5.3 Infinite periodic lattices
5.3.1 The infinite Hubbard chain
In Fig. 5.5 we present results for several equilibrium properties of the half-filled 1D Hubbard model as functions of the temperatureT for representative values of the Coulomb-repulsion strengthU/t. These results were obtained fromFT-LDFTin com-bination with the scaling approximation (5.33) for the correlation contributionGc[γ12] to the free-energy functional using a 7-site ring as reference system. The 7-site ring has been chosen, since it is the largest 1D reference system for which the compu-tations within our current implementation ofFT-LDFTare feasible. Furthermore, in order to account for long-range correlations a sufficiently large reference system is required, such that one generally expects to obtain more accurate results if a larger reference system is used. Further below we will quantify to which extent the equilib-rium properties obtained in the framework ofFT-LDFTdepend on the choice of the reference system. In order to assess the accuracy of theFT-LDFT results shown in Fig.5.5, we compare them with the exact finite-temperature solution of the 1D Hub-bard model [56]. The free energy F, shown in Fig. 5.5 (a), results directly from the minimization of the free-energy functional (5.52) and the comparison with the corre-sponding exact analytical solution demonstrates thatFT-LDFTis remarkably accurate in the whole range of temperaturesT and Coulomb-repulsion strengthsU/t. Indeed, the deviation between the exact andFT-LDFTresults is smaller than the width of the individual lines on the scale used in Fig.5.5 (a). The temperature dependence ofF is in fact reproduced to such a high level of detail that accurate results are also obtained for its derivatives with respect to the Coulomb-repulsion strength and the temperature.
Therefore, we obtain accurate results also for the average number of double occupa-tions D = ∂F/∂U, the entropyS = −∂F/∂T, and the specific heatCV = T∂S/∂T, which are shown in Figs.5.5 (d)–(f).
For the double occupationsD, shown in Fig.5.5 (d), we find a non-vanishing value in the ground state (T = 0), which is the result of the competition between electronic delocalization, driven by hybridization, and localization, which reduces the local
0 1 2 3
−F/Nat
(a)
1D
n↑=n↓= 1/2
U/t=4
20
−0.6
−0.4
−0.2 0.0 0.2 0.4 0.6
E/Nat
(b)
U/t=20
4 FT-LDFT
exact [56]
0.2 0.4 0.6 0.8 1.0
−K/Nat
U (c)
/t= 5 4 6 78 10 12 16 20
0.00 0.05 0.10 0.15
D/Na
(d) U/t=4
5 6 78
10 12 16 20
0.0 0.5 1.0 1.5 2.02.0 2.52.5 3.0
kBT/t
0.00 0.25 0.50 0.75 1.00 1.25
S/NakB
(e) U/t=4
20
U/t
45 6
78 10
1216 20
0.0 0.5 1.0 1.5 2.02.0 2.52.5 3.0
kBT/t
0.0 0.1 0.2 0.3 0.4 0.5
CV/NakB
(f) 5 U/t=4 6
7 8
10 12 16 20
Figure 5.5:Equilibrium properties of the half-filled infinite 1D Hubbard chain as functions of the temperatureT for representative values of the Coulomb-repulsion strengthU/t. Results obtained byFT-LDFTin combination with the scaling approximation (5.33) using a 7-site ring as reference system (full curves) are compared with the exact solution of Jüttneret al.[56]
(open circles): (a) free energyF, (b) total energyE, (c) kinetic energyK, (d) average number of double occupationsD, (e) entropyS, and (f) specific heatCV.
Coulomb energy. Thus, the average number of double occupations in the ground state decreases monotonously as the Coulomb-repulsion strengthU/t increases. Clearly, the same is also true for any finite temperature, since the creation of double occu-pations requires more energy, and thus higher temperatures, if the Coulomb repul-sions are stronger. Remarkably, the minimum ofDis not attained in the ground state, but at a finite temperature which increases slightly with increasingU/t. This ef-fect is caused by low-lying spin excitations from theantiferromagnetic (AFM)ground state which leads to a partialferromagnetic (FM) alignment of the spins, such that the number of hopping processes permitted by the Pauli principle decreases. As a result, the charge fluctuations which give rise to the binding energy in the ground state are partly suppressed as the temperature increases, which is accompanied by a decrease of the double occupations generated by them. As the temperature increases beyond a critical threshold, charge excitations in the upper Hubbard band give rise to a renewed increase ofD and thus to the formation of a minimum at a finite temper-ature which increases withU/t. Finally, in the high-temperature limitkBT/U → ∞ the electronic motion becomes essentially uncorrelated, and thus one hasD = Na/4 regardless of the Coulomb-interaction strength (U kBT). Notice thatFT-LDFT re-produces the decrease of the double occupations due to the spin excitations from the AFMground and the corresponding minimum inD very accurately for all values of the Coulomb-interaction strengthU/t. Also the increase ofDin the high-temperature regime is very well reproduced. In fact, the relative error∆D = |Dex−DFT-LDFT|/Dex in the double occupations obtained within the framework ofFT-LDFTis in average1 just about 2% and always below 8% in the complete range of parameters shown in Fig.5.5 (d).
