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Ising antiferromagnets in a nonzero uniform magnetic field
This content has been downloaded from IOPscience. Please scroll down to see the full text.
Download details:
IP Address: 200.52.182.236
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ournal of Statistical Mechanics:An IOP and SISSA journalJ Theory and Experiment
LETTER
Ising antiferromagnets in a nonzerouniform magnetic field
Chi-Ok Hwang1,2, Seung-Yeon Kim3, Daeseung Kang4 andJin Min Kim2
1 National Institute for Mathematical Sciences, Daejeon 305-340, Korea2 Department of Physics and Computer-Aided Molecular Design ResearchCenter, Soongsil University, Seoul 156-743, Korea3 School of Liberal Arts and Sciences, ChungJu National University, Chungju380-702, Korea4 Department of Electrical Engineering, Soongsil University, Seoul 156-743,KoreaE-mail: [email protected], [email protected], [email protected],[email protected] and [email protected]
Received 28 November 2006Accepted 16 April 2007Published 8 May 2007
Online at stacks.iop.org/JSTAT/2007/L05001doi:10.1088/1742-5468/2007/05/L05001
Abstract. We evaluate the density of states g(M,E) as a function of energy Eand magnetization M of Ising models on square and triangular lattices, using theexact enumeration method for small systems and the Wang–Landau method forlarger systems. From the density of states the average magnetization per spin,m(T, h), of the antiferromagnets has been obtained for any values of temperatureT and uniform magnetic field h. Also, based on g(M,E), the behaviour ofm(T, h) is understood microcanonically. The microcanonical approach revealsthe differences between the unfrustrated model (on the square lattice) and thefrustrated one (on the triangular lattice).
Ising antiferromagnets in a nonzero uniform magnetic field
Since the Onsager solution [1] of the square-lattice Ising model in the absence of amagnetic field, the Ising model has played a central role in our understanding of phasetransitions and critical phenomena. However, the Ising model in a nonzero magneticfield (except for the one-dimensional case) has been one of the most intriguing andoutstanding unsolved problems in the study of phase transitions. The phase transitions ofthe ferromagnets occurs only with no magnetic field [2, 3], whereas the phase transitionsof the antiferromagnets can occur even in a nonzero magnetic field [4]–[9]. Therefore,to understand the important properties of antiferromagnets, Ising-type antiferromagets(the simplest antiferromagnets) in a uniform magnetic field have been studied for severaldecades (with few successes) [6]–[9].
In 1960, Fisher [4] introduced a solvable toy model with non-magnetic spins (whosemagnetic moment is zero) on the square-lattice sites and magnetic spins on the bondswhere magnetic (or non-magnetic) spins do not interact with each other, and thevertical (horizontal) interactions between magnetic spins and non-magnetic spins areferromagnetic (antiferromagnetic). The exact solution of the square-lattice Fisher modelis known because this simple model in a uniform magnetic field is transformed into theOnsager solution of the square-lattice Ising model in the absence of a magnetic field.The solvable kagome-lattice Fisher model was also studied in 2005 [5]. However, theproperties of Ising antiferromagnets in a nonzero magnetic field are not known except infew cases, such as the critical line (the critical temperature as a function of a magneticfield) for square lattice [7, 8]. In the case of a triangular-lattice Ising antiferromagnet, ithas been a very difficult task to understand the critical-line property, especially near zerotemperature in a small magnetic field h [10]–[12]. Only in a recent study [9], have Qianet al shown that a renormalization-mapping method is successful in understanding thecritical-line properties of a triangular-lattice Ising antiferromagnet near zero temperaturein a magnetic field.
Furthermore, the properties of thermodynamic functions such as average magneti-zation as a function of temperature and magnetic field have been little known for Isingantiferromagnets in a uniform magnetic field. Recently, microcanonical approaches havebeen widely used to understand phase transformations in small systems such as nanosys-tems, atomic clusters, nuclear systems, biological systems, and astrophysical objects [13].These approaches are mainly based on the density of states g(E) as a function of energy Eonly. However, the more general microcanonical approaches based on the density of statesg(M, E) as a function of both energy E and magnetization M have not been studied well.In this paper, we develop the microcanonical approach based on g(M, E) and apply thismethod for understanding the behaviour of the average magnetization m(T, h) of Isingantiferromagnets in a uniform magnetic field.
