Power-Law Decay Exponents of Nambu-Goldstone Transverse Correlations
aa r X i v : . [ m a t h - ph ] S e p Power-Law Decay Exponents of Nambu-GoldstoneTransverse Correlations
Tohru Koma Abstract:
We study a quantum antiferromagnetic Heisenberg model on a hypercubiclattice in three or higher dimensions d ≥
3. When a phase transition occurs with thecontinuous symmetry breaking, the nonvanishing spontaneous magnetization which is ob-tained by applying the infinitesimally weak symmetry breaking field is equal to the max-imum spontaneous magnetization at zero or non-zero low temperatures. In addition, thetransverse correlation in the infinite-volume limit exhibits a Nambu-Goldstone-type slowdecay. In this paper, we assume that the transverse correlation decays by power law withdistance. Under this assumption, we prove that the power is equal to 2 − d at non-zero lowtemperatures, while it is equal to 1 − d at zero temperature. The method is applied alsoto a quantum XY model and a classical Heisenberg model at non-zero low temperaturesin three or higher dimensions. The resulting power is given by the same 2 − d at non-zerolow temperatures. Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588,JAPAN, e-mail: [email protected]
Introduction
In general, when phase transitions occur with continuous symmetry breaking in many-body systems, there appear Nambu-Goldstone modes [1, 2, 3, 4] which are reflected ina slow decay of the transverse correlations. In order to elucidate the universal natureof Nambu-Goldstone modes, we study classical and quantum lattice spin systems whichexhibit phase transitions at zero or low non-zero temperatures with continuous symmetrybraking [5, 6, 7, 8, 9]. In the previous paper [10], we studied a quantum antiferromagneticHeisenberg model as an example, and proved that, when the spontaneous magnetizationis non-vanishing at zero or non-zero low temperatures, the transverse correlations in theinfinite-volume limit exhibit a Nambu-Goldstone-type slow decay with distance. However,this does not necessarily imply a power-law decay of the transverse correlations with largedistance. In this paper, we make a natural assumption that the transverse correlationsshow a power-law decay with large distance. Under this assumption, we can determine thepower of the decay of the transverse correlations. More precisely, for the antiferromagneticHeisenberg model in three or higher dimensions d ≥
3, the power is given by 2 − d at non-zero temperatures, while it is given by 1 − d at zero temperature. Our approach is alsoapplied to a quantum XY model and a classical Heisenberg model at non-zero temperaturesin three or higher dimensions. The resulting power is given by the same 2 − d . In this and the following two sections, we will treat only a quantum Heisenberg antifer-romagnet which has reflection positivity [6, 7]. The treatment of a quantum XY modeland a classical Heisenberg model both of which have reflection positivity [5, 6] is givenin Sec. 5 because our approach to these two models is the same as that to the quantumHeisenberg antiferromagnet.Let Γ be a finite subset of the d -dimensional hypercubic lattice Z d , i.e., Γ ⊂ Z d , with d ≥