The temperature dependence of the kinetic energy K = −tNaz γ12, shown in Fig. 5.5 (c), is obtained directly from the minimization of the free-energy func-tional (5.52). In the ground state we find the minimal value ofK, which is the result of theNNcharge fluctuations in theAFMground state. These charge fluctuations are accompanied by double occupations, which have more impact on the energy if the Coulomb repulsions are stronger. Thus, we find a monotonously decreasing kinetic energy in the ground state and also at any finite temperature asU/t increases. As already discussed above, increasing temperatures give rise to the excitation of low-lying collective spin waves from theAFMground state, causing a partial suppression of the ground-state charge fluctuations, and thus a rather rapid increase ofK is ob-served in the low-temperature regime (kBT . t/2). Notice that K increases more rapidly in the low-temperature regime if the Coulomb repulsions are stronger. This is
1Here and in the following the average error is calculated as the deviation between the results of FT-LDFTand the given benchmark, averaged among the whole dataset. TheFT-LDFTresults are always distributed equally spaced on the scale shown and the benchmark data are interpolated to the same regular grid in order to compute the errors.
due to the fact that the low-lying spin excitations are described by theAFM Heisen-berg model (2.52) with coupling constant J = 2t2/U, and thus the bandwidth of the spin waves narrows asU/t increases. As the temperature further increases, anti-bonding Bloch-states are progressively excited and finally, in the high-temperature limitkBT/t → ∞, bonding and antibonding Bloch-states are equally occupied, result-ing in an unbound state withγij = 0 for alli , j. Thus, the kinetic energy vanishes for all values ofU kBT in this limit. The comparison with the corresponding exact results demonstrates that the increase of the kinetic energy caused by the low-lying spin excitations from theAFMground state, as well as the transition to an unbound state as the temperature increases is very accurately reproduced withinFT-LDFTin the complete range of temperatures and Coulomb-interaction strengths. The relative deviation∆K = |(Kex−KFT-LDFT)/Kex|between the kinetic energy obtained from FT-LDFTand the exact solution is in average about 4%, and a maximum deviation of 8%
is found for strong couplingU/t = 20 atkBT = 0.2t. Clearly, the temperature depen-dence of the total energyE = K +U D, shown in Fig.5.5 (b), results from the kinetic energy and the double occupations already discussed. Therefore,FT-LDFTyields very accurate results also for the total energyEin the complete range from the ground state to the high-temperature limit and for all values of the Coulomb-interaction strength.
The temperature dependence of the entropy S is shown in Fig. 5.5 (e). We find a rapid increase of the entropy at low temperatures if the Coulomb repulsion is strong (U/t & 10), which results from the excitation of low-lying collective spin waves from the AFM ground state. A further entropy increase at higher tempera-tures is the result of charge-transfer excitations from the lower to the upper Hubbard-band, which involves the creation of double occupations. For weaker Coulomb repul-sions (U/t . 8), the energy scales of spin and charge excitations have a noticeable overlap, such that a rather continuous increase of the entropy is observed in this case.
Notice, however, that FT-LDFTfails to reproduce the linear entropy increase in the regime of very low temperatures (kBT . 0.2t).2 This is an artefact of the finite size of the reference system and results from the gap between the ground state and the lowest-lying excited states, as well as from degeneracies of the ground state. How-ever, forkBT & 0.2t the entropy obtained in the framework of FT-LDFTfollows the exact analytical result very closely and an average relative deviation of 0.6% is found, which never exceeds 6% for all values of the Coulomb-repulsion strength shown in
2Notice that theNNhopping integraltin transition metals is typically of the order 0.1–0.5 eV, such that comparable temperaturesT ∼ t/kB = 1000–6000 K are actually quite large for typical ex-perimental setups. However, here and in the following the term “low temperature” refers to tem-peratureskBT which are small when compared to the energy scales specified by the model under consideration, such as the bandwidthw ∼4dtfor a lattice inddimensions, the Coulomb-repulsion strengthU, or the effective exchange-coupling constantJ =2t2/U which is relevant in the strongly-interacting Heisenberg limit.
