Thermodynamic geometry of Ising ferromagnet in an external magnetic field: A self-consistent field theory calculation

Physica A 526 (2019) 121173

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Thermodynamic geometry of Ising ferromagnet in an external magnetic field: A self-consistent field theory calculation Rıza Erdem Department of Physics, Akdeniz University, 07058, Antalya, Turkey

highlights • • • •

We present thermodynamic geometry of ferromagnetic Ising model. We introduce a metric in the phase space of magnetization versus spin correlation. We use this metric to calculate the thermodynamic curvature scalar. Continuous/discontinuous phase transitions of the system are studied using the curvature scalar in the presence of external magnetic field.

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Article history: Received 24 October 2018 Received in revised form 18 March 2019 Available online 25 April 2019 Keywords: Thermodynamic curvature Ferromagnetic Ising model Pair approximation Self-consistent field theory

a b s t r a c t We present a complete geometrical description for the ferromagnetic Ising model in the pair approximation as introduced by Balcerzak (2003) using self-consistent field theory. A metric is defined in a two-dimensional phase space of magnetization (M) and nearestneighbour correlation function (C ). Based on the metric elements an expression for the thermodynamic Ricci scalar (R) is derived in terms of the lattice coordination number q. We study R as the temperature (T ), magnetic field (h) and exchange energy coupling (J) are varied and show that there are T and h dependent critical properties for q = 6. By direct comparison, we demonstrate that the special case q = 2 provides a consistent behaviour with the already known exact formula in Janyszek and Mrugała work (1989) for the one-dimensional Ising model. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The thermodynamics of magnetic systems is now rather understood in the framework of statistical mechanics. Particularly the phase transitions of well-known spin models appear in the literature in many aspects [1–11]. Some of the results obtained within statistical theories and published in the literature are being verified from the geometrical point of view. Therefore, recent efforts on magnetic phase transitions are devoted to the geometrical perspective. Several authors started with the geometrization of one-dimensional (1-D) Ising model in the presence of magnetic field. A curvature scalar (R) (a useful quantity which is also called the thermodynamic curvature) was firstly calculated on a two-dimensional parameter space of exchange coupling (J) and external magnetic field (h). According to the calculations in Refs. [12–14], a singularity of R (or a divergence to infinity) in the vicinity of the critical point with zero-temperature and zero-magnetic field was observed. Since this exact result was known, the applications of geometrical methods to other magnetic models have become widespread [15–25]. From these applications, it is now clear that the thermodynamic curvature has significant role in describing both classical and quantum spin systems. However, the magnetic field effects E-mail address: [email protected]. https://doi.org/10.1016/j.physa.2019.121173 0378-4371/© 2019 Elsevier B.V. All rights reserved.

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R. Erdem / Physica A 526 (2019) 121173

on geometrical properties of these models have not been explored rigorously. As far as we know, only few works predicted these effects in some spin models such as 1-D Ising model [12], Pauli paramagnetic gas [16] and mean-field Curie–Weiss (CW) model of ferromagnetism [12,22]. In this work, we investigate the influence of magnetic field on the thermodynamic curvature expression for the ferromagnetic Ising model in pair approximation (FIMPA) by using self-consistent field (SCF) theory [2]. Firstly, we have made use of a metric defined on a two-dimensional phase space of magnetization and spin correlation function to derive the curvature scalar when the external magnetic field (h) exists. This type of metric definition has recently been introduced by us for the geometrical analysis of Ising model with nearest-neighbour ferromagnetic and antiferromagnetic interactions near the Curie (Tc ) and Néel (TN ) temperatures, respectively. From the analysis, the divergence of R to infinity in the vicinity of these temperatures on both sides has been shown analytically [26,27]. But, the effect of h has not been taken into account in both papers while only the effect of temperature (T ) was given, hence, the continuous (second-order) phase transition was studied geometrically. In most of the works on spin models based on approximate theories, a finite jump (across 2-ferromagnetic phase coexistence curve) in magnetization occurs at h = 0 when T < Tc [28]. This is called the discontinuous (first-order) phase transition. In order to capture this behaviour geometrically using the SCF theory, we have calculated the curvature scalar as a function of h and showed that these transitions are seen as the crossing of the various branches of R curves on the coexisting phases. This paper is organized as follows: In the next section, we describe the model briefly and review its equilibrium solution in the pair approximation. Next, in Section 3, we present some basics of the thermodynamic geometry and show the derivation of curvature scalar for the spin model under consideration. Numerical results and discussion on R are given in Section 4. Finally, summary and some concluding remarks are presented in Section 5. 2. The model, equilibrium formulation and phase transitions We consider the spin-1/2 Ising model in a longitudinal magnetic field, which can be described through the Hamiltonian [1–3] Nq/2

