# Does a $d$-dimensional stat-mech theory necessarily have a $(d-1)$-dimensional quantum theory equivalence?

A $$d$$-dimensional stat-mech theory on a lattice usually can be represented by a $$d$$-dimensional tensor network. Taking a row/slice of tensors ($$M$$ tensors or sites) as the transfer matrix (MPO in 2$$d$$ or PEPO in 3$$d$$ in the tensor network language) $$\hat{T} = \sum_i\lambda_i \vert R_i\rangle \langle L_i \vert$$, the partition function can be written as $$Z=Tr({\hat{T}}^L)=\sum_i\lambda_i^{M L},$$ where $$\hat{T}$$ may not be a symmetric (Hermitian) matrix. If we can write $$\hat{T} = U e^{-\hat{H}} U^{-1}$$ with a Hermitian matrix/operator $$\hat{H}$$, we can say it has a quantum equivalent theory whose dynamics is generated by $$\hat{H}$$. This requires $$\hat{T}$$ contains only non-negative real spectra.

As the partition function $$Z$$ is a positive number, the leading eigenvalue $$\lambda_0$$ is also guaranteed to be positive when $$L$$ becomes large. It seems that from the partition function alone, one can not conclude the strong limitation to the spectra of $$\hat{T}$$. The free energy density is mostly sensitive to low energy (a few first largest ones) spectra of $$\hat{T}$$, so that it seems the thermodynamic physical quantities are determined by the low energy spectra alone. Particularly, for tensor networks, it is more convenient to discuss systems in thermodynamic limit directly, so that the practical calculation aims to find the fixed point (leading eigenvector pair) of the transfer matrix $$\hat{T}$$, which is represented by one lower dimension tensor network states.

My questions are:

1. whether the positive low energy spectra already ensures $$Z$$ to be a well-defined stat-mech theory?

2. whether there is a stat-mech theory that has no quantum theory equivalence?

3. will this be true (a stat-mech theory has a quantum theory equivalence) for stat-mech theories which have both scale invariance and rotation symmetry at least in 2d, for which an underlying conformal field theory description exists?

• Do you assume unitarity for the quantum theories? There are many known statistical mechanics models that can not be mapped to unitary quantum systems. Jan 10, 2022 at 16:02
• @MengCheng Thanks for the point. Yes, I assumed unitary quantum theories in my question. So the models you mentioned serve as counterexamples. Could you give some examples or references/links? Jan 10, 2022 at 21:37

There are many critical statistical mechanics models described by non-unitary conformal field theories. One family of examples can be obtained from Potts model. For integer $$Q>1$$, $$Q$$-state Potts model can be perfectly mapped to quantum spin chains. However, it is possible to represent the partition function as a loop model where $$Q$$ can be any positive real number (actually any complex number, but then $$Z$$ would be complex). More details about the model can be found e.g. http://www.phys.ens.fr/~jacobsen/AIMES/Potts.pdf . The $$Q\rightarrow 1$$ limit becomes bond percolation, which at the critical point is described by a CFT with vanishing central charge, and as far as I can tell has no unitary quantum mechanical mapping. The same can be said for other non-integer values of $$Q$$ (see for example https://arxiv.org/abs/1808.04380)