Inequality for local energy of Ising models with quenched randomness and its application
JJournal of the Physical Society of Japan
Inequality for local energy of Ising model with quenched randomnessand its application
Manaka Okuyama ∗ and Masayuki Ohzeki , , Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 226-8503, Japan Sigma-i Co. Ltd., Tokyo 108-0075, Japan
In this study, we extend the lower bound on the average of the local energy of the Ising modelwith quenched randomness [J. Phys. Soc. Jpn.
1. Introduction
Spin-glass models describe magnetic materials that randomly interact spatially. The meanfield theory of spin-glass models, e.g., the Sherrington–Kirkpatrick model, has been solvedrigorously by the full replica symmetry breaking solution; however, it is extremely di ffi cultto obtain analytical results for finite-dimensional models, except on the Nishimori line. Al-though analytical approaches for two-dimensional systems are slightly progressing, thosefor three-dimensional systems have been primarily neglected, except numerical analysis.In ferromagnetic spin models, correlation inequalities play an important role in non-perturbative analysis and yield rigorous results for unsolvable models. Correlation inequal-ities are also valid for the Ising model in a random field. A recent study proved based onthe Fortuin–Kasteleyn–Ginibre inequality that the random-field Ising model comprising two-body interactions for all the lattice and field distributions does not have a spin-glass phase.Therefore, it is expected that the concept of correlation inequalities will be important for arigorous analysis of spin-glass models, and their establishment for spin-glass models is a very ∗ [email protected] 1 / a r X i v : . [ c ond - m a t . d i s - nn ] A p r . Phys. Soc. Jpn. important problem.Some previous studies have been conducted on the correlation inequalities in spin-glassmodels. A recent study
8, 9) exhibited that the response of the quenched average of a parti-tion function with respect to the variance is generally positive, which is considered as thecounterpart of the Gri ffi ths first inequality in spin-glass models. In addition, for various bondrandomness, including Gaussian and binary distribution types, it is shown that the counterpartof the Gri ffi ths second inequality holds on the Nishimori line.
10, 11)
However, correlation in-equalities, as in the case of ferromagnetic spin models, have not been obtained in general, anda rigorous analysis based on them is yet to be conducted satisfactorily for spin-glass models.In this study, we obtain a lower bound on the quenched average of the local energy for theIsing model with quenched randomness. The result of a previous study that was limited to asymmetric distribution is generalized to an asymmetric distribution. Furthermore, as a simpleapplication of the acquired inequality, we obtain the correlation inequalities for a Gaussiandistribution. We demonstrate that the expectation of the square of the correlation functiongenerally has a finite lower bound at any temperature. Thus, we prove that the spin-glassorder parameter has a finite lower bound in the Ising model in a Gaussian random field,regardless of the forms of the other interactions.The organization of this paper is as follows. In Sec. II, we define the model and present themethod to obtain the lower bound on the average of the local energy for the Ising model withquenched randomness. In Sec. III, we describe the application of the acquired inequality whenthe randomness of the interactions follows a Gaussian distribution. Finally, our conclusion ispresented in Sec. IV.
2. Lower bound on local energy for asymmetric distribution of randomness
Following Ref., we consider a generic form of the Ising model, H = − (cid:88) B ⊂ V J B σ B , (1) σ B ≡ (cid:89) i ∈ B σ i , (2)where V is the set of sites, the sum over B is over all the subsets of V in which interactionsexist, and the lattice structure adopts any form. The probability distribution of a randominteraction J B is represented as P B ( J B ). The probability distributions can be generally di ff erentfrom each other, i.e., P B ( x ) (cid:44) P C ( x ), and are also allowed to present no randomness, i.e., P B ( J B ) = δ ( J − J B ). /