3. For each site x = ( x (1) , x (2) , . . . , x ( d ) ) ∈ Γ, we associate three component quantumspin operator S x = ( S (1) x , S (2) x , S (3) x ) with magnitude of spin, S = 1 / , , / , , . . . . Moreprecisely, the spin operators, S (1) x , S (2) x , S (3) x , are (2 S + 1) × (2 S + 1) matrices at the site x .They satisfy the commutation relations,[ S (1) x , S (2) x ] = iS (3) x , [ S (2) x , S (3) x ] = iS (1) x , and [ S (3) x , S (1) x ] = iS (2) x , and ( S (1) x ) + ( S (2) x ) + ( S (3) x ) = S ( S + 1) for x ∈ Γ. For the finite lattice Γ, the wholeHilbert space is given by H Γ = O x ∈ Γ C S +1 . More generally, the algebra of observables on H Γ is given by A Γ := O x ∈ Γ M S +1 ( C ) , where M S +1 ( C ) is the algebra of (2 S + 1) × (2 S + 1) complex matrices. When two finitelattices, Γ and Γ , satisfy Γ ⊂ Γ , the algebra A Γ is embedded in A Γ by the tensor2roduct A Γ ⊗ I Γ \ Γ ⊂ A Γ with the identity I Γ \ Γ . The local algebra is given by A loc = [ Γ ⊂ Z d : | Γ | < ∞ A Γ , where | Γ | is the number of the sites in the finite lattice Γ. The quasi-local algebra is definedby the completion of the local algebra A loc in the sense of the operator-norm topology.Consider a d -dimensional finite hypercubic lattice,Λ := {− L + 1 , − L + 2 , . . . , − , , , . . . , L − , L } d ⊂ Z d , (2.1)with a large positive integer L and d ≥
3. The Hamiltonian H (Λ) ( B ) of the Heisenbergantiferromagnet on the lattice Λ is given by H (Λ) ( B ) = H (Λ)0 − BO (Λ) , (2.2)where the first term in the right-hand side is the Hamiltonian of the nearest neighborspin-spin antiferromagnetic interactions, H (Λ)0 := X { x,y }⊂ Λ: | x − y | =1 S x · S y , (2.3)and the second term is the potential due to the external magnetic field B ∈ R with theorder parameter, O (Λ) := X x ∈ Λ ( − x (1) + x (2) + ··· + x ( d ) S (3) x . Here, we impose the periodic boundary condition.The thermal expectation value at the inverse temperature β is given by h· · ·i (Λ) B,β := 1 Z (Λ) B,β
Tr ( · · · ) e − βH (Λ) ( B ) , (2.4)where Z (Λ) B,β is the partition function. The infinite-volume thermal equilibrium state isgiven by ρ B,β ( · · · ) = weak ∗ - lim Λ ր Z d h· · ·i (Λ) B,β . (2.5)Here, if necessary, we take a suitable sequence of the finite lattices Λ in the weak ∗ limit sothat the expectation value converges to a linear functional [10]. Similarly, we write ρ ,β ( · · · ) := weak ∗ - lim B ց ρ B,β ( · · · ) . (2.6)Then, the spontaneous magnetization m s ,β is given by m s ,β := lim Λ ր Z d | Λ | ρ ,β ( O (Λ) ) (2.7)with the hypercubic lattice Λ of (2.1) with the even side length 2 L . Because of thetranslational invariance with period 2, the right-hand side dose not depend on the size ofthe lattice Λ. 3he ground-state expectation value is given by ω (Λ) B ( · · · ) := lim β →∞ h· · ·i (Λ) B,β . (2.8)We also write ω ( · · · ) := weak ∗ - lim B ց weak ∗ - lim Λ ր Z d ω (Λ) B ( · · · ) . (2.9)The corresponding spontaneous magnetization m s at zero temperature is given by m s := lim Λ ր Z d | Λ | ω ( O (Λ) ) . (2.10)In the previous paper [10], we proved that the Nambu-Goldstone transverse correla-tions show a slow decay with large distance. In this paper, we require slightly strongerassumptions at zero and non-zero temperatures as follows: Assumption 2.1
