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Utilizing the fact that there is a correspondence between a $d$ dimensional quantum system and a $d+1$ dimensional classical system (c.f. Trotter Decomposition), my question regards what the critical temperature of the classical system corresponds to in the quantum system.

For concreteness, consider the 2D square lattice Ising model with Hamiltonian $$H = -J\sum_{\left<ij\right>}\sigma_i\sigma_j$$ where $\sigma_i = \pm 1$ and the sum is over nearest neighbors, which has a phase transition at a nonzero temperature near $T_c/J \sim 2.2$. The Ginzburg-Landau action for the classical model in the continuum limit is

$$S = \int d^2x \left[\frac{1}{2}\left(\frac{\partial \phi}{\partial x}\right)^2 + \frac{1}{2}\left(\frac{\partial \phi}{\partial y}\right)^2 + \frac{r}{2}\phi^2 + \frac{u}{4!}\phi^4\right]$$

where $r \sim (T_c - T)$ is negative below $T_c$ and we interpret $\phi$ as the order parameter. Let's consider one of the dimensions to be the imaginary time direction of the 1D quantum Ising model, and we will allow the spatial direction to be discrete, which leads to

$$S = \int d\tau \sum_i \left[\frac{1}{2}\left(\frac{\partial \phi_i}{\partial \tau}\right)^2 + \frac{1}{2}\left(\phi_{i+1}-\phi_i\right)^2 + \frac{r}{2}\phi^2 + \frac{u}{4!}\phi^4\right]$$

We can reinterpret $\phi_i$ as the position of a particle moving in a double-well potential (for $r<0$), and further interpret this action as describing a set of particles in the same double-well potential. The pairwise interaction prefers for the particles to be nearby, indicating that there should be a state where the particles are localized in one well. In the language of the 2D model, this corresponds to the presence of a stable ordered phase below $T_c$. (Juxtapose this to the 1D classical model which does not order at any finite $T$. This model would correspond to a single particle in a double well potential which has quasi-classical instanton solutions representing tunneling between the wells. These correspond to the fact that the 1D ordered groundstate is unstable to perturbation at finite $T$. Such solutions do not arise in 1D or "quantum gas" model).

My question is, what does the critical temperature of the classical model correspond to in the quantum model? What does it mean for the quantum model that the classical model has a phase transition?

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What does it mean for the quantum model that the classical model has a phase transition?

When the classical 2d model has a phase transition, the dual 1D quantum model will also have a phase transition.

What does the critical temperature of the classical model correspond to in the quantum model?

The answer is given after the first (unnumbered) equation in the first page of the following notes.

The answer, repeated here: $N δτ = β$, where N the number of particles, $ δτ $ the infinitesimal time step for the imaginary-time path integral of the quantum partition function and $β $ the inverse temperature of the classical model.

Further details for the mapping can be found in any book on quantum phase transitions. See for example, S. Sachdev’s book, “Quantum Phase Transitions”.

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