Electronic Entropy in Green's-Function Calculations at Finite Temperatures

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II. THEORETICAL FORMALISM (a) A. Electron densities and energies

The majority of usual quantities in ab initio electron theory of solids, such as the valence contribution to electron densities in the real space or to the sum of occupied C one-particle eigenvalues, can be written as integrals over the real energy axis

+∞ Q = D(E)f(E)dE, (1) C1 Z−∞ where the function D(E) is closely related to a projected or total density of states (DOS) and E is a real energy variable. The function f(E) denotes the Fermi-Dirac dis- tribution f(E)= {1 + exp[β(E − µ)]}−1, where µ is the chemical potential and β refers to the reciprocal value (b) −1 of the ﬁnite temperature T as β = (kBT ) , where kB is the Boltzmann constant. In Green’s-function techniques, C C the function D(E) can be written as -- +

i C> D(E) = lim [Γ(E + iε) − Γ(E − iε)] , (2) ε→0+ 2π C C where the function Γ(z) of a complex variable z depends linearly on the resolvent G(z) = (z − H)−1 of the un- C derlying one-particle Hamiltonian H. Note that H and C< G(z) are operators (matrices), whereas D(E) and Γ(z) are usual real and complex functions, respectively. We C -- C+ assume that the function Γ(z) is analytic everywhere in the complex plane with exception of real energies belong- ing to the spectrum of H. The standard transformation of Eq. (1) into a complex integral starts from the form 1 FIG. 1. (a) The integration contours C and C1 in Eq. (3) and Q = Γ(z)f(z)dz, (3) Eq. (5), respectively. The horizontal double line marks the 2πi ZC valence part of the spectrum of Hamiltonian H, the crosses denote the Matsubara points zk and the squares denote the where the complex contour C is drawn around the va- ∗ points ZM and ZM for M = 2. (b) The integration contours lence part of the spectrum of H as shown in Fig. 1a. The C, C+, C−, C>, and C< for entropy calculation. modification of Eq. (3) rests on the well-known properties of the functions Γ(z) and f(z), such as their analyticity, −1 ∗ the periodicity of f(z) with an imaginary period 2πiβ , where the contour C1 starts at ZM and ends at ZM , see the existence of simple poles of f(z) at the Matsubara Fig. 1a, −1 points zk = µ + πiβ k, where k runs over all odd inte- odd gers, an exponential decay of f(z) for ℜ(z) → +∞, and −1 an exponential decay of [f(z)−1] for ℜ(z) → −∞, where Q2 = β Γ(zk), (6) ℜ(z) denotes the real part of z. |kX|<2M The final form of Q, described in the literature12, em- −1 where the sum runs over all Matsubara points zk between ploys a point ZM = µ +2πiβ M, where M is a positive Z and Z∗ , and integer; note that there are exactly M Matsubara points M M +∞ zk lying inside the segment (µ,ZM ). This yields: i sign(ξ) dξ Q3 = Γ(ZM + ξ) 2π −∞ 1 + exp(β|ξ|) Q = Q1 + Q2 + Q3, (4) Z i +∞ sign(ξ) dξ − Γ(Z∗ + ξ) , (7) with the individual terms given by 2π M 1 + exp(β|ξ|) Z−∞ 1 where the integrations over a real variable ξ correspond Q1 = Γ(z)dz, (5) 2πi to complex integrals along the horizontal dashed lines ZC1 3 in Fig. 1a. Note that the contributions over ξ > 0 in where the omitted term includes a regular part and a Eq. (7) refer to integrations of f(z)Γ(z), whereas those weak (logarithmic) singularity. Third, along the vertical over ξ < 0 refer to integrations of [f(z) − 1]Γ(z). By line z = µ + iη (η is real), the limits of σ±(µ ± ε + iη) for ∗ ∗ + employing the rule Γ(z )=Γ (z), the integrations in Q1 ε → 0 can be considered; their difference equals and Q3 and the summation in Q2 can be performed with τ(µ + iη) = lim [σ+(µ + ε + iη) − σ−(µ − ε + iη)] arguments z of Γ(z) lying only in the upper (or lower) ε→0+ complex half-plane, see Ref. 12 for more details. =2πi [[βη/(2π)]], (13) Numerically, the integral in Q1 (5) can be evaluated by using a finite number of nodes along the path C1 and where [[u]] denotes the integer nearest to the real quan- corresponding weights, and the integrals in Q3 (7) can tity u. This result means that τ(µ + iη) is a piecewise be obtained from a finite number of terms of the Tay- constant function of η with discontinuities at η = πβ−1k, lor expansion of Γ(z) at the point ZM . The coefficients where k is an odd integer (i.e., whenever µ + iη = zk). of this expansion are obtained from the values of Γ(z) The last property reflects the fact that the derivatives of in a few points z lying in the neighborhood of ZM and σ+(z) and σ−(z) coincide mutually in the whole complex the remaining integrals can be reduced (for odd positive plane except at the Matsubara points zk, where second- integers j) to order poles of both derivatives are located. Fourth, along horizontal lines z = E +2πiβ−1m, where m is an integer, +∞ j |u| du −j it holds Rj = =2 j! (1 − 2 ) ζ(j + 1), (8) 1+e|u| −∞ −1 2πim Z σ±(E +2πiβ m)= σ(E) ± , (14) ∞ −a 1 + exp[±β(E − µ)] where ζ(a) = n=1 n is the Riemann’s zeta-function (for a> 1). proving explicitly that the functions σ (z) are not peri- P ± odic with the period 2πiβ−1, in contrast to the function f(z). B. Entropy The expression for the entropy (9) can be written as a complex integral The entropy corresponding to a positive temperature S 1 and the one-particle Hamiltonian H is given by = Γ(z)σ(z)dz (15) kB 2πi C +∞ Z S = kB D(E)σ(E)dE, (9) along the same path C as in Eq. (3). The deformation Z−∞ of the contour C has to be performed separately on both where D(E) is the total DOS of the system related to sides of the vertical line ℜ(z)= µ, see Fig. 1b. This leads the Green’s function (resolvent) by Eq. (2) with Γ(z) = to the form Tr{G(z)}, and where the function σ(E) is defined as S = S+ + S− + S> + S< (16) σ(E)= −f(E) ln[f(E)] − [1 − f(E)] ln[1 − f(E)]. (10) with the individual terms given by