We assume that for non-zero temperatures β − satisfying m s ,β > , thecorresponding transverse correlation decays by power law, i.e., (cid:12)(cid:12) ρ ,β ( S (1) x S (1) y ) (cid:12)(cid:12) ∼ K ( β ) | x − y | d − η (2.11) for a large | x − y | with an exponent η , where K ( β ) is a positive function of β .Remark: Theorem 2.5 in the previous paper [10] rules out the possibility of a rapid decay o ( | x − y | − ( d − ) for the transverse correlation ρ ,β ( S (1) x S (1) y ) at non-zero temperatures, where o ( ε ) denotes a quantity q ( ε ) such that q ( ε ) /ε is vanishing in the limit ε ց
0. Therefore,the exponent η must satisfy 2 − d < η ≤ Assumption 2.2
We assume that m s > , and that the corresponding transverse corre-lation decays by power law at zero temperature, i.e., (cid:12)(cid:12) ω ( S (1) x S (1) y ) (cid:12)(cid:12) ∼ Const . | x − y | d − η ′ (2.12) for a large | x − y | with an exponent η ′ .Remark: Similarly, Theorem 2.3 in the previous paper [10] rules out the possibility of arapid decay o ( | x − y | − ( d − ) for the transverse correlation ω ( S (1) x S (1) y ) at zero temperature.This implies that the exponent η ′ must satisfy 2 − d < η ′ ≤ Theorem 2.3
Under Assumption 2.1, the following is valid: If the spontaneous magne-tization m s ,β is nonvanishing at non-zero temperatures β − in dimensions d ≥ , then thecorresponding transverse correlation decays by power law, ρ ,β ( S (1) x S (1) y ) ∼ ( − x (1) + ··· + x ( d ) ( − y (1) + ··· + y ( d ) K ( β ) | x − y | d − , (2.13) for a large | x − y | , where K ( β ) is a positive function of β . Namely, the exponent is givenby η = 0 . heorem 2.4 Under Assumption 2.2, the following is valid: If the spontaneous magneti-zation m s is nonvanishing at zero temperature in dimensions d ≥ , then the correspondingtransverse correlation decays by power law, ω ( S (1) x S (1) y ) ∼ ( − x (1) + ··· + x ( d ) ( − y (1) + ··· + y ( d ) K | x − y | d − , (2.14) for a large | x − y | , where K is a positive constant. Namely, the exponent is given by η ′ = 1 . Consider first the case of non-zero temperatures, and we will give a proof of Theorem 2.3in this section. To begin with, we recall the previous result of [10] about the transversecorrelation. From Theorem 2.5 of [10], the rapid decay o ( | x − y | − ( d − ) for the transversecorrelation is excluded. This implies that the exponent η of (2.11) must satisfy 2 − d <η ≤
0. Therefore, in order to prove Theorem 2.3, it is sufficient to show η ≥ Lemma 3.1
