-\frac{2k-1}{(k-1{)}^2}{\left(\frac{1}{2}-\frac{k}{2}\left|\frac{\phi }{\pi }-2\right|\right)}^2,$$

$$X_{(1,-1)}^{\ast }(\phi )=X_{(1,-1)}^{\mathrm{eff}}(\phi )-\frac{2k-1}{(k-1{)}^2}{\left(1-\frac{k}{2}\left|\frac{\phi }{\pi }-1\right|\right)}^2,|\phi -\pi |>\frac{2\pi }{k}.$$

$$K^{\ast }(\phi )=\frac{i}{2(\pi -2\gamma )}(|\phi |-{\phi }_c)=i\left(\frac{k|\phi |}{2\pi }-\frac{1}{2}\right)\mathrm{for}{\phi }_c\le \phi \le {\phi }_{c,2}$$

$$\mathrm{Im}(K^{\ast }(\phi ))=\frac{4{\gamma }^2}{{\pi }^2-4{\gamma }^2}=\frac{(k-1{)}^2}{2k-1}\mathrm{for}\phi \ge {\phi }_{c,2}.$$

Full text

Integrable two-dimensional vertex models and the corresponding $(1+1)$-dimensional quantum spin chains have provided many examples for lattice realizations of conformal field theories (CFTs) and thereby led to important insights into the critical properties of strongly correlated systems. For lattice models with finite dimensional representations for the local degrees of freedom and Hermitian Hamiltonian the unitary CFTs appearing in the continuum limit can be identified by relating Virasoro representations to the discrete finite size spectrum of the former Blöte et al. (1986); Affleck (1986).

Recently, a growing number of lattice models related to order-disorder transitions in quantum Hall systems, the anti-ferromagnetic Potts model, intersecting loops or physical properties of two-dimensional polymers, has been found whose continuum limit – although having a finite-dimensional quantum space for the local degrees of freedom on the lattice – are CFTs with a non-compact target space Essler et al. (2005); Ikhlef et al. (2008); Frahm and Martins (2011, 2012); Vernier et al. (2014); Frahm and Martins (2015); Vernier et al. (2016). This manifests itself in the emergence of a continuous component in the spectrum of scaling dimensions in the thermodynamic limit and, possibly, a non-analytic dependence of the effective central charge of the lattice model on the twist of boundary conditions related to discrete levels appearing in the spectrum of the lattice model when the corresponding operator in the CFT becomes normalizable.

For some of these lattice models the spectral data obtained from their exact solution have allowed to identify the CFT describing their continuum limits:

• •

An integrable vertex model built from the three-dimensional fundamental and dual representations of the superalgebra $sl(2|1)$ was shown to flow to a $SU(2|1)$ Wess-Zumino-Novikov-Witten (WZNW) model at level $k=1$ Essler et al. (2005); Saleur and Schomerus (2007). For a physical modular invariant partition function one is forced to consider continuous values of the $sl(2|1)$ charge quantum number leading to continua of scaling dimensions.

• •

A staggered six-vertex model related to the anti-ferromagnetic Potts model Ikhlef et al. (2008) (which also appears in the phase diagram of a staggered superspin chain built from four-dimensional representations of $U_{q}[sl(2|1)]$, see Ref. Frahm and Martins (2012)) has been proven to provide a realization of the $SL(2,\mathbb{R})/U(1)$ Euclidean black hole sigma model Witten (1991); Dijkgraaf et al. (1992). Here the identification was based on the density of states in the continuous spectrum Ikhlef et al. (2012); Candu and Ikhlef (2013); Frahm and Seel (2014).