Since complex logarithm is a multivalued function, the k≷0,odd S≷ i real function σ(E) can be directly continued analytically = Γ(z)τ(z)dz + (zk − µ)Γ(zk), kB 2π C into its complex counterpart σ(z) only in a stripe around Z ≷ |k|<2M −1 X the real energy axis, namely, for |ℑ(z)| < πβ , where S 1 ℑ(z) denotes the imaginary part of z. [Here and below, ± = Γ(z)σ (z)dz, (17) k 2πi ± we assume the branch cut of ln(w) along the real nega- B ZC± tive half-axis, w< 0.] For present purposes, let us define where the paths C+, C−, C>, and C< are depicted in continuations σ−(z) and σ+(z) of σ(E) that are analytic Fig. 1b and where the functions σ±(z) (11), τ(z) (13), in the entire half-planes ℜ(z) <µ and ℜ(z) >µ, respec- and the singular behavior of σ±(z) (12) have been used. tively: The contribution S+ is calculated from Eq. (14) for ±t m = M and m = −M, and similarly for S−. This yields σ (z)= + ln(1 + e∓t), t = β(z − µ), (11) together ± 1+e±t +∞ S+ + S− 1 and let us discuss briefly their properties. First, the func- = − ℑ[Γ(ZM + ξ)] σ(µ + ξ) dξ kB π tion σ+(z) decays exponentially for ℜ(z) → +∞ and the Z−∞ +∞ function σ−(z) decays exponentially for ℜ(z) → −∞. sign(ξ) dξ − 2M ℜ[Γ(ZM + ξ)] . (18) Second, it can be shown that σ+(z) and σ (z) possess − −∞ 1 + exp(β|ξ|) the same leading term of their singular behavior near the Z Matsubara points zk: Numerically, the integrals in Eq. (18) can again be obtained from a finite number of terms of the Taylor expan- −1 σ±(z) = (µ − zk)(z − zk) + ..., (12) sion of Γ(z) at the point ZM . The encountered integrals 4 are Rj (8) and (for even non-negative integers j) resulting right-hand side of Eq. (21) is defined unambigu- ously again. Finally, the additional numerical effort to +∞ ln(1 + eu) ln(1 + e−u) N = uj + du calculate the entropy is negligible as compared to other j 1+eu 1+e−u computations, since the complex energy arguments in- Z−∞ j +2 volved (which comprise the M lowest Matsubara points = R +1 . (19) j +1 j zk, the point ZM , and a few points near ZM ) enter the self-consistent electron-density calculations as well (see The evaluation of S> and S< in Eq. (17) is greatly end of Section II A). simplified by the fact that the function τ(z) defined along the paths C> and C< is piecewise constant, see Eq. (13). If we denote by Φ(z) a primitive function to Γ(z), so that III. NUMERICAL IMPLEMENTATION dΦ(z)/dz = Γ(z), we get

odd The developed formalism has been implemented on a S> model level and in the self-consistent scalar-relativistic = Φ(zk) − MΦ(ZM ) 17 kB TB-LMTO method in the atomic sphere approximation 0 + S< complex weights. In the illustrating example presented =2ℜ [Φ(zk) + (zk − µ)Γ(zk)] k kB in Section IV, several thousands of vectors were used ( 0

-2 0.5 (a) B / k ) x x -4 ν=2 S |/S x -6 0

(|S-S ν=4 10 -1 (b) ν ν

log -8 =2 =4 ν=6 ν=6 ν=8 ν ) =8 x -3 -10 |/S x 0 5 10 15 -5 (|S-S

M 10 -7 log

FIG. 2. Relative deviations of the calculated entropy S with respect to the exact value Sx = kB ln 2 for the isolated -9 eigenvalue E0 = µ and for diﬀerent degrees ν of the Tay- lor polynomial as functions of the number M of Matsubara 0 0.5 1 1.5 2 points. Note the logarithmic scale on the vertical axis. ∆ kBT /

The relative difference between S and Sx is displayed in Fig. 2 as a function of M for several values of ν. One FIG. 3. (a) The temperature dependence of the exact en- can see that the accuracy of the approximate scheme is tropy Sx for a model Lorentzian DOS of width ∆, see text quite high, the only exceptions being the cases with very for details. (b) Relative deviations of the calculated entropy S with respect to Sx as functions of temperature for different small M or ν. The studied model is very simple indeed; degrees ν of the Taylor polynomial and for two values of the note however that an isolated eigenvalue belongs to the number M of Matsubara points: M = 5 (solid symbols) and strongest singularities to be encountered in the electronic M = 12 (open symbols). spectra. Another simple model is described by a Lorentzian DOS parametrized by its center E0 and width ∆, so in Eq. (18) by using the Taylor expansion polynomials. 2 2 −1 that D(E) ∝ [(E − E0) + ∆ ] and, consequently, Note that the same source of inaccuracy refers also to −1 Γ(z) = (z − E0 + i∆) and Φ(z) = ln(z − E0 + i∆) the quantity Q3 (7). There are two particular questions for ℑ(z) > 0. We have chosen a slight offset of the cen- related to this point, namely, (i) the effect of the finite ter E0 with respect to the chemical potential µ, namely, degree ν of the Taylor polynomial, and (ii) the role of E0 − µ = 0.4∆, and have assumed all parameters (E0, the numerical procedure to extract the coefficients of the ∆, µ) as T -independent. The exact entropy Sx, obtained polynomial. The results found for the above simple mod- by a highly accurate real-energy numerical quadrature els and presented in Fig. 4 provide a partial answer to according to Eq. (9), is shown in Fig. 3a as a function both questions. First, it is seen that the increase in ν of T . The temperature dependence of the relative dif- reduces in general the relative deviations of entropy, but ference between the approximate S and the exact Sx is the obtained trends are not strictly monotonic. Second, presented in Fig. 3b for two values of M (M = 5 and the numerically obtained coefficients lead essentially to M = 12) and for several values of ν. One can see that the same values of S as the exact coefficients (see the the relative deviations are essentially independent of T , values marked by red crosses and open boxes in Fig. 4). despite the pronounced increase of the entropy with in- These facts represent undoubtedly positive features of creasing temperature. One can also observe an increase the presented formalism from the practical point of view; in the relative accuracy due to higher values of M and ν, however, certain caution is needed in attempts to in- indicating that modest numbers M and ν are sufficient crease the accuracy by using too high degrees ν. First, for practical applications of the developed formalism. the convergence radius of the Taylor series of Γ(ZM + ξ) The origin of numerical inaccuracy in the present eval- is inevitably finite due to the branch cuts (and possible uation of the entropy can easily be identified (disregard- poles) of Γ(z) on the real energy axis. This means that ing the well-known convergence issues with respect to there is no strict convergence of S to Sx for ν → ∞, the BZ sampling): it is the treatment of integrations at least for a fixed number M of the Matsubara points. 6

2.0 2.4 -4 (a) )

x -6 ) |/S B

x 2.2

1.5 B µ

-8 ( Fe S / k (|S-S M 10

log -10 1.0 2.0 -12 8.0 0.8 2 4 6 8 ν (b)

4.0 0.4 FIG. 4. Relative deviations of the calculated entropy S with respect to the corresponding exact value Sx as functions of the degree ν of the Taylor polynomial: for the isolated eigenvalue with M = 5 (solid circles) and for the Lorentzian DOS (mRy) tot with kBT = ∆ and M = 12 (open boxes). The red crosses 0.0 0.0 F (mRy) mark deviations obtained in the latter case with the Taylor E expansion coeﬃcients extracted by a numerical procedure.

-4.0 -0.4 TABLE I. Calculated local magnetic moment MFe and electronic entropy S (per atom) for three temperatures T in the -2 -1 0 1 2 3 DLM state of bcc Fe. The values of MFe in parenthesis are from Ref. 7. Bext (mRy)

T (K) MFe (µB) S/kB 0 2.02 (1.96) 0.00 FIG. 5. Calculated dependence of various quantities on the 2000 1.92 (1.85) 0.89 random external magnetic field Bext for bcc Fe in the DLM 4000 1.41 (1.30) 2.09 state at T = 4000 K: (a) the local magnetic moment MFe (open circles) and the electronic entropy S (solid triangles), and (b) the total energy Etot (open boxes) and the free energy F (solid diamonds). The quantities S, Etot and F are given Second, the procedure to extract the Taylor coefficients per one atom; the vertical scales in panel (b) have been shifted from several values of the function Γ(z) in neighborhood to a common zero at Bext = 0. of z = ZM leads to a set of ν linear equations for ν unknown variables. This linear problem (Vandermonde system) is ill-conditioned20 which can prevent its stable set to ν = 6. For the very high temperatures considered numerical solution for large values of ν. here, small numbers of the Matsubara points were suf- As an illustrating application to a realistic system, we ficient: we set M = 8 for T = 2000 K and M = 4 for have considered bcc iron in the disordered-local-moment T = 4000 K. Calculated values of the self-consistent local (DLM) state21. This system at very high temperatures magnetic moments MFe and of the electronic entropy S (up to T = 6000 K) and under strong pressures attracts (per atom) are shown in Table I for three selected tem- ongoing interest in the context of physical properties of peratures. One can see a sizeable reduction of the local the Earth’s core6,7,22,23. Here we focus only on the ef- moment due to increasing temperature as expected; the fect of elevated temperatures and treat thus bcc iron of values of MFe in this work are slightly higher than the val- a density corresponding to ambient conditions (Wigner- ues reported in Ref. 7. The electronic entropy increases Seitz radius s = 2.65 a.u.) within the local spin-density with increasing temperature, which is another expected approximation with the local exchange-correlation poten- trend of finite-temperature behavior. tial parametrized according to Ref. 24. The valence spdf- In order to make better assessment of internal consis- basis was used in the TB-LMTO-ASA method and the tence of the formulated entropy S with other electronic CPA; the degree of the Taylor expansion polynomial was quantities, such as the electron densities, local magnetic 7 moments MFe and total energies Etot, we have studied V. CONCLUSION these quantities as functions of a randomly oriented external magnetic field Bext coupled to the electron spin. We have revised the evaluation of physical quantities in [For simplicity, the quantity Bext includes the Bohr mag- finite-temperature Green’s-function techniques with par- neton µB, so that ±Bext describes the spin-dependent ticular attention paid to the electronic entropy. We have shift added to the spin-polarized exchange-correlation shown that usual obstacles encountered in entropy calcu- potential.] The application of this random external field lations (branch cuts or ambiguity of complex logarithm) is equivalent to an effect of the constraint in the fixed- can be removed completely, which leads to a simple fi- 23 spin-moment method applied to the DLM state . Var- nal expression without an explicit contour integration. ious calculated quantities for the case of T = 4000 K The final result can be implemented both in semiempir- are displayed in Fig. 5. One can observe that the mag- ical TB schemes as well as in ab initio Green’s function netic moment MFe, the entropy S, and the total energy approaches based on the KKR or the LMTO methods, Etot are monotonic functions of the external field Bext optionally also with the CPA for chemically disordered throughout the studied range. However, the free energy systems. The finite numerical accuracy of the developed F = Etot − TS exhibits a clear minimum at Bext = 0, formalism, which is due to a standard auxiliary Taylor which represents a necessary condition for an internally expansion of the resolvent, seems to be well under con- consistent theory at finite temperatures. Note that the trol. The electronic entropy thus need not be avoided total energy Etot is expressed only in terms of quantities in Green’s function techniques, but it should rather be discussed in Section II A, i.e., it is fully independent of employed directly for reliable computations of other ther- the treatment of the electronic entropy S described in modynamic potentials. Section II B.

ACKNOWLEDGMENTS

This work was supported ﬁnancially by the Czech Sci- ence Foundation (Grant No. 18-07172S).

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