The following bound is valid: ( − x (1) + ··· + x ( d ) ( − y (1) + ··· + y ( d ) h S (1) x S (1) y i (Λ) B,β ≥ for any x, y ∈ Λ .Proof: We first introduce a unitary transformation U which causes π rotation of the spins S x about the 2 axis for the sites x with odd ( x (1) + · · · + x ( d ) ). As a result, the Hamiltonianis transformed into [6]˜ H (Λ) ( B ) := U ∗ H (Λ) ( B ) U = X | x − y | =1 ( − S (1) x S (1) y + S (2) x S (2) y − S (3) x S (3) y ) − B X x S (3) x . (3.2)The left-hand side of (3.1) can be written( − x (1) + ··· + x ( d ) ( − y (1) + ··· + y ( d ) h S (1) x S (1) y i (Λ) B,β = hh S (1) x S (1) y ii (Λ) B,β , (3.3)where we have written hh· · ·ii (Λ) B,β := 1˜ Z (Λ) B,β
Tr( · · · ) e − β ˜ H (Λ) ( B ) with ˜ Z (Λ) B,β := Tr e − β ˜ H (Λ) ( B ) . Therefore, it is sufficient to show that the right-hand side of(3.3) is nonnegative.The Hamiltonian ˜ H (Λ) ( B ) of (3.2) can be decomposed into two parts,˜ H (Λ) ( B ) = ˜ H (Λ)XY + ˜ H (Λ)Ising ( B ) , with ˜ H (Λ)XY := − X | x − y | =1 ( S (1) x S (1) y − S (2) x S (2) y )5nd ˜ H (Λ)Ising ( B ) := − X | x − y | =1 S (3) x S (3) y − B X x S (3) x . As usual, we write S ( ± ) x := S (1) x ± iS (2) x . Then, one has S (1) x S (1) y − S (2) x S (2) y = 12 (cid:2) S (+) x S (+) y + S ( − ) x S ( − ) y (cid:3) . (3.4)All the matrix elements of this right-hand side are nonnegative in the usual real, or-thonormal basis which diagonalizes S (3) x . Therefore, all the matrix elements of − ˜ H (Λ)XY arenonnegative. On the other hand, one has e − β ˜ H (Λ) ( B ) = lim M →∞ h e − β ˜ H (Λ)XY /M e − β ˜ H (Λ)Ising ( B ) /M i M from the above decomposition of the Hamiltonian ˜ H (Λ) ( B ). Therefore, from the aboveobservation, this left-hand side e − β ˜ H (Λ) ( B ) has only nonnegative matrix elements. Sinceall the matrix elements of S (1) x are also nonnegative in the same basis, these observationsimply the right-hand side of (3.3) is nonnegative.By using the assumption (2.11), the positivity (3.1) and the relation (3.3), we have( − x (1) + ··· + x ( d ) ( − y (1) + ··· + y ( d ) ρ ,β ( S (1) x S (1) y ) = ˜ ρ ,β ( S (1) x S (1) y ) ≥ C | x − y | d − η (3.5)for any x, y satisfying | x − y | ≥ r with a large positive constant r , where C is a positiveconstant which may depend on β , and we have written˜ ρ ,β ( · · · ) := weak ∗ - lim B ց weak ∗ - lim Λ ր Z d hh· · ·ii (Λ) B,β . (3.6)For an operator A , we introduce three quantities as [6] g (Λ) B,β ( A ) := 12 h hh AA ∗ ii (Λ) B,β + hh A ∗ A ii (Λ) B,β i , (3.7) b (Λ) B,β ( A ) := ( A ∗ , A ) (Λ) B,β := 1˜ Z (Λ) B,β Z ds Tr h A ∗ e − sβ ˜ H (Λ) ( B ) A e − (1 − s ) β ˜ H (Λ) ( B ) i and c (Λ) B,β ( A ) := hh [ A ∗ , [ ˜ H (Λ) ( B ) , A ]] ii (Λ) B,β . (3.8)The method of the reflection positivity [6] is applicable to the present system with theHamiltonian ˜ H (Λ) ( B ). As a result, the following bound [6] is valid: (cid:16) σ (cid:16) d X i =1 ∂ i h i (cid:17) , σ (cid:16) d X i =1 ∂ i h i (cid:17)(cid:17) (Λ) B,β ≤ β − d X i =1 X x ∈ Λ | h i ( x ) | (3.9)6here h i are complex-valued functions on Λ, ∂ i h j ( x ) := h j ( x + e i ) − h j ( x ) with the unitlattice vector e i whose k -th component is given by e ( k ) i = δ i,k , and σ ( f ) = X x ∈ Λ f ( x ) S (1) x for a function