H = −J

∑

Si Sj − h

⟨ij⟩

N ∑

Si ,

(1)

i=1

where Si is the spin variable (at site i) which take on two values ±1/2, J is the exchange coupling energy for the nearestneighbour spins only, N denotes the total number of lattice sites (or spins), q is their coordination number (i.e. the number of nearest neighbours) and h = −g µB H z is the external magnetic field H z applied along the z-direction with µB Bohr magneton of each spin and g electronic g factor. The properties of the system at equilibrium are generally determined self-consistently using Gibbs energy calculations. A good reference for these calculations is Balcerzak [2] whose notation is used here. Assuming the magnetization per lattice site and spin-pair correlation function for nearest-neighbour spins defined by M = ⟨Si ⟩,

C = ⟨Si Sj ⟩,

(2)

the magnetic Gibbs energy functional is given by the following general formula [2]:

ψ = U − NhM − NT σ ,

(3)

where T is the absolute temperature, U is the internal energy and σ is the entropy: U =−

N 2

qJC ,

(4)

q σP − (q − 1)σS . (5) 2 The single-site and pair entropies in Eq. (5) are found, respectively, from the pair approximation definitions as follows:

σ =

σS = −kB

σP = −kB

[(

1 2

[(

1 4

)

(

+ M ln

1 2

) +M +

) + M + C ln

(

1 4

(

1 2

)

(

− M ln

) +M +C

( +2

1 4

1 2

)] −M

) − C ln

(

, 1 4

(6)

) −C

( +

1 4

) − M + C ln

(

1 4

)] −M +C

,

(7)

where kB is called as the Boltzmann constant. In the pair approximation, the variables M and C are treated as two independent variational parameters with respect to which functional (3) can be minimized simultaneously. Hence, the conditions

∂ψ = 0, ∂M

∂ψ = 0, ∂C

(8)

R. Erdem / Physica A 526 (2019) 121173

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Fig. 1. (a) Temperature (T ) dependence of M at various field values for q = 6 and J = 1. (b) same as (a) but for C .

yield the self-consistent equations M=

C =

sinh α

1

2 cosh α + e−J /2kB T 1 cosh α − e−J /2kB T 4 cosh α + e−J /2kB T

,

(9)

,

(10)

at equilibrium. Here, the following notations are introduced for simplicity:

α=

1 qkB T

[(q − 1)r + h] ,

(11)

where

( r = kB T ln

1 2 1 2

+M −M

) .

(12)

Having the Gibbs energy formula (Eq. (3)), together with the extremum results (Eqs. (9) and (10)), all thermodynamical quantities starting from magnetic specific heat (including the thermodynamic potentials and isothermal susceptibility) have been expressed by M and C and discussed previously in Ref. [2]. Thus, the knowledge of both parameters for given T and h is fundamental. Some examples of solutions of Eqs. (9) and (10) are briefly provided in Figs. 1–3. In Fig. 1, the magnetization M and spin correlation C vs. temperature T are plotted for the simple cubic lattice structure with q = 6. For comparison of static properties we have drawn the curves for four different values of the magnetic field. In the subsequent two figures, we have shown M and C vs. h curves (Fig. 2) and M and C vs. J curves (Fig. 3) for three different temperatures. In the absence of external magnetic field (h = 0), M and C become saturated as M → ±1/2 and C → +1/4 when T approaches zero while they decrease continuously with increasing temperature and one of them (M) converges to