9. Phys. Soc. Jpn.
The correlation function for a set of fixed interactions, { J B } , is expressed as (cid:104) σ A (cid:105) { J B } = Tr σ A exp (cid:0) β (cid:80) B ⊂ V J B σ B (cid:1) Tr exp (cid:0) β (cid:80) B ⊂ V J B σ B (cid:1) . (3)The configurational average over the distribution of the randomness of the interactions iswritten as E (cid:2) g ( { J B } ) (cid:3) = (cid:89) B ⊂ V (cid:90) ∞−∞ dJ B P B ( J B ) g ( { J B } ) . (4)For example, the quenched average of the correlation function is obtained as E (cid:2) (cid:104) σ A (cid:105) { J B } (cid:3) = (cid:89) B ⊂ V (cid:90) ∞−∞ dJ B P B ( J B ) Tr σ A exp (cid:0) β (cid:80) B ⊂ V J B σ B (cid:1) Tr exp (cid:0) β (cid:80) B ⊂ V J B σ B (cid:1) . (5)Our result is the following theorem. Theorem 2.1
When the distribution function of the randomness satisfiesP A ( − J A ) = exp( − β N L J A ) P A ( J A ) , (6) then for any even function f ( J A ) ≥ , the system defined above satisfies the following inequal-ity: E (cid:2) − J A f ( J A ) (cid:104) σ A (cid:105) { J B } (cid:3) ≥ E (cid:34) − J A f ( J A ) tanh( β J A ) − J A f ( J A )(1 − e − β N L J A ) 1sinh(2 β J A ) (cid:35) . (7)We note that the right-hand side of Eq. (7) does not depend on the other interactions. Whenthe distribution function is symmetric, i.e., P A ( − J A ) = P A ( J A ) ( β N L =
0) and f ( J A ) =
1, Eq.(7) is reduced to E (cid:2) − J A (cid:104) σ A (cid:105) { J B } (cid:3) ≥ E (cid:2) − J A tanh( β J A ) (cid:3) , (8)which is in accordance with the result in Ref. In this case, an intuitive explanation of theinequality is possible: the local energy is generally larger than or equal to the energy in theabsence of all the other interactions. However, for β N L (cid:44)
0, it is di ffi cult to provide an intuitiveexplanation, because the second term in the right-hand side of Eq. (7) does not have a physicalrelevance. On the other hand, the right-hand side of Eq. (7) can be rewritten as E (cid:34) − J A f ( J A ) tanh( β J A ) − J A f ( J A )(1 − e − β N L J A ) 1sinh(2 β J A ) (cid:35) = E (cid:2) − J A f ( J A ) tanh( β J A ) (cid:3) − (cid:90) ∞ dJ A P A ( J A ) J A f ( J A ) e − β N L J A (1 − e β N L J A ) β J A ) ≤ E (cid:2) − J A f ( J A ) tanh( β J A ) (cid:3) , (9)which suggests that the local energy can be lower than the energy in the absence of all theother interactions, unlike in the case of β N L =
0. We also note that the second term in the /
9. Phys. Soc. Jpn. right-hand side of Eq. (7) is numerically very small. Thus, to establish the bound for β N L (cid:44) Proof.
We define Z ( β, J A ) and (cid:104) σ A (cid:105) J A as Z ( β, J A ) = (cid:88) { σ } exp β (cid:88) B ⊂ V \ A J B σ B + β J A σ A , (10) (cid:104) σ A (cid:105) J A = (cid:80) { σ } σ A exp (cid:16) β (cid:80) B ⊂ V \ A J B σ B + β J A σ A (cid:17)(cid:80) { σ } exp (cid:16) β (cid:80) B ⊂ V \ A J B σ B + β J A σ A (cid:17) . (11)We note that (cid:104) σ A (cid:105) J A = (cid:104) σ A (cid:105) { J B } but (cid:104) σ A (cid:105) − J A (cid:44) (cid:104) σ A (cid:105) { J B } . Subsequently, we obtain Z ( β, J A ) Z ( β, − J A ) = cosh(2 β J A ) + (cid:104) σ A (cid:105) − J A sinh(2 β J A ) = e β N L J A + Γ ( β, − J A ) sinh(2 β J A ) J A ≥ , (12) Z ( β, − J A ) Z ( β, J A ) = cosh(2 β J A ) − (cid:104) σ A (cid:105) J A sinh(2 β J A ) = e − β N L J A + Γ ( β, J A ) sinh(2 β J A ) J A ≥ , (13)where Γ ( β, J A ) is defined as Γ ( β, J A ) ≡ − J A (cid:104) σ A (cid:105) J A + J A tanh( β J A ) + (1 − e − β N L J A ) J A sinh(2 β J A ) . (14)Since Eq. (12) is the reciprocal of Eq. (13), we obtain e − β N L J A Γ ( β, − J A ) = − e − β N L J A Γ ( β, J A ) e − β N L J A + Γ ( β, J A ) sinh(2 β J A ) J A . (15)Then, from Eq. (14), we immediately obtain E (cid:2) − J A f ( J A ) (cid:104) σ A (cid:105) { J B } (cid:3) = E (cid:34) f ( J A ) Γ ( β, J A ) − J A f ( J A ) tanh( β J A ) − J A f ( J A )(1 − e − β N L J A ) 1sinh(2 β J A ) (cid:35) . (16)Furthermore, for any even function f ( J A ) ≥