In this study, we deal with a square-lattice Ising antiferromagnet (representingunfrustrated system) and a triangular-lattice Ising antiferromagnet (representingfrustrated system). The Hamiltonian H of the Ising model in a uniform magnetic field H isH = −JE−HM for the system size N = L2. Here E =
∑〈i,j〉 σiσj is the exchange energy,
M =∑N
i=1 σi the total magnetization, J the coupling constant (J > 0 for ferromagnetsand J < 0 for antiferromagnets), 〈i, j〉 denotes distinct pairs of nearest-neighbour sites,and σi = ±1. The partition function of the system is given by
Ising antiferromagnets in a nonzero uniform magnetic field
Figure 1. Entropy S(M,E) = ln[g(M,E)] for (a) the 10× 10 square-lattice Isingmodel (exact values) and (b) the 12 × 12 triangular-lattice Ising model (Wang–Landau results).
where Etot = E − hM , h = H/(−J), g(M, E) is the density of states, and a = eJ/kBT .The average of a physical quantity A is
〈A(T, h)〉 =
∑M,E Ag(M, E)aEtot
Z. (2)
We have evaluated the exact integer values for g(M, E) of Ising models on L × Lsquare lattices (up to L = 10) and triangular ones (up to L = 9) with periodic boundaryconditions by using the microcanonical transfer matrix [3, 8, 14]5. For bigger lattices (wetested up to L = 32 for square lattices and L = 30 for triangular ones), we obtaing(M, E) by implementing the Wang–Landau algorithm [15]6. As a test, we calculatedm(T, h) by the Wang–Landau algorithm for 10 × 10 square and 9 × 9 triangular lattices;it was noticed that they agree well with m(T, h) from the exact results, indicating thatthe Wang–Landau algorithm works well. Figure 1 shows the microcanonical entropy [13]
5 For the first time, the exact g(M,E)′s for triangular lattices with periodic boundary conditions and for a 10×10square lattice are calculated.6 This is the first time that g(M,E) s from both the exact method and the Wang–Landau algorithm have beencompared, though g(E) has been compared before [15]. They are in good agreement (the typical average relativeerror for an L = 9 triangular lattice is around 0.3%).
Ising antiferromagnets in a nonzero uniform magnetic field
Figure 2. Average magnetization m(T, h) as a function of T (in unit of |J |/kB)for given values of h, of the AF Ising models on (a) the square lattice with L = 10and (b) the triangular lattice with L = 12. For h = 0, all the magnetization iszero. The critical magnetic field hc is 4 (square lattice) and 6 (triangular lattice).At T = 0, m(T, h) is 0 (square lattice) and 1/3 (triangular lattice) for h < hc,while it is 1 for h > hc. For h = hc, m(T, h) goes to a nontrivial constant atT = 0.
S(M, E) = ln[g(M, E)] for the Ising model on 10 × 10 square and 12 × 12 triangularlattices.
From the partition function Z(T, h), by using equation (2), the average magnetizationm(T, h) = 〈M〉/N of the antiferromagnetic (AF) Ising model has been evaluated for thesquare-lattice AF Ising model and the triangular-lattice AF Ising model. Figure 2 showsm(T, h) as a function of T for different values of h.
The overall behaviour of m(T, h) on the square lattice in figure 2 is similar to that ofthe solvable toy models, Fisher models [4, 5]7. However, the overall behaviour of m(T, h) onthe triangular lattice in the figure is quite different from that of the Fisher models. Whenh = 0, the magnetization for all temperature is zero for both lattices. The critical magneticfields are hc = 4 for the square lattice and hc = 6 for the triangular lattice [11, 16], whereflipping spins not neighbouring one another does not change Etot. In the case of h > hc,as T decreases, m(T, h) increases monotonically to 1. When h = hc, the magnetizationat T = 0 goes to a nontrivial value of the critical magnetization mc ≡ m(0, hc). For theunfrustrated (square lattice) system (figure 2(a)), when 0 < h < hc, m(T, h) increases
7 Also, the overall behaviour of the critical line for a square-lattice Ising antiferromagnet [7, 8] is similar to thatof the Fisher models [4, 5].
Ising antiferromagnets in a nonzero uniform magnetic field
Figure 3. Top views of figure 1 for (a) a square lattice and (b) a triangular lattice.A given value of h determines the corresponding line satisfying E = hM + Emin
tot .
to a maximum and decreases rapidly to 0 as T decreases. Therefore m(T, 0 < h < hc)becomes 0 at both zero and infinite temperatures. m(∞, h) = 0 comes from random spinorientations but m(0, h) = 0 from AF order. For the frustrated (triangular lattice) system(figure 2(b)), more complex behaviour is observed when 0 < h < hc. For 3 < h < hc, thebehaviour of m(T, h) is similar to that of the unfrustrated system except for m(0, h) = 1/3,due to frustration. For 0 < h < 3, m(T, h) increases monotonically to 1/3 as T decreases,and in particular, it increases rapidly at lower temperatures for small h.