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R. Erdem / Physica A 526 (2019) 121173

Fig. 2. (a) Magnetic field (h) dependence of M at various temperatures for q = 6 and J = 1. (b) same as (a) but for C .

zero. This corresponds to a continuous (or a second-order) phase transition at a critical temperature (T = Tc ) from the ferromagnetic phase (M > C > 0 or M < 0, C > 0) to paramagnetic phase (M = 0, C > 0), as shown by solid green curves in Fig. 1(a) and (b). In the presence of an external magnetic field (h ̸ = 0), the phase transition has been removed (see solid red and blue curves in the same figures) [2]. These solutions are usually known as the stable branch. There exists another set of solutions (or unstable branch) which are obviously seen from the dotted curves in Fig. 1. On the other hand, when the spin system is in ordered phase with T < Tc , upon decreasing the magnetic field there is a first-order phase transition between spin-up (M > 0) and spin-down (M < 0) states at h = 0 with a discontinuous change in the stable magnetization M, illustrated by the solid blue curve in Fig. 2(a). When T ≈ Tc , where the first-order phase transition terminates, the system has a second-order phase transition at h = 0 (green curve in Fig. 2(a)) and the system is paramagnetic for all temperatures with T > Tc (red curve in Fig. 2(a)). Another aspect of these properties are given in Fig. 2(b), where the corresponding C vs. h curves are drawn. From this figure, one can see that spin correlation is always positive function of h and independent of the orientation of the magnetic field, C (−h) = C (h). In both figures, we have also indicated the metastable and unstable solutions by the dashed and dotted blue curves, respectively. The metastable branch is the natural extension of the stable one. Between the two turning points the system becomes unstable. These two solutions constitute the spinodal curve and form a basis for the hysteretic behaviour of the M − h curves in the spin system. In the next figure (Fig. 3), we show M vs. J and C vs. J curves which are calculated using Eqs. (9) and (10) for q = 6 and h = 0.05 at several temperatures. For any temperature, the stable branches of M and C become saturated as M → +1/2 and C → +1/4, respectively, while J increases. One can also see from the figure that there are metastable (dashed curves) and unstable (dotted curves) states in M < 0, C > 0 region. In this case, M → −1/2 and C → +1/4 as J → +∞. In fact, among well-known spin systems, dependence of order parameter on exchange energy constant was shown before in an Ising chain on a dimerized lattice. A numerical calculation of the relaxation process was given in terms of the J-dependent lattice deformation in Ref. [29]. Motivated by this investigation, it would be interesting to reflect the behaviour given in

R. Erdem / Physica A 526 (2019) 121173

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Fig. 3. (a) Exchange coupling (J) dependence of M at various temperatures for q = 6 and h = 0.05. (b) same as (a) but for C .

Fig. 3 to the geometrical quantities (thermodynamic curvature), and for example to provide a base for future applications of the present geometrical theory to the dimerized spin models. 3. Geometrical structure of thermodynamic state space and derivation of the curvature scalar for the Ising model In order to identify the geometrical structure of a n-dimensional thermodynamic state space, firstly a metric (or a line element) is defined as follows ds2 = Gij dxi dxj ,

(13)

where x (i = 1, 2, . . . , n) denote the various thermodynamic coordinates (for magnetic systems thermodynamic quantities per unit spin is appropriately used) and Gij are the components of a covariant metric tensor. Above metric introduces the concept of a distance in the space of equilibrium thermodynamics. In other words, a large distance between two states means a small probability that these are related by a thermal fluctuation. Two different definitions for Eq. (13) were introduced by Weinhold [30] and Ruppeiner [13,31]. Although these are closely related to each other in the framework of distribution functions of statistical mechanics, their metric structures are respectively the Hessian matrix of the internal energy (U) and negative entropy (σ ) with respect to the extensive thermodynamic variables: i

