Dimensional reduction in quantum field theories at finite temperature and density

In this work we present two correspondences between the massless Gross-Neveu model with one or two coupling constants in $1+1$ dimensions and nonrelativistic field theories in $3+1$ dimensions. It is shown that on a mean-field level the massless Gross-Neveu model can be mapped onto BCS theory provided that translational invariance of the condensate is assumed. The generalized massless Gross-Neveu model with two coupling constants is mapped onto a quasi one-dimensional extended Hubbard model used in the description of spin-Peierls systems. It is shown that the particle hole symmetry of the Hubbard model implies self-consistency of the condensate. The dimensional reduction allows an identification of the phase diagrams of the models.

Figure 1

Phase diagram of the genGN model as derived in 21.

Figure 2

Phase diagram of the massless GN model assuming only translationally invariant phases. The continuous line denotes a second order and the dashed line a first order transition. The thin continuous lines mark boundaries of metastability.

Figure 3

Phase diagram of the massless GN model assuming only translationally invariant phases for different cutoff values (inner to outer) $\Lambda =2.5$, 4 and $\infty$. The thin continuous lines mark boundaries of metastability.

Figure 4

Slice through the Fermi surface of the quasi one-dimensional Hubbard model.

Figure 5

Second order phase transition sheet of the genGN model for the cutoff values $\Lambda =4$, 5 and $\infty$.

Figure 6

Phase diagram of the inorganic spin-Peierls cuprate ${\mathrm{CuGeO}}_3$. The fit is done with $\xi \approx 2$. The theoretical curve is fit to the scale of the data. The data is taken from 49.