• •

The $a_{2}^{(2)}$ model (equivalent to the 19-vertex Izergin-Korepin model Izergin and Korepin (1981)) in ’regime III’ has a non-compact continuum limit again described by the $SL(2,\mathbb{R})/U(1)$ black hole sigma model Vernier et al. (2014). Here the spectrum of discrete levels appearing in the lattice model with twisted boundary conditions has been found to be consistent with the predicted appearance of levels related to the principal discrete representations of $SL(2,\mathbb{R})$ states Hanany et al. (2002); Ribault and Schomerus (2004). Similarly, the scaling limit of the general $a_{N-1}^{(2)}$ model has been shown to be a $SO(N)/SO(N-1)$ gauged WZNW model Vernier et al. (2016).

In this paper we are studying the continuum limit of the mixed $U_{q}[sl(2|1)]$ superspin chain based on the three dimensional atypical representation and its dual (“quark” and “antiquark” , labeled $3$ and $\bar{3}$ in the following) Gade (1999), an anisotropic deformation of the $sl(2|1)$ superspin chain mentioned above. In previous work on this model, and similar as in the isotropic case, the existence of an exact zero energy state and of continua of scaling dimensions have been established Frahm and Martins (2011). Here we generalize the lattice model to include a twist allowing for an adiabatic change from periodic to anti-periodic boundary conditions for the fermionic degrees of freedom. Using the Bethe ansatz solution of this model we find the exact dependence of some low lying levels on the twist angle. Translating our results into the context of the field theory describing the continuum limit of the model we find that under the spectral flow states from the continuous part of the spectrum in the Neveu-Schwarz sector are mapped onto discrete levels in the Ramond sectors, and vice versa.

The construction of an integrable mixed superspin chain with alternating three-dimensional quark and anti-quark representations of $U_{q}[sl(2|1)]$ is based on four $\mathcal{R}$-matrices acting on tensor products $3\otimes 3$, $3\otimes\bar{3}$, $\bar{3}\otimes 3$, and $\bar{3}\otimes\bar{3}$ satisfying Yang-Baxter equations

[TABLE]

with $\omega_{j}\in\{3,\bar{3}\}$ for $j=1,2,3$ Gade (1999) (see also Perk and Schultz (1981); Kulish and Sklyanin (1982)). Two families of row-to-row transfer matrices acting on a Hilbert space $(3\otimes\bar{3})^{\otimes L}$ are constructed as the supertrace over auxiliary spaces $A\cong\mathbb{C}^{3}$ of ordered products of these $\mathcal{R}$-matrices as Kulish (1986)

[TABLE]

Here $\gamma$ parametrizes the deformation parameter $q=\exp(-i\gamma)$, more details can be found in Ref. Frahm and Martins (2011).111Note, the definition of $\gamma$ in Ref. Frahm and Martins (2011) differs from the one used here by a factor of 2. In Eq. (2) boundary conditions are controlled by the diagonal twist matrices $\mathcal{G}^{(\omega)}(\alpha)=\exp\left(2i\alpha J_{3}^{(\omega)}\right)$, $J_{3}^{(\omega)}$ being the diagonal generator of the spin-subalgebra in the representation $\omega$. For $\alpha=0$ the lattice model obeys periodic boundary conditions while for $\alpha=\pm\pi$ bosons are periodic while fermions obey antiperiodic boundary conditions. In the field theory describing the continuum limit of the lattice model these cases correspond to the Ramond (R) and Neveu-Schwarz (NS) sector, respectively.

As a consequence of (1) the two transfer matrices commute, $\left[\tau^{(\omega_{1})}(\lambda),\tau^{(\omega_{2})}(\mu)\right]=0$ for all $\lambda$, $\mu$ and $\omega_{j}\in\{3,\bar{3}\}$. Local integrals of motion are generated by the double row transfer matrix

[TABLE]

For example, the Hamiltonian of the mixed $U_{q}[sl(2|1)$ superspin chain is

[TABLE]