Gij = ∂i ∂j U ,

(14)

1 Gij = −k− B ∂i ∂j σ ,

(15)

where ∂i = ∂/∂ x . Ruppeiner’s metric is also defined in terms of the Hessian of a thermodynamic potential (or free energy) as indicated in the third line of Table II in Ref. [13]: i

Gij = −β∂i ∂j φ,

(16)

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R. Erdem / Physica A 526 (2019) 121173

where β = 1/kB T and φ = ψ/N. In terms of Gij the Christoffel symbols can be found by the formula

Γjki =

1 2

Gil ∂k Glj + ∂j Glk − ∂l Gjk ,

(

)

(17)

where Gil denotes the inverse of the metric tensor Gij . The curvature tensor may be written in terms of the Christoffel symbols as i i Γjkm . Γjlm − Γml Rijkl = ∂k Γjli − ∂l Γjki + Γmk

(18)

Then the Ricci tensor is defined by Rij = Rninj ,

(19)

and after another contraction of the Ricci tensor indexes, follows the Ricci curvature scalar R = Gij Rij .

(20)

The case R = 0 corresponds to a flat geometry and a non-interacting spin model. But, a nonzero R (arising from any equilibrium state space geometry) encodes the microscopic interactions between the spins underlying any magnetic system. Therefore, the curvature scalar measures the complexity of the system and plays a central role in any attempt to look at magnetic phase transitions from geometrical perspective. For the FIMPA introduced in Ref. [2], we parametrize a two-dimensional Riemannian manifold by (x1 , x2 ) = (M , C ). This type of manifold choice is different from (x1 , x2 ) = (M , T ) or (x1 , x2 ) = (M , h) where the metric geometry becomes trivial since the free energy expression is explicitly a linear function of T or h without any additional term in the Hamiltonian [22]. This is not the case for the novel definition where no additional term in the Hamiltonian is necessary and the function ψ is a nonlinear function of C . Also, for a nontrivial geometric analysis, Ruppeiner metric seems a better and more natural choice than Weinhold metric since the metric components in Eq. (14) vanish which results in zero curvature or flat geometry. With the choice of Ruppeiner’s geometric approach and order parameter manifold, one can find the components of the metric tensor directly using the pair entropy (7) in (15) as follows: p G11

−1 ∂

= −kB

σp

∂M2

p

1 G12 = −k− B

p

2

1 G22 = −k− B

( =

1 4

)−1 +M +C

( +

(

1 4

)−1 +M +C

−M +C

4

2 ∂ 2 σp 1 ∂ σp = Gp21 = −k− = B ∂M∂C ∂C∂M

∂ 2 σp = ∂C2

)−1

1

( +2

(

1 4

)−1

−C

4

(21)

+M +C

)−1

1

,

( −

( +

1 4

1 4

)−1 −M +C

)−1 −M +C

,

(22)

.

(23)

where the derivatives are evaluated in the equilibrium state. With the metric tensor defined in relations (21)–(23) and using (17)–(20) we have calculated the Ricci scalar as Rp = −1/2. According to this result, the manifold (M, C ) is a curved two-dimensional phase space which has a constant negative curvature. Hence, above metric choice presents no critical properties for the spin model under consideration. For a geometrical analysis of critical properties, we have to define another Ruppeiner metric. We use (3) in (16) (or (5) in (15)) to obtain the components of such a metric as follows:

[(

)−1

)−1 ]

G11

∂ 2φ = −β = (q − 1) ∂M2

[(

G12

∂ 2φ ∂ 2φ 1 = −β = G21 = −β =− q ∂M∂C ∂C∂M 2

[(

(

G22

∂ 2φ 1 = −β 2 = − q ∂C 2

1 2

1 4

+M

( +

1 2

)−1 +M +C

−M

1 4

+2

1 4

1

[(

− q 2

4

)−1

(

+M +C

)−1 −C

1

−

( +

1 4

)−1 +M +C

1 4

( +

)−1 ] −M +C

)−1 ] −M +C

.

