Detecting phases in one-dimensional many-fermion systems with the functional renormalization group

The functional renormalization group (FRG) has been used widely to investigate phase diagrams, in particular the one of the two-dimensional Hubbard model. So far, the study of one-dimensional models has not attracted as much attention. We use the FRG to investigate the phases of a one-dimensional spinless tight-binding chain with nearest and next-nearest neighbor interactions at half filling. The phase diagram of this model has already been established with other methods, and phase transitions from a metallic phase to ordered phases take place at intermediate to strong interactions. The model is thus well suited to analyze the potential and the limitations of the FRG in this regime of interactions. We employ flow equations that are exact up to second order in the interaction, which implies that we take into account the frequency dependence of the two-particle vertex as well as the feedback of the dynamic self-energy. For intermediate nearest neighbor interactions, our scheme captures the phase transition from a metallic phase to a charge density wave with alternating occupation. The critical interaction, at which this transition occurs, is underestimated due to our approximations. Similarly, for intermediate next-nearest neighbor interactions, we observe a transition to a charge density wave with occupation pattern $...00110011...$. We show that taking into account a feedback of the two-particle vertex in the flow equation is essential for the detection of those phases.

Figure 1

Sketch of the one-dimensional tight-binding chain with the hopping $t$, the nearest neighbor interaction $U$, and the next-nearest neighbor interaction $U^{\prime }$.

Figure 2

Phase diagram of the model Eq. (1) as reported by Mishra et al. The above plot was sketched after reading off the approximate location of the transition lines from Fig. 1 of Ref. [28]. The full lines indicate the phase transitions, see the text for further details. The sketches indicate the fermion distribution on the sites in the CDW phases or the strength of the bonds in the BO phase. The dashed lines show the cuts of the phase diagram studied in this paper.

Figure 3

Flow equations for the self-energy ${\mathrm{\Sigma }}^{\mathrm{\Lambda }}$ and the two-particle vertex ${\mathrm{\Gamma }}^{\mathrm{\Lambda }}$, neglecting the three-particle vertex.

Figure 4

Difference between the ED and the FRG result for the expectation value $\langle c_2^{†}{c}_3\rangle$ for a system with $N=10,L=2,U^{\prime }=0$.

Figure 5

Phase transition to the CDW-I, comparison between the static and the dynamic result for $U^{\prime }=0,N=82,L=10,S=0.001$.

Figure 6

Phase transition to the CDW-I for different system sizes, $U^{\prime }=0,S=0.001$.

Figure 7

Comparison of different choices of the small initial symmetry breaking term. $U^{\prime }=0,N=82,L=10$.

Figure 8

Maximum of the absolute value of all components of $P$ and $X$ without an initial symmetry breaking term. For $U$ larger than shown, it is not possible to solve the flow equations down to ${\mathrm{\Lambda }}_{\text{f}}=0$.

Figure 9

Comparison of the results for the phase transition from the LL phase to the CDW-I for different FRG feedback schemes. $U^{\prime }=0,N=42,S=0.001$. The inset shows the maximum of all components of $P$ and $X$.

Figure 10

Comparison between the real-space FRG and the momentum space scheme with $S=0.001$. For the real space FRG, the system parameters are $N=258,L=10$.

Figure 11

Phase transition LL-CDW-I for $U^{\prime }=0.5,S=0.001$.

Figure 12

Phase transition to the CDW-II, see text for further explanations. OBC, $U=0,N=40,L=10$.

Figure 13

Phase transition from the LL to the BO phase, $N=42,L=10,S=0.001$. The triangles pointing up indicate color coded the transition LL–BO phase, and the triangles pointing down BO–CDW-II according to Ref. [28]. For $U^{\prime }$ larger than shown in the different curves, the flow equations cannot be integrated down to ${\mathrm{\Lambda }}_{\text{f}}=0$.