For a staggered model as the one considered here we can introduce another combination of the single-row transfer matrices, i.e. $\tau^{(3)}(\lambda)[\tau^{(\bar{3})}(\lambda)]^{-1}$, whose logarithm, unlike (3), is odd under the exchange of the quark and antiquark representation. Among the conserved quantities generated by this object, and motivated by the analysis of a staggered six-vertex model by Ikhlef et al. Ikhlef et al. (2012), see also Refs. Candu and Ikhlef (2013); Frahm and Seel (2014), we consider the ‘quasi momentum operator’

[TABLE]

The transfer matrices (2) (and therefore $\mathcal{H}$ and $\mathcal{K}$) can be diagonalized using the nested algebraic Bethe ansatz Kulish and Reshetikhin (1983); Babelon et al. (1982). The resulting expressions obtained within this framework depend on the choice of grading for the underlying superalgebra Kulish (1986); Essler and Korepin (1992); Foerster and Karowski (1993); Pfannmüller and Frahm (1996, 1997). As in Refs. Essler et al. (2005); Frahm and Martins (2011) we use $[p_{1},p_{2},p_{3}]=[0,1,0]$. In the sector of the Hilbert space with fixed quantum numbers related to the $U(1)$ subalgebras of $U_{q}[sl(2|1)]$, i.e. charge $b=(N_{1}-N_{2})/2$ and $z$-component of the spin $j_{3}=L-(N_{1}+N_{2})/2$ with non-negative integers $N_{1,2}$ the spectrum of the transfer matrices can be parametrized $N_{1}+N_{2}$ complex rapidities $\{\lambda_{j}^{(1)}\}_{j=1}^{N_{1}}$ and $\{\lambda_{j}^{(2)}\}_{j=1}^{N_{2}}$ solving the Bethe equations,

[TABLE]

The corresponding eigenvalues of the Hamiltonian (4) and the quasi momentum (5) are

[TABLE]

Previous studies of this model Essler et al. (2005); Frahm and Martins (2011, 2012) have shown that the spectrum of the transfer matrix (3) in the sector with charge $b=0$ (i.e. $N_{1}=N_{2}=N$) contains (after scaling by a factor of two) that of the antiferromagnetic spin-$1$ XXZ Heisenberg model with twist $\varphi=\pi+\alpha$ Zamolodchikov and Fateev (1980): for a subset of eigenstates of the mixed superspin chain the Bethe roots degenerate as $\lambda_{j}^{(1)}=\lambda_{j}^{(2)}=\lambda_{j}$. These corresponding eigenvalues can be identified with eigenenergies of the spin-$1$ chain in the sector with $j_{3}=L-N$, see Refs. Frahm and Martins (2011, 2012).222The eigenvalues $K$ of the quasi momentum operator, Eq. (7), are odd under the interchange $\{\lambda_{j}^{(1)}\}\leftrightarrow\{\lambda_{j}^{(2)}\}$ changing the sign of the charge $b$. As an immediate consequence $K=0$ for the levels from the spin-$1$ XXZ chain subset of the spectrum. At low energies the operator content of the effective field theory for the spin-$1$ XXZ model is given in terms of composites of a $U(1)$ Kac-Moody field and Ising operators Alcaraz and Martins (1989, 1990); Frahm et al. (1990). This provides a subset of the finite size spectrum of the superspin chain Essler et al. (2005); Frahm and Martins (2011). After rescaling of the finite size energy gaps, $\Delta E\to(L\Delta E/2\pi v_{F})\equiv X$ ($v_{F}=\pi/\gamma$ is the Fermi velocity of the gapless excitations), the corresponding effective scaling dimensions of the superspin chain (4) are

[TABLE]

where the integer $m$ is the quantum number $j_{3}$ of the corresponding state in the superspin chain while $w$ is related to the vorticity of that state. Note that the energy of the level $(m,w)=(0,-1)$ in the Ramond sector ($\varphi=\pi$) vanishes identically. The Bethe root configuration corresponding to this exact zero mode in the $(b,j_{3})=(0,0)$ sector of the periodic superspin chain is highly degenerate, i.e. $\lambda_{j}^{(a)}\equiv 0$ for all $j=1,\ldots,L$ and $a=1,2$. This is the ground state of the periodic superspin chain (and the related spin-$1$ XXZ model with anti-periodic boundary conditions where the zero-mode has been shown to be a singlet under an exact dynamical lattice supersymmetry Hagendorf (2015)) for anisotropies $0\leq\gamma\leq\pi/4$.