Now, we investigate the relation of the average magnetization to the microcanonicalentropy S(M, E), that is, the density of states g(M, E). Figure 3 shows the top views ofS(M, E), which we call ME diagrams. As shown in figure 3, the ME diagram has threevertices for the square lattice and four vertices for the triangular lattice. In the diagram,the top right (left) vertices correspond to the state of all spins up (down). In figure 3(a)the bottom vertex corresponds to the unfrustrated AF ground states. The bottom linein figure 3(b) corresponds to the frustrated AF ground states, and the right (left) vertexof the bottom line to the threefold-degenerate ordered state with m = 1/3(−1/3). Notethat the slope of the line connecting the top right vertex and the bottom (bottom right)vertex in the square (triangular) lattice is 4 (6), the critical magnetic field hc. On the MEdiagram, the critical magnetic fields are geometrically manifested as the slopes of the edgelines. For h > hc (h < hc), only the top right (bottom or bottom right) vertex gives the
Ising antiferromagnets in a nonzero uniform magnetic field
nonzero g(M, E) satisfying E = hM + Emintot . There are a tremendous number of ground
states (g(M, E)′s for many (M, E) pairs) along the edge lines with slopes hc satisfyingE = hcM + Emin
tot . Here, Emintot is the minimum energy among Etot = E − hcM for all
possible (M, E) pairs. Also, there are a huge number of frustrated AF ground states onthe bottom line (whose slope is h = 0) in figure 3(b) [18].
From the ME diagram (figure 3), we can understand the properties of magnetizationm(T, h) of the AF Ising model in a magnetic field (figure 2). At T = 0 (a = 0), fromequation (2) the average magnetization per site becomes
〈m(T = 0, h)〉 =
∑{M,E;Emin
tot } Mg(M, E)
N∑
{M,E;Emintot } g(M, E)
. (3)
Here, N = L2 is the system size. The summation is over all (M, E) pairs satisfyingEmin
tot = E − hM , that is, all minimum ground state(s). The unity magnetizationm(0, h > hc) of figure 2 results from the state of all spins up (AF paramagnetic orFM ground state) at the top right vertex of the microcanonical entropy plane. Forh = hc, the different states from numerous (M, E) points along the right edge linecontribute to the ground states ranging from the FM ground state to AF ground states.These states determine the critical magnetization mc = m(0, hc) [19], estimated to be0.5482(4) for square lattice and 0.675 27(6) for triangular lattice. The magnetizationm(0, 0 < h < hc) = 0 (1/3) shown in figure 2(a) (figure 2(b)) results from the AF groundstates at the bottom (right) vertex of the microcanonical entropy plane. The line definedby E = hM + Emin
tot , in figure 3, determines the ground-state properties of the AF Isingmodel at the given value of h.
At T = 0, m(0, h) is related to the corresponding g(M, E) on the microcanonicalentropy plane. Here, for T > 0 we try to understand m(T, h) via the correspondingg(M, E) on the microcanonical entropy plane. From the canonical probabilityg(M, E)aEtot/Z for a given value of h, heuristically it is easily inferred that the g(M, E)s of (M, E) pairs satisfying E = hM + Etot, for Emin
tot < Etot < 0, will determine m(T, h)for T > 0. The change of the y-intercept Etot from Emin
tot to 0 for T > 0 is the paralleltranslation of the line E = hM+Emin
tot at a given value of h. On a single line E = hM+Etot
with the slope h and a given intercept Etot, we can find the unique point (M0, E0) withthe maximum entropy S(M0, E0), which we call the maximum point for short. For a givenvalue of h, the magnetization behaviour of the AF Ising model is determined mainly by themaximum points. The maximum trajectory is made from the maximum points by varyingEtot for the given value of h. Figure 4 shows the maximum trajectories for different valuesof h. The behaviours of these trajectories are the microcanonical origin of the canonicalmagnetization behaviours in figure 2. For a given magnetic field we can understand theproperties of the magnetization following the trajectory of the maximum points. As anexample, for the triangular h = 4 case, by following the maximum trajectory and observingthe change of M divided by N in the ME diagram we can understand the magnetizationbehaviour that the magnetization increases up to a certain point and decreases to m = 1/3as temperature decreases to zero.
In this work, with both the exact enumeration method for small systems and theWang–Landau method for larger systems, the density of states g(M, E) for energy Eand magnetization M of the Ising models on square and triangular lattices has beenevaluated. Using the density of states, we obtain the canonical magnetization per spin
Ising antiferromagnets in a nonzero uniform magnetic field
Figure 4. Maximum trajectories on the ME diagram for different values of h ofthe AF Ising models on (a) the square lattice with L = 10 and (b) the triangularlattice with L = 12. Given h, there is a family of lines E = hM +Etot on the MEdiagram whose intercept Etot changes from Emin
tot to 0. Each line E = hM + Etot
with slope h and intercept Etot has a maximum point (M0, E0) where the entropyis the maximum on the line. The maximum trajectory is made from the maximumpoints by varying Etot for the given value of h.
of the Ising antiferromagnets on square and triangular lattices in a uniform magneticfield at any values of temperature. Also, the microcanonical properties of the averagemagnetization are understood based on the density of states (the so-called ME diagram).The microcanonical approach reveals the differences between the unfrustrated model (onthe square lattice) and the frustrated one (on the triangular lattice).
In addition, we should note that our microcanonical approach for the averagemagnetization property is general enough to be applicable to other thermodynamicquantities of any spin lattice system in a magnetic field.
This work was supported by the Korea Research Foundation Grant funded by the KoreanGovernment (MOEHRD) (KRF-2005-005-J01103).
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