0, we obtain E (cid:2) f ( J A ) Γ ( β, J A ) (cid:3) ≥
0, because E (cid:2) f ( J A ) Γ ( β, J A ) (cid:3) = (cid:90) ∞−∞ dJ A P A ( J A ) f ( J A ) E (cid:2) Γ ( β, J A ) (cid:3) (cid:48) = (cid:90) ∞ dJ A P A ( J A ) f ( J A ) E (cid:2) Γ ( β, J A ) + exp( − β N L J A ) Γ ( β, − J A ) (cid:3) (cid:48) = (cid:90) ∞ dJ A P A ( J A ) f ( J A ) E Γ ( β, J A ) sinh(2 β J A ) J A e − β N L J A + Γ ( β, J A ) sinh(2 β J A ) J A (cid:48) ≥ , (17) /
9. Phys. Soc. Jpn. where E [ · · · ] (cid:48) denotes the configurational average over the randomness of the interactionsother than J A . We used Eq. (15) in the third identity and Eq. (13) in the last inequality. Thus,Eqs. (16) and (17) yield Eq. (7). (cid:50)
3. Application to Gaussian spin-glass model
In this section, we present the application of Eq. (7) to a spin-glass model with a Gaussiandistribution. We note that the result for β N L = is su ffi cient to obtain the inequalitiesthat are presented in this section.First, we consider the distinct case, P A ( J , A − J A ) = P A ( J , A + J A ). Then, we obtain thefollowing result: Corollary 3.1
When the distribution function of the randomness satisfiesP A ( J , A − J A ) = P A ( J , A + J A ) , (18) then for any even function f ( J A ) ≥ , the system defined above satisfies the following inequal-ity: E (cid:2)(cid:0) J , A − J A (cid:1) f ( J A − J , A ) (cid:104) σ A (cid:105) { J B } (cid:3) ≥ E (cid:2)(cid:0) J , A − J A (cid:1) f ( J A − J , A ) tanh( J A − J , A ) (cid:3) . (19) Proof.
We regard P A ( J , A + J A ) as a new probability distribution P (cid:48) A ( J A ), where P (cid:48) A ( J A ) issymmetric. Therefore, using Eq. (7) for β N L =
0, we obtain E (cid:2)(cid:0) J , A − J A (cid:1) f ( J A − J , A ) (cid:104) σ A (cid:105) { J B } (cid:3) = (cid:90) ∞−∞ dJ A P A ( J , A + J A ) E (cid:104) − J A f ( J A ) (cid:104) σ A (cid:105) J A + J , A (cid:105) (cid:48) ≥ (cid:90) ∞−∞ dJ A P A ( J , A + J A ) ( − J A ) f ( J A ) tanh( J A ) = E (cid:2)(cid:0) J , A − J A (cid:1) f ( J A − J , A ) tanh( J A − J , A ) (cid:3) . (20) (cid:50) In the following, using Eq. (19), we obtain several inequalities.
Next, we consider the case where all the interactions follow a Gaussian distributionwith mean J , B and variance Λ B . All the J , B and Λ B can adopt di ff erent values. We denotethe configurational average over the distribution of the randomness of the interactions as E [ · · · ] { J , B , Λ B } . Then, we obtain the following result: Corollary 3.2
For the quenched average of the square of the correlation function, we obtain /
9. Phys. Soc. Jpn. a lower bound, E (cid:104) tanh ( β J A ) (cid:105) { , Λ A } ≤ E (cid:104) (cid:104) σ A (cid:105) { J B } (cid:105) { J , B , Λ B } . (21)We note that the left-hand side of Eq. (21) is independent of mean { J , b } . Inequality (21)indicates that the expectation of the square of the correlation function is generally a finitenon-zero value, regardless of the other interactions. This behavior is quite di ff erent fromthose of the correlation functions of ferromagnetic models and may reflect the fact that thecounterpart of the Gri ffi ths second inequality has not been established in spin-glass models. Proof.