In the large $L$ limit the possible root configurations solving the Bethe equations (6) can be classified using the string hypothesis. A class of low energy excitations in the zero charge sector $b=0$ has been identified with collections of $O(L)$ ’strange $2$-strings’ consisting of complex conjugate pairs of rapidities $\lambda^{(1)}=\left(\lambda^{(2)}\right)^{*}$. These strange strings come in two types, i.e. $(\pm)$ with $\mathrm{Im}(\lambda^{(1)})=\pm\gamma/2$. Solutions to the Bethe equations consisting of $N_{\pm}$ type-$(\pm)$ strange strings with $N_{+}+N_{-}=L-j_{3}$ but $\Delta N=N_{+}-N_{-}\neq 0$ have been found to form a continuous component of the finite size spectrum starting at levels (8) with $m+w\in 2\mathbb{Z}$ Essler et al. (2005); Frahm and Martins (2011).

Based on these insights from the lattice model it has been argued that the continuum limit of the isotropic (i.e. $\gamma=0$) superspin chain flows to a $SU(2|1)$ WZNW model at level $k=1$ Essler et al. (2005); Saleur and Schomerus (2007). The generic class I irreducible representations of the affine superalgebra $\widehat{sl}(2|1)_{k=1}$ are built over the typical representations $[b,j=\frac{1}{2}]$ with charge $b\in\mathbb{C}$ and spin $j=\frac{1}{2}$. To obtain a physical modular invariant partition function from the characters for these representations which can be compared with the spectrum of the lattice model one is forced to consider continuous values of the charge $b$ which, after analytical continuation $b\to i\beta$ does yield continua starting at scaling dimensions $X^{\mathrm{eff},R}(m,w)\equiv X^{\mathrm{eff}}_{(m,w)}(\varphi=\pi)\in\mathbb{N}+\frac{1}{4}$ in the Ramond sector Saleur and Schomerus (2007). Note, however, that this proposal does not capture the primary fields with integer scaling dimension $X^{\mathrm{eff},R}$ (i.e. $m+w$ odd in (8)), in particular the $sl(2|1)$ singlet state with $X^{\mathrm{eff},R}=0$. Since modular invariance appears to preclude the appearance of this singlet on its own it has been argued in Ref. Saleur and Schomerus (2007) that this state is an artifact of the lattice model which disappears in the continuum limit once the partition function is properly normalized.

Here we will not try to extend this picture to describe the continuous spectrum of critical exponents of the anisotropic superspin chain. Instead we focus on another characteristic feature of certain conformal field theories with non-compact target space, i.e. the appearance of a discrete component of the spectrum. To be specific we study the spectrum of the $U_{q}[sl(2|1)]$ superspin chain as a function of the twist $\varphi=\alpha+\pi$ which, in the continuum limit, corresponds to the spectral flow between the Neveu-Schwarz sector and the Ramond sector.

The ground state of the former is found in the sector with quantum numbers $(b,j_{3})=(0,0)$. For even $L$ it is realized in the XXZ subspace of this sector and the corresponding Bethe root configuration can be mapped to that of the singlet ($j_{3}=0$) ground state of the periodic spin-$1$ chain. Its effective scaling dimension is $X^{\mathrm{eff},NS}(0,0)\equiv X^{\mathrm{eff}}_{(0,0)}(\varphi=0)=-\frac{1}{4}$ (leading to an effective central charge $c^{\mathrm{eff}}=3$). For odd $L$ the lowest state in the NS sector is doubly degenerate. One of the Bethe root configurations for this state consists of $N_{\pm}=(L\pm 1)/2$ type-$(\pm)$ strange strings, i.e. $\Delta N=1$. Its scaling dimension exhibits a logarithmic dependence on the system size, $X^{*}=-\frac{1}{4}+O((\Delta N/\log L)^{2})$ Essler et al. (2005); Frahm and Martins (2011), consistent with the emergence of a continuum of levels starting at $X^{\mathrm{eff},NS}(0,0)=-\frac{1}{4}$ in the thermodynamic limit.