Fig.5.5 (e)as long askBT > 0.2t.
The specific heatCV, shown in Fig. 5.5 (f), displays a most interesting tempera-ture dependence. For intermediate Coulomb-repulsion strengths (U/t . 5) we find a broad peak in the specific heat which splits into two separate peaks asU/tincreases.
The peak appearing at low temperatures corresponds to spin excitations in the lower Hubbard-band, while the one at higher temperatures is caused by charge fluctua-tions, which give rise to increasing Coulomb interactions. The low-lying spin excita-tions are governed by theAFMHeisenberg model (2.52) with exchange-coupling con-stantJ =2t2/U. Therefore, we expect to find the low-temperature peak in the specific heat, which marks the Néel transition from theAFMground-state to theparamagnetic (PM)phase, at a temperaturekBTN ∝ t2/U which scales like the effective exchange-coupling constant J with the Coulomb-interaction strength. In fact, in the strong-coupling limitU/t → ∞we expect to find the low-temperature peak in CV at the temperaturekBTN = J =2t2/U where the transition between theAFMandPMphases occurs in the one-dimensional spin-1/2 Heisenberg model [119, 120]. Figure 5.6 (a) shows the Néel-transition temperatureTN inferred from the low-temperature peak in the specific heatCV as a function of the Coulomb-repulsion strengthU/t. The com-parison with the Néel-transition temperature derived from the exact solution [56]
reveals thatFT-LDFTin combination with the scaling approximation (5.33) not only reproduces the qualitative behaviorTN ∝t2/U correctly, but also yields very accurate values for the transition temperature, such that the relative error inTN is in average about 13% and never exceeds 14.3% for all values shown in Fig.5.6 (a). Furthermore, the convergence to the asymptotic behaviourkBTN =2t2/U in the strongly-correlated limitU/t → ∞is very well reproduced withinFT-LDFT.
Another interesting feature of the specific heat is the almost unique high-temperature crossing point of the curves forU .8, which occurs atkBT '1.4t. This nearly universal crossing point has attracted much attention in the past [121, 122], since it not only occurs in the Hubbard model but has been also observed experimen-tally in the specific-heat curvesCV(T)of strongly correlated systems at different pres-sures. This includes normalfluid3H as well as heavy-fermion systems such as CeAl3 and UBe13 [123–126]. The comparison with the corresponding exact results shown in Fig.5.5 (f)demonstrates thatFT-LDFTreproduces the nearly universal crossing point of the specific-heat curves very accurately for the 1D Hubbard model.
For U/t & 6 we find a second peak in CV at high temperatures, which corre-sponds to charge excitations across the Hubbard gap. In Fig.5.6 (b)the corresponding charge-excitation temperatureTC, inferred from the position of the high-temperature peak, is shown as a function of the Coulomb-repulsion strengthU/t. Since the high-temperature peak inCV is caused by charge excitations in the upper Hubbard-band, which lead to the creation of double occupations, we expect that the charge-excitation temperatureTC scales linearly with the Coulomb-repulsion strengthU. In fact, the
5 10 15 20
U/t
0.10 0.15 0.20 0.25 0.30 0.35 0.40
kBTN/t
FT-LDFT (a)
exact [56]
2t/U
5 10 15 20
U/t
1 2 3 4
kBTC/t
(b)
1D
n↑=n↓= 1/2
0.21U/t
Figure 5.6: (a) Néel-transition temperatureTN and (b) charge-excitation temperatureTC of the half-filled infinite 1D Hubbard chain as functions of the Coulomb-repulsion strengthU/t. The critical temperatures are inferred, respectively, from the positions of the low- and high-temperature peaks in the specific heatCV. Results obtained byFT-LDFTin combination with the scaling approximation (5.33) using a 7-site ring as reference system are indicated by blue crosses, while red plus-symbols correspond to the critical temperatures derived from the exact solution of Jüttneret al.[56]. The gray solid line in (a) marks the asymptotic behaviourkBTN = 2t2/U of the Néel-transition temperature in the strongly-correlated limitU/t → ∞, as inferred from the 1D Heisenberg model [119,120]. The gray dashed line in (b) marks the corresponding strong-correlation asymptotekBTC = 0.21U of the charge-excitation temperature, which is derived from the specific heat of the doublons (see AppendixF).
comparison with the temperatureTC derived from the exact finite-temperature solu-tion of the 1D Hubbard-model [56] reveals thatFT-LDFTnot only yields the expected behaviorTC ∝U but also reproduces the value ofTCalmost exactly. The relative error inTCis in average only 0.6% and never exceeds 1.5% in the complete range of parame-ters shown in Fig.5.6 (b). In the strongly-correlated limitU/t → ∞, where the energy scales of spin and charge excitations are widely separated, the dominant contribu-tion to the specific heat at temperatureskBT ∼U results from the charge fluctuations and the accompanying fluctuations in the average number of double occupations. In this case, we can infer the asymptotic behaviourkBTC = 0.21U from the structure-independent specific heat of the doublons, which we have calculated in AppendixF.
