Finite Volume Effects and Phase Transition

In a first step, the numerical results for the spin density wave constellation will now be discussed in detail, subsequently corresponding results for the CDW state are represented. In the following the temperature is set toβ = 20κ⁻¹ and the chemical potential is fixed to µ = 0 in order to compare with recent results from Ref. [54]

for the wholeU-V coupling plane and especially with Ref. [84] and Ref. [23] for the case of vanishing nearest neighbor potentialV = 0.

With the current numerical setup we are able to reach huge lattice sizes in com-parison to calculations on the basis of Monte Carlo methods. Since the lattice size in real space determined byN² reflects the real size of the graphene sheet, we expect equal results for equal lattice sizes here. However, correction terms due to the finite lattice in the imaginary time formalism might differ. Therefore we should make sure to reach the limitN_t→ ∞ and preferablyN → ∞ likewise.

In Fig. 6.7 the order parameter in dependence of the on site coupling U for zero nearest-neighbor coupling V = 0 is shown for a fixed time lattice Nt = 500 for different lattice sizes in momentum spaceN = 18−216.

As one can see in Fig. 6.7 we found almost no volume effects for lattice sizes greater or equal to N = 54 for this setup. In contrary, lattice sizes of N = 18 are shown to be way too small in order to estimate the phase transition. As a good tradeoff, in order to minimize finite volume effects concerning the spatial lattice and similarly choose a possibly smallN to allow for larger time lattice sizes Nt, we fix the number of unit cells toN = 90 in the following.

Figure 6.8: Here the order parameter (Eq. (6.29)) for different lattice sizes in imag-inary time direction Nt is shown for a fixed spatial lattice withN = 90 and a fixed inverse temperature ofβ = 20κ⁻¹. The obtained values were extrapolated toN_t→ ∞as shown in Fig. 6.9 and fitted to a polynomial function of first-order.

6.4 RESULTS WITHTRIVIALWAVEFUNCTIONRENORMALIZATIONS

In Fig. 6.8 an analogical consideration is represented for various lattice sizes in time direction between Nt= 500 andNt= 2300.

Here the results for different lattice sizes N_t were additionally extrapolated to Nt→ ∞ by a polynomial least-square fit of second degree,

f(x) =a(1/N_t)²+b(1/N_t) +c .

Such a fit is exemplary shown in Fig. 6.9 forU = 3.8κ, where the extracted order parameter is plotted against the inverse lattice size 1/N_t. The value forN_t→ ∞is then finally determined by the parameterc, for a fitted intercept smaller than zero the order parameter was accordingly set to zero.

Figure 6.9: The polynomial fit of second degree to the order parameter is illustrated in dependence of the inverse number of lattice points in imaginary time direction 1/N_t. Exemplary, an on-site coupling ofU = 3.8κwas chosen.

These values extrapolated to N → ∞ are represented as black dots in Fig. 6.8 where they were additionally fitted to a straight line. The interception with the x-axis consequently determines the critical coupling forV = 0 to be

U_c= 3.51 κ . (6.30)

This is in rough accordance to the results from Ref. [84] and Ref. [23] were a critical coupling between U_c ∼3.7−3.8 κ was found. The deviation could still be caused by the vertex approximation of the applied Dyson-Schwinger framework, but also finite volume and temperature effects are not excluded. As it was shown in the last section, a Ball Chiu vertex ansatz would not change the result, a vertex dressing as discussed in Sec. 4.4 should therefore be taken into account. However, the achieved accordance with the results from Quantum Monte Carlo algorithms

6 FREQUENCYDEPENDENTSOLUTION

within the frequency dependent ansatz is more than satisfactory for the considered SDW case and really overcomes the observed overestimation of the critical on-site coupling as obtained in the last chapter with the static approach of the screening function.

Figure 6.10: The semimetal to antiferromagnetic insulator transition is shown as contour plot forβ= 20κ⁻¹within theU-V coupling plane for a spatial lattice size fixed to N = 90.

Figure 6.11: The order parameter is plotted in theU-V coupling plane for the case of a CDW configuration. Analogically to Fig. 6.11, the calculations were made for β= 20κ⁻¹ for a fixed spatial lattice size N = 90, whereas the imaginary time lattice has been extrapolated toN_t→ ∞for lattice sizes between N_t= 100−1000.

6.4 RESULTS WITHTRIVIALWAVEFUNCTIONRENORMALIZATIONS

As one can conclude from Fig. 6.8 and Fig. 6.9 the calculation for such large lattices in imaginary time direction is not necessary for the considered temperature regime.

An analog evaluation for time lattices betweenN_t= 100 andN_t= 1000 was therefore applied to the whole coupling regime characterized by U and V. The received phase diagram, where the order parameter as given in Eq. (6.29) is represented in dependence of the control parametersU and V, is shown in Fig. 6.10.

This results finally show up a very nice qualitative as well as quantitative accor-dance with the outcomes from Ref. [54]. The great advantage of the Dyson-Schwinger framework is the accessibility of the coupling regime beyond V = 1/3 U since the theory exhibits no sign problem here. Apart from that, the numerical setup estab-lished in Sec. 3.5 enables the calculation for very large lattice sizes.

In order to eventually access the phase transition with regard to both phases, the SDW as well as the SDW phase, an anolog consideration was made for the numerical system fixed to the CDW case. The result is represented in Fig. 6.11. In contrary to the SDW state the phase transition concerning the CDW state seems to be in nice quantitative and qualitative agreement with the results obtained in the static approximation.

Figure 6.12: The resulting phase diagram is plotted on the extended Hubbard plane.

The critical coupling pairs as well as the indicated errorbars were ob-tained with the same techniques as already pointed out in Sec. 4.3.

All in all, the considered screening effects from the π−bands itself seem to be of much higher influence regarding the SDW case. In complete analogy to Fig. 5.9 the presented phase transitions are summarized in Fig. 6.12 where the individual areas of the SDW, CDW and SM phase are sketched. The regime of coexistence is shaded

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in gray and limited by the spinodal lines as before. The first-order line,V = 1/3U, separating CDW and SDW constellations is still valid as shown in Sec. 4.4.