Adiabatically changing the twist angle we can follow this state under the spectral flow. For $|\varphi|<\varphi_{c}=\pi/k$ its scaling dimension $X^{*}_{(m,w)=(0,0)}(\varphi)$ stays within the continuum above $X^{\mathrm{eff}}_{(0,0)}(\varphi)$. As the twist approaches $\pm\varphi_{c}$ the strange string with largest roots goes to $\infty$. Beyond $\pm\varphi_{c}$ the root configuration changes and the finite size scaling dimension of the state deviates from (8). Unlike the higher excitations ($|\Delta N|>1$) within this class of Bethe states it splits off from the emerging continuum above $X^{\mathrm{eff}}_{(0,0)}(\varphi)$, see Fig. 1. Based on our finite size data we conjecture the following $\varphi$-dependence of the scaling dimension of the scaling dimension:

[TABLE]

where $\varphi_{c,2}=\pi-\frac{\pi(k-1)}{k(2k-1)}$.

Hence we find that one state from the continuum above $X^{\mathrm{eff},NS}(0,0)$ evolves, under the spectral flow $\varphi=0\ldots\pi$, to a discrete level with dimension

[TABLE]

in the Ramond sector. The Bethe root configuration for this state for $\varphi=\pi$ consists of $(L-1)/2$ of the usual $2$-strings (consisting of two complex conjugate rapidities from the same level Takahashi and Suzuki (1972)) and, in addition, one single root at $\infty$ on either level, i.e.333This configuration has already been observed for $L=3$ in Ref. Frahm and Martins (2011) but not considered further.

[TABLE]

Note that there are strong logarithmic finite size corrections to scaling to (10), similar as for the states in the continuum, see Figure 2.

Increasing the twist further we find that $X^{*}_{(0,0)}(\varphi)=X^{*}_{(0,0)}(2\pi-\varphi)$. For $\varphi>2\pi-\varphi_{c,2}$ this level coincides with

[TABLE]

and disappears in the continuum above $X^{\mathrm{eff}}_{(0,-2)}(\phi)$ at $\varphi=2\pi-\varphi_{c}$.

Concluding our analysis of the scaling dimensions under the spectral flow we note that starting from the lowest state in the continuum above $X^{\mathrm{eff}}_{(1,-1)}(\varphi=\pi)$ (for even $L$) we have observed another discrete level in the spectrum of the superspin chain

[TABLE]

Let us now analyze the quasi momentum of the state with dimension $X^{*}$ (9) identified above. An operator similar to (5) has first been introduced for a $Z_{2}$-staggered six-vertex model where its (real) eigenvalues have been identified with the quantum number parametrizing the spin $j$ of the $SL(2,\mathbb{R})$ affine primaries from the continuous series, $j=-1/2+iK$ Ikhlef et al. (2012). Based on this identification the staggered six-vertex model has be shown to be described by the $SL(2,\mathbb{R})/U(1)$ Euclidean black hole CFT in the continuum limit.

For the superspin chains considered here the quasi momentum allows to label the continua of scaling dimensions in a similar way Frahm and Hobuß : in fact starting with the lowest state in the NS sector but outside of the XXZ subspace the corresponding eigenvalue $K$ is found to take real values for $|\varphi|<\varphi_{c}$, see Figure 3.

The amplitude vanishes as $1/\log L$ in the thermodynamic limit. This agrees with the density of states observed in the continua Essler et al. (2005); Frahm and Martins (2011).

For $|\varphi|>\varphi_{c}$, however, i.e. as the discrete level (9) leaves the continuum of scaling dimensions, the quasi momentum becomes imaginary with a linear dependence on the twist angle $\varphi$

[TABLE]

(we have chosen the branch of the logarithm in (5) such that $\mathrm{Im}(K)\in[0,(k-1)/2]$.) Corrections to scaling in this expression are small which allows to observe this behaviour already for $L=3$, see Figure 3. As for the scaling dimension (9) the $\varphi$-dependence of the quasi momentum changes when the twist is increased beyond $|\varphi|=\varphi_{c,2}$: while $K$ remains purely imaginary its value saturates at

[TABLE]

Note that $\mathrm{Im}(K^{*}(\pi))$ is smaller than the maximum $\gamma/(\pi-2\gamma)=(k-1)/2$ which is possible according to the definition (5).

The emergence of a discrete level out of the continuum at finite twist is strongly reminiscent to the situation in the $a_{2}^{(2)}$ spin chain Vernier et al. (2014): in the black hole CFT describing the low energy behaviour of this lattice model it is understood as a consequence of the inclusion of principal discrete representations of $SL(2,\mathbb{R})$ as Kac-Moody primaries in the coset theory. For operators corresponding to normalizable states in the parent WZNW theory of the model the spin of these representations (related to the momentum along the non-compact direction of the target space, an infinite cigar for the $SL(2,\mathbb{R})/U(1)$ sigma model) is restricted to values $j\leq-1/2$. To satisfy this bound a finite, non-zero twist has to be applied Hanany et al. (2002); Ribault and Schomerus (2004). For the bosonic $SL(2,\mathbb{R})/U(1)$ coset at level $k$ the spectrum of discrete representations also needs to be truncated by the ’unitarity bound’ $j\geq-(k-1)/2$ to guarantee non-negative conformal weights Dijkgraaf et al. (1992); Maldacena and Ooguri (2001); Hanany et al. (2002).

The appearance of the discrete level $X^{*}$ (9) in the spectrum of the superspin chain at twist $\varphi_{c}$ accompanied with the change of quasi momentum from real to imaginary can be interpreted in a similar way: states in the continuum of scaling dimensions have real $K$, the discrete levels are characterized by imaginary quasi momentum. The observed bound $\mathrm{Im}(K)\geq 0$ for states in the lattice model can be attributed to the normalizability of the corresponding primaries. Continuing (14) to $|\varphi|<\varphi_{c}$ would yield imaginary quasi momenta $-1/2\leq\mathrm{Im}(K)<0$ which is not realized in the spectrum of the superspin chain. As for a restriction of $\mathrm{Im}(K)$ from above (similar to the unitarity bound in the string theory) our data for the lattice model do not provide a conclusive answer. We would need to see a level being ’absorbed’ by the continuum at such a bound under the spectral flow (as is the case for the $a_{2}^{(2)}$ spin chain Vernier et al. (2014)). It would be tempting to associate this bound with $\mathrm{Im}(K)\leq(k-1)/2$ as implied by our choice of the branch for the logarithm in (5). Since the quasi momentum for the discrete level studied above saturates at the value (15) below $(k-1)/2$, however, our data cannot support this conjecture.

To summarize, we have studied the spectral flow for a staggered superspin chain with continuous and discrete components in the spectrum of scaling dimensions under a twist in the boundary conditions. Based on the exact solution of this model we identified a state from the continuous part of the spectrum in the Neveu-Schwarz sector (i.e. with anti-periodic boundary conditions for the fermionic degrees of freedom) which under variation of the twist flows to a discrete level in the Ramond sector which has not been identified previously. Levels can be attributed to the continuous or discrete part of the spectrum based on their quasi momentum. The appearance of discrete levels in the spectrum of the lattice model has been argued to be related to the normalizability of primary fields in the non-rational conformal field theory describing the continuum limit.