From Fig.5.6 (b)we conclude that the charge-excitation temperatureTCof the 1D Hub-bard model converges rapidly to the strongly-correlated behaviorkBTC = 0.21U and thatFT-LDFTis able to reproduce this rapid convergence with astonishing accuracy.
It is most remarkable that FT-LDFT in combination with the scaling approxima-tion (5.33) is able to reproduce the gradual separaapproxima-tion of spin and charge degrees
0 1 2 3
kBT/t
0.02 0.04 0.06 0.08 0.10 0.12
D/Na
(a)
1D
n↑=n↓=1/2 U/t = 8
exact
0 1 2 3
kBT/t
0.0 0.1 0.2 0.3 0.4 0.5
CV/NakB
(b)
exact
Narf
2 3 4
5 6 7
Figure 5.7:Equilibrium properties of the half-filled infinite 1D Hubbard chain withU/t =8 obtained fromFT-LDFTin combination with the scaling approximation (5.33) using 1D rings withNarf =2–7 sites as reference systems. Results are shown for the temperature dependence of (a) the average number of double occupationsD and (b) the specific heatCV. The thick black curves mark the corresponding exact results for the infinite 1D Hubbard chain [56].
of freedom as the Coulomb-repulsion strength increases. In fact, this subtle effect of strong electronic correlations has, to our knowledge, not been reproduced in the framework ofDFTbefore, neither on the qualitative and even less on the quantitative level. Clearly, it is one of the major advantages ofFT-LDFTin combination with the scaling approximation (5.33) that the approximate functionalGc[γ12]is derived from aninteractingsystem which already incorporates the effects of electronic correlations, such as the separation of spin and charge degrees of freedom. In contrast, most ap-proximations in conventionalDFTare based on the homogeneous electron gas, which does not incorporates most of the crucial effects of electronic correlations.
Before we apply the methods ofFT-LDFTto the Hubbard model in two and three dimensions, let us briefly assess the importance of the reference system and investi-gate how the equilibrium properties obtained from the scaling approximation (5.33) are influenced by the choice of the reference system. To this aim we compare in Fig.5.7 the average number of double occupationsD and the specific heatCV of the half-filled infinite 1D Hubbard chain withU/t = 8 obtained from the scaling ap-proximation (5.33) using 1D rings with Narf = 2–7 sites as reference systems. From Fig.5.7 (a)we conclude that the average number of double occupations depends rather weakly on the choice of the reference system in the whole range from the ground state to the high-temperature limit. As expected, the most noticeable deviations are observed in the low-temperature regime (kBT . t), where correlation effects play a
crucial role. At higher temperatures, the dependence ofD on the choice of the refer-ence system becomes rather negligible and the average number of double occupations of the infinite 1D Hubbard chain is reproduced very accurately. The situation is dif-ferent in the case of the specific heatCV shown in Fig. 5.7 (b), where the extent of the low-temperature peak strongly depends on the chosen reference system. This is, however, to be expected, since the specific heatCV = −T ∂2F/∂T2 is a second-order derivative of the free-energy and thus depends sensitively on minor changes of the free-energy functional (5.52) in the vicinity of the minimum. Notice, however, that the Néel-transition temperatureTN is nevertheless fairly well reproduced with a maximal relative error of 28% when a 5-site ring is used as reference system. The only excep-tion to this occurs forNarf = 3, where the low-temperature peak in the specific heat degenerates into a shoulder. For higher temperatures (kBT & t) we observe a rapid converge to the exact specific heat of the infinite 1D Hubbard chain as the size of the reference system increases, such that the relative error in the charge-excitation tem-peratureTC never exceeds 6.8% if rings with three or more sites are used as reference systems.
We conclude that FT-LDFTin combination with the scaling approximation (5.33) tends to yield more accurate results as the size of the reference system increases.
However, it should be noted that a reference system whose symmetries and local topology matches those of the target system should always be preferred.