1 4

,

)−1 ] −M +C

,

(24)

(25)

(26)

Based on (24)–(26) and using (17)–(20) we have found the following simple formula for the curvature scalar R which can be expressed in terms of the known equilibrium values of M and C determined by expressions (9) and (10), respectively:

R=−

(4M 2 − 1)(4qM 2 − 8M 2 − 8qC + 8C + q) (4qM 2 − 8M 2 − 4qC + 4C + 1)2

.

(27)

Some examples of numerical calculations related with Eq. (27) and the discussion of theoretical results will be given in the next section.

R. Erdem / Physica A 526 (2019) 121173

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4. Numerical results and discussion In this section, we investigate the main characteristics of the thermodynamic curvature scalar for Ising model under the pair approximation in the ferromagnetic (ordered) and paramagnetic (disordered) phase regions as well as near the phase transition points (both continuous and discontinuous). Schematic illustrations of the thermal, magnetic field and exchange coupling behaviours for R (with J > 0) are presented, respectively, in Figs. 4–6. Especially, we have examined the magnetic field effects on R near the discontinuous phase transitions. Besides, we have compared in Fig. 7 the R vs. T and R vs. h curves obtained from our method for the 1-D Ising chain with those of exact formulation results [12]. Firstly, curvature scalar (R) versus temperature calculations are performed at several h values for J = 1 and the results are displayed in Fig. 4(a) and (b). Four curves corresponding to the cases h = 0, 0.03, 0.05 and 0.1 are drawn in Fig. 4(a) for the simple cubic lattice structure with q = 6. It is pointing out that a divergence of R occurs (R → +∞) around T = Tc when h = 0 (green curve) as expected. This result is consistent with the curvature around the Curie temperature belonging to the mean-field CW ferromagnetic model [12,22]. In the presence of an external field h ̸ = 0 the divergence of R has been removed and maxima of the curves (or peaks) are observed. Comparing the curves for h = 0.03, 0.05 and 0.1 we see that with the increase of h the maxima become smaller and shift towards higher temperatures. For the sake of comparison, we have also calculated R vs. T for the same system, but using different lattice structures (with q = 4, 6, 8 corresponding to the square lattice, simple cubic lattice, body centred cubic lattice structures, respectively), whose results are displayed in Fig. 4(b). We have considered h = 0.05 for all lattice types in the figure. On increasing lattice coordination number q the peaks become larger and shift towards higher temperatures. It is worthwhile to mention that a similar picture of R vs. T curves (Fig. 4(a)) can also be seen in most references on various static or dynamic quantities, such as magnetic susceptibility [2] and relaxation times [32]. A common feature of these quantities are that they diverge to infinity at the critical point (with allowance for different values of the critical exponents) in the absence of magnetic field or show a field-dependent peak property when h ̸ = 0. In h = 0 case, a standard relation for the curvature exponent is expressed in terms of α (exponent that is related to specific heat) by λ = α − 2 which demonstrates the critical behaviour of R [17,19,23]. This yields the scaling R ≈ ε α−2 where α ≥ 0 and ε = (Tc − T )/Tc . It should be noted from Ref. [2] that the specific heat displays no singular part but shows a finite jump at Tc when h = 0. This property is expressed as α = 0 for the FIMPA. Hence, setting α = 0 we reach R ≈ ε −2 . For a second-order transition the Ricci scalar depends also on the correlation volume: R ≈ ξ d , where d is the dimensionality of the system, ξ = ε −υ is the correlation length