f on Λ. Here, · · · denotes complex conjugate. In addition to this, thefunction b (Λ) B,β ( A ) satisfies [11] b (Λ) B,β ( A ) ≥ g (Λ) B,β ( A )] g (Λ) B,β ( A ) + βc (Λ) B,β ( A ) , (3.10)where we have used the inequalities (34) and (A10) in [6], and t − (1 − e − t ) ≥ (1 + t ) − for t > . Using the inequality (3.10), the function g (Λ) B,β ( A ) is estimated as g (Λ) B,β ( A ) ≤ (cid:26) b (Λ) B,β ( A ) + q [ b (Λ) B,β ( A )] + βb (Λ) B,β ( A ) c (Λ) B,β ( A ) (cid:27) . (3.11)First, we recall the following well known facts [5]: Note that h ∂ i ϕ, ψ i = X x ∈ Z d [ ϕ ( x + e i ) − ϕ ( x )] ψ ( x )= X x ϕ ( x + e i ) ψ ( x ) − X x ϕ ( x ) ψ ( x )= X x ϕ ( x ) ψ ( x − e i ) − X x ϕ ( x ) ψ ( x )= − X x ϕ ( x )[ ψ ( x ) − ψ ( x − e i )]for functions, ϕ and ψ , on Z d with a compact support. Therefore, the adjoint ∂ ∗ i of ∂ i isgiven by ∂ ∗ i ψ ( x ) = − [ ψ ( x ) − ψ ( x − e i )] . The corresponding Laplacian ∆ is given by∆ ϕ ( x ) = d X i =1 [ ϕ ( x + e i ) + ϕ ( x − e i ) − ϕ ( x )] . Actually, one has − ∆ = d X i =1 ∂ ∗ i ∂ i = d X i =1 ∂ i ∂ ∗ i . The inverse of ∆ is given by∆ − ( x, y ) = 1(2 π ) d Z π − π dk · · · Z π − π dk d e ik ( x − y ) E k , E k := 2 d X i =1 [cos k i − . Clearly, ∆ − is well defined for d ≥ h i = − ∆ − ∂ ∗ i χ Ω with the characteristic function χ Ω of a cubeΩ with the sidelength R . Then, one has d X i =1 ∂ i h i = − d X i =1 ∂ i ∆ − ∂ ∗ i χ Ω = χ Ω . (3.12)This implies σ d X i =1 ∂ i h i ! = σ ( χ Ω ) = X x ∈ Z d χ Ω ( x ) S (1) x = X x ∈ Ω S (1) x . (3.13)Further, d X i =1 X x ∈ Z d | h i ( x ) | = d X i =1 h ∆ − ∂ ∗ i χ Ω , ∆ − ∂ ∗ i χ Ω i = d X i =1 h χ Ω , ∂ i ∆ − ∂ ∗ i χ Ω i = h χ Ω , ( − ∆ − ) χ Ω i . (3.14)Since ( − ∆ − )( x, y ) ∼ | x − y | − d for a large | x − y | , h χ Ω , ( − ∆ − ) χ Ω i ≤ C R d +2 (3.15)for a large sidelength R of the box Ω, where C is a positive constant.Next, in order to handle the finite lattice Λ with the periodic boundary condition, weset h (Λ) := χ Ω − | Ω || Λ | χ Λ (3.16)for Ω ⊂ Λ. Clearly, one has P x ∈ Λ h (Λ) ( x ) = 0. Therefore, one can define h (Λ) i := − (∆ (Λ) ) − ∂ ∗ i h (Λ) , (3.17)where ∆ (Λ) is the Laplacian for the finite lattice Λ with the periodic boundary condition.Further, one obtains X i ∂ i h (Λ) i = h (Λ) and X i h h (Λ) i , h (Λ) i i = h h (Λ) , [ − (∆ (Λ) ) − ] h (Λ) i . Substituting these into (3.9), we have( σ ( h (Λ) ) , σ ( h (Λ) )) (Λ) B,β ≤ β − h h (Λ) , [ − (∆ (Λ) ) − ] h (Λ) i . (3.18)8n the right-hand side, h h (Λ) , [ − (∆ (Λ) ) − ] h (Λ) i → h χ Ω , ( − ∆ − ) χ Ω i as Λ ր Z d . (3.19)Since h (Λ) is given by (3.16), the left-hand side is written( σ ( h (Λ) ) , σ ( h (Λ) )) (Λ) B,β = ( σ ( χ Ω ) , σ ( χ Ω )) (Λ) B,β − | Ω || Λ | ( σ ( χ Ω ) , σ ( χ Λ )) (Λ) B,β − | Ω || Λ | ( σ ( χ Λ ) , σ ( χ Ω )) (Λ) B,β + | Ω | | Λ | ( σ ( χ Λ ) , σ ( χ Λ )) (Λ) B,β . (3.20)The first term in the right-hand side is written( σ ( χ Ω ) , σ ( χ Ω )) (Λ) B,β = ( A Ω , A Ω ) (Λ) B,β = b (Λ) B,β ( A Ω ) , (3.21)where we have written A Ω := σ ( χ Ω ) = X x ∈ Λ χ Ω ( x ) S (1) x = X x ∈ Ω S (1) x for short. In order to