For the Gaussian distribution with mean J , B and variance Λ B , and f ( J A ) =
1, Eq.(19) is reduced to E (cid:2) ( J , A − J A ) (cid:104) σ A (cid:105) { J B } (cid:3) { J , B , Λ B } = E (cid:104) − J A (cid:104) σ A (cid:105) { J B + J , B } (cid:105) { , Λ B }≥ E (cid:2) − J A tanh( β J A ) (cid:3) { , Λ B } . (22)Furthermore, conducting integration by parts, we obtain Eq. (21). (cid:50) A similar calculation is possible for higher order terms. Taking f ( J A ) = J A in Eq. (19), weobtain E (cid:104) − ( J A − J , A ) (cid:104) σ A (cid:105) { J B } (cid:105) { J , B , Λ B } = E (cid:104) − J A (cid:104) σ A (cid:105) { J B + J , B } (cid:105) { , Λ B }≥ E (cid:104) − J A tanh( β J A ) (cid:105) { , Λ B } . (23)Conducting integration by parts and using Eq. (21), we obtain the lower bound on the expec-tation of the fourth power of the correlation function, E (cid:34) β Λ A (cid:16) β Λ A − (cid:17) (cid:16) (cid:104) σ A (cid:105) { J B + J , B } − tanh ( β J A ) (cid:17) + tanh ( β J A ) (cid:35) { , Λ A } ≤ E (cid:104) (cid:104) σ A (cid:105) { J B } (cid:105) { J , B , Λ B } . (24)For 8 β Λ B ≥
3, Eqs. (21) and (24) yield E (cid:104) tanh ( β J A ) (cid:105) { , Λ A } ≤ E (cid:104) (cid:104) σ A (cid:105) { J B } (cid:105) { J , B , Λ B } . (25)Thus, for a su ffi ciently high temperature, the quenched average of the fourth power of thecorrelation function has a non-zero lower bound. Finally, we demonstrate that the spin-glass order-parameter in the Ising model in a Gaus-sian random field generally adopts a finite value at any temperature, regardless of the formsof the other interactions.We consider the case where a random field, { h i } , is independently applied to all the sites, /
9. Phys. Soc. Jpn. where { h i } follows a Gaussian distribution with mean J and variance Λ . The Hamiltonian isobtained as H = − (cid:88) B ⊂ V J B σ B = − (cid:88) B ⊂ V \{ h i } J B σ B − N (cid:88) i = h i σ i , (26)where interaction J B other than { h i } takes any form. Then, Eq. (21) is reduced to E (cid:104) tanh ( β h i ) (cid:105) { , Λ } ≤ E (cid:104) (cid:104) σ i (cid:105) { J B } (cid:105) { J , Λ } , (27)which suggests that the quenched average of the square of the local magnetization has a non-zero value.Furthermore, because the same inequality holds for all the sites, we obtain the followingresult: Corollary 3.3
For the spin-glass order-parameter, q,q = N (cid:88) i E (cid:104) (cid:104) σ i (cid:105) { J B } (cid:105) { J , Λ } , (28) the system (26) satisfies the following inequality: E (cid:104) tanh ( β h i ) (cid:105) { , Λ } ≤ q . (29)Thus, when a Gaussian random field is applied, the spin-glass order-parameter generally hasa non-zero lower bound. In ferromagnetic models, the ferromagnetic order parameter, i.e.,magnetization, has a finite value when a magnetic field is applied. Equation (29) suggeststhat a similar phenomenon occurs in the Ising model in a Gaussian random field. This is anatural consequence; however, the existence of a finite lower bound is not obvious.In addition, we note that Eq. (29) does not indicate that there is a spin-glass phase in theIsing model in a Gaussian random field.
4. Conclusions
In this study, we have obtained the lower bound on the local energy of the Ising modelwith quenched randomness. We emphasize that the acquired inequality (7) is independent ofthe other interactions. Our result is a natural generalization of Ref. in which a symmetricdistribution was considered.Applying the obtained inequality to a Gaussian spin-glass model, we determine that theexpectation of the square of the correlation function generally has a finite lower bound at any /
9. Phys. Soc. Jpn. temperature. Thus, the spin-glass order-parameter in the Ising model in a Gaussian randomfield generally adopts a finite value at any temperature, which is a natural but not a obviousresult.It is an interesting question whether a similar inequality as Eq. (21) will hold for a gen-eral distribution function of the random interactions. Our proof relies on the property of aGaussian distribution, and we have not obtained proofs for other distributions.
Acknowledgment
The authors thank Shuntaro Okada for useful discussions. The present work was finan-cially supported by JSPS KAKENHI Grant Nos. 18H03303, 19H01095, and 19K23418, andJST-CREST grant (No. JPMJCR1402) of the Japan Science and Technology Agency. /
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