and υ is the correlation length exponent. In this case λ = −dυ . It has been shown in most references that the mean-field behaviour sets in when the upper critical dimension (d = 4) is reached [33]. In our model calculations, both M and C are evaluated self-consistently by the improved mean-field theory. Following this start and using the connection R ≈ ε −dυ , our curvature calculations reproduce the value of υ = 1/2 for α = 0 and λ = −2. This result fits in well with mean-field predictions in Refs. [34,35]. On the other hand, the extrema of the curves observed for h ̸ = 0 in Fig. 4(a) and (b) determine the line of curvature maxima thus indicating the loci of correlation length maxima or the Widom line for the FIMPA. In Fig. 4(c) we construct two Widom lines predicted for each of lattice structures corresponding to curves in Fig. 4(b). These start at the critical points indicated by the small arrows in Fig. 4(b) and extend into the regions h > 0 and h < 0 in the h – T plane. We compare our results for Widom lines with those of mean-field CW model where a (M, T ) manifold is used [22]. We find a striking agreement between the methods, thus supporting the power of (M, C ) manifold choice in the geometrical investigations. To demonstrate another role of the external magnetic field, we keep the temperature of the spin system described in Fig. 4(a) fixed and change the value of magnetic field. Our results for T = 1.180 (blue curve), T = Tc ≈ 1.232 (green curve) and T = 1.280 (red curve) are presented in Fig. 5. Similar to the spin correlation function (Fig. 2(b)), the curvature function is also positive and symmetric on both sides of h = 0 line for all temperatures. This means that the curvature scalar is independent of the orientation of the magnetic field, R(−h) = R(h), which is in agreement with the Riemannian geometry of one-dimensional Ising model [12]. As evidenced from the coloured curves in Fig. 5(a), R vs. h calculations behave differently in the ferromagnetic and paramagnetic phases. When the temperature is less than the critical temperature (T < Tc ), the curvature for each temperature (or isothermal R) has two physical branches (blue solid and dashed curves) which cross at h = 0. According to Ruppenier’s conjecture [13], this implies equality of correlation lengths at h = 0 which is interpreted as the discontinuous phase transition between the phases M > 0 and M < 0. It is also interesting to show the behaviour of R along the unstable solutions given in Fig. 2. This is plotted as the blue dotted curve in the central part of Fig. 5(a). In this case, a minimum of R can be seen explicitly when h = 0 and the values of h at which the unstable and metastable branches of R (dashed blue curves) join correspond to the turning points of magnetic isotherms in Fig. 2(a). On the other hand, on approaching the critical temperature (T ≈ Tc ), a divergence to infinity in R appears again, just as in Fig. 4(a). This can be regarded as the geometrical signature of the continuous phase transition which undergoes between the phases M > 0 and M < 0. In the paramagnetic phase (T > Tc ), there is no R-crossing and no phase transition. But, the curvature shows a maximum with respect to h, which is always at h = 0. In order to explore these maxima in more detail, we have obtained the curves for T = 1.28 by changing slightly the exchange coupling energy (Fig. 5(b)). We see a strong agreement between the intersection of R curves in the present investigation (Fig. 5(a)) and the microscopic characterization based on R for the first-order phase transitions in other works such as the classical spin models (in the mean-field CW approximation) [22], liquid–gas systems [22,36] and black holes [37]. In the mean-field