estimate the second term in the right-hand side, we note that [6] | ( σ ( χ Ω ) , σ ( χ Λ )) (Λ) B,β | ≤ q hh σ ( χ Ω ) ii (Λ) B,β q hh σ ( χ Λ ) ii (Λ) B,β = q hh A ii (Λ) B,β q hh A ii (Λ) B,β . (3.22)Using the translation invariance and Schwarz inequality, we have1 | Λ | hh A ii (Λ) B,β = 1 | Γ || Λ | hh A Γ A Λ ii (Λ) B,β ≤ | Γ || Λ | q hh A ii (Λ) B,β q hh A ii (Λ) B,β for Γ ⊂ Λ. This implies [12] 1 | Λ | hh A ii (Λ) B,β ≤ | Γ | hh A ii (Λ) B,β . (3.23)Therefore, 1 | Γ | ˜ ρ ,β ( A ) = weak ∗ - lim B ց weak ∗ - lim Λ ր Z d | Γ | hh A ii (Λ) B,β ≥ weak ∗ - lim B ց weak ∗ - lim Λ ր Z d | Λ | hh A ii (Λ) B,β , (3.24)where the two weak ∗ limits are taken so that hh· · ·ii (Λ) B,β converges to ˜ ρ ,β ( · · · ). Further, bythe argument in Sec. 7 of [10], we obtainlim Γ ր Z d | Γ | ˜ ρ ,β ( A ) = 0 . Therefore, we have weak ∗ - lim B ց weak ∗ - lim Λ ր Z d | Λ | hh A ii (Λ) B,β = 0 . (3.25)9rom the above inequality (3.24). Namely, the long-range order of the transverse corre-lations is vanishing. From (3.22) and (3.25), the second term in the right-hand side of(3.20) is vanishing in the double weak ∗ limit. Similarly, the third and fourth terms arealso vanishing. Combining these observations, (3.18), (3.19) and (3.21), we obtainweak ∗ - lim B ց weak ∗ - lim Λ ր Z d b (Λ) B,β ( A Ω ) ≤ β − h χ Ω , ( − ∆ − ) χ Ω i . (3.26)The right-hand side is estimated by (3.15). As a result, we haveweak ∗ - lim B ց weak ∗ - lim Λ ր Z d b (Λ) B,β ( A Ω ) ≤ C β − R d +2 . (3.27)From (3.8), one has weak ∗ - lim B ց weak ∗ - lim Λ ր Z d c (Λ) B,β ( A Ω ) ≤ C R d . (3.28)Further, from (3.6) and (3.7), we have˜ ρ ,β ( A ) = weak ∗ - lim B ց weak ∗ - lim Λ ր Z d g (Λ) B,β ( A Ω ) . (3.29)This left-hand side satisfies C R d +2 − η ≤ ˜ ρ ,β ( A ) (3.30)from the lower bound for the correlation (3.5). Substituting these into the inequality(3.11), we obtain C R d +2 − η ≤ (cid:20) C β − R d +2 + q C β − R d +2) + C R d +2 · C R d (cid:21) (3.31)for a large R . This implies η ≥
0. Since η ≤ η = 0. Next consider the transverse correlation at zero temperature. To begin with, we recallthat the ground-state expectation value is given by ω (Λ) B ( · · · ) = lim β →∞ h· · ·i (Λ) B,β = lim β →∞ Z (Λ) B,β
Tr ( · · · ) e − βH (Λ) ( B ) . (4.1)From the bound (3.9), one has lim β →∞ b (Λ) B,β ( A Ω ) = 0 . Therefore, the key bound (3.11) is replaced by [13]lim β →∞ g (Λ) B,β ( A Ω ) ≤ lim β →∞ q βb (Λ) B,β ( A Ω ) c (Λ) B,β ( A Ω ) . (4.2)The same argument as in the case of non-zero temperatures yields C R d +2 − η ′ ≤ p C C R d +1 (4.3)for a large R . Here, the positive constant C may be different from that in the case ofnon-zero temperatures. This inequality implies η ′ ≥
1. However, η ′ ≤ η ′ = 1 at zero temperature.10 Other models
In this section, we will treat the quantum XY model and the classical Heisenberg modelin three or higher dimensions d ≥
3. For the transverse correlation at non-zero tempera-tures, we can obtain the same exponent η = 0 as that of the quantum antiferromagneticHeisenberg model under the power-law decay assumