Draft:The c-d conjecture

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In an arXiv preprint.., José Ignacio Latorre and Germán Sierra made the following conjecture about the upper bound of the central charge for one-dimensional quantum critical lattice Hamiltonians with nearest-neighbor interactions:

• If the local Hilbert space dimension of the lattice model is $d$, the maximal central charge that the model can reach is $c_{\mathrm{m}\mathrm{a}\mathrm{x}}=d-1$.

Examples

The currently known examples are consistent with this conjecture.

The upper bound is saturated for the SU($n$) Uimin-Lai-Sutherland model, whose low-energy effective theory is the SU($n$) level 1 Wess-Zumino-Witten model. The local Hilbert space dimension of the lattice model is $d=n$, and the SU($n$) level 1 Wess-Zumino-Witten model has central charge $c=n-1$.

The Reshetikhin models are a family of integrable models with SO($n$) symmetric nearest-neighbor interactions. The local Hilbert space dimension of the Reshetikhin models is $d=n$, which transforms under the vector representation of SO($n$). These models are critical and their low-energy effective theory is the SO($n$) level 1 Wess-Zumino-Witten model with central charge $c=n/2$

The spin-$s$ XXX models are a family of integrable models with SU($2$) symmetric nearest-neighbor interactions. The local Hilbert space dimension of the spin-$s$ XXX models is $d=2s+1$, which transforms under the spin-$s$ irreducible representation of SU($2$). These models are critical and their low-energy effective theory is the SU($2$) level $2s$ Wess-Zumino-Witten model with central charge $c=\frac{3s}{s+1}$.

The $Z_Q$ parafermion models are a family of integrable self-dual models with $Z_Q$ symmetric nearest-neighbor interactions. The local Hilbert space dimension of the $Z_Q$ parafermion models is $d=Q$. These models are critical and their low-energy effective theory is the $Z_Q$ parafermion conformal field theory with central charge $c=\frac{2(Q-1)}{Q+2}$. For $Q=2$ and $3$, the models correspond to the critical Ising model and the critical $Z_3$ Potts model, respectively. For $Q=4$, the model corresponds to a special point of the critical Ashkin-Teller model