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R. Erdem / Physica A 526 (2019) 121173

Fig. 4. (a) Temperature (T ) dependence of R at various field values for q = 6 and J = 1. (b) same as (a) but for various lattice coordination numbers (q) with h = 0.05. (c) Loci of maxima of R for the case (b) on the h – T plane.

CW case, the numerical values for R were produced using the Riemann manifold (x1 , x2 ) = (M , T ) and the two-phase coexistence was firstly observed as the crossing of two physical branches of R by Dey and co-workers. We see a remarkable resemblance of Fig. 4 in Ref. [22] to Fig. 5(a) in this work. In the same publication, the authors extended their curvature analysis (calculated on temperature–volume manifold) to look at behaviour of R with respect to the pressure near the liquid–liquid phase co-existence in a toy model. Again, the isothermal R-crossing occurred below the criticality. In another

R. Erdem / Physica A 526 (2019) 121173

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Fig. 5. (a) Magnetic field (h) dependence of R at various temperatures for q = 6 and J = 1. (b) Same as (a) but for various J values with T = 1.28.

article, Ruppeiner and co-workers [36] predicted using standard metric definitions in thermodynamic geometry for the first-order liquid–gas phase transition terminating in a critical point, that R ≈ vg where vg is the molecular volume in the gas phase related with the first derivative of the Gibbs potential which has a finite discontinuity corresponding to the first-order phase transition [38]. It should be mentioned that above prediction loses its significance and becomes R ≈ ξ 3 (ξ correlation length) around the critical temperature. A good view similar to Fig. 5(a) is also found in a recent work on black holes [37]. The patterns of R-crossing (corresponding to the liquid–gas like first-order phase transition) were depicted in the plots of R vs. black hole temperature and they completely disappeared as the magnetic charge increased resulting in a divergence of R. Based on the equilibrium results presented as coloured curves for different temperatures in Fig. 3, a numerical analysis of Eq. (19) shows that the isothermal R exhibits a local maximum with respect to J, whose locus can naturally be interpreted as the Widom line in the T − J plane which is not included in this work. The temperature-dependent peak grows in height and shifts towards right in the ferromagnetic phase with M > 0. After this maxima, the curvature decreases continuously to R = 0, shown in Fig. 6(a). Another property of the maxima in R is given in Fig. 6(b). For increasing field values, we have found the shifts of the peaks towards lower J values. Finally, it is of great interest to compare the calculated curvature data (SCF) with the exact formula (EF) for the onedimensional Ising chain presented by Janyszek and Mrugała [12], corresponding to our case with q = 2. In Fig. 7, we show temperature- and magnetic field-dependent behaviours of R using two separate curves obtained by the present investigation (blue curve labelled as SCF) and exact calculation (red curve labelled as EF). To see how well the SCF calculation reproduces the EF results we have calculated both curves for the same set of exchange energy values given in Ref. [12]. The curvature as a function of temperature has a divergence to infinity at T = 0 when h = 0 (Fig. 7(a)). This is the behaviour similar to that of the exact curvature formula. We can also see the dependence of R on h through the SCF curve in Fig. 7(b). It is symmetric function of h as expected and this agrees to the exact result too. One difference between two calculations is that a remarkable shift (or a discrepancy) exists at lower temperatures in Fig. 7(a) and around h = 0 in Fig. 7(b).

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R. Erdem / Physica A 526 (2019) 121173

Fig. 6. (a) Exchange coupling (J) dependence of R at various temperatures for q = 6 and h = 0.05. (b) same as (a) but for various field values with T = 0.8.

5. Summary and conclusions In this paper, we have studied the thermodynamic geometry of Ising model with ferromagnetic exchange interactions using the SCF theory. Firstly, we have reviewed the system at equilibrium under the pair approximation. Then, the equilibrium solutions of self-consistent equations were reflected to the scalar curvature R of a two-dimensional phase space of M and C. The discontinuous phase transitions appeared as the intersection of two branches of R in the R-h plane while the continuous phase transitions are displayed as the singularities (divergence) in the R-T and R-h plots, as predicted in most references. Besides these findings, maxima (or peaks) of R are observed near or away from the criticality in all figures including R-J plots. Based on the previous discussions [22] we have used the locus of a maximum of R to predict the Widom lines in h-T plane. In order to explore these lines in more detail, their behaviours can be contrasted in a further study with several thermodynamic response functions, such as specific heat and isothermal susceptibility from Ref. [2]. It would also be great interest to use a similar metric in thermodynamic curvature calculations in other spin models which undergo continuous/ discontinuous phase transitions. The metric can be defined on any phase space of various order parameters and hence one can obtain more information about the attractive/repulsive character of microscopic interactions from the sign of curvature scalar (R < 0 or R > 0). Because, the situation is still unclear in the magnetic systems while much is known in the case of fluid systems [39–41]. Acknowledgements The author thanks Dr. U. Camcı for the critical reading of paper, and acknowledges the support from the Scientific Research Projects Coordination Unit of Akdeniz University.

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Fig. 7. (a) Temperature (T ) dependence of R for q = 2, J = 1.2 and h = 0. (b) Magnetic field (h) dependence of R for q = 2 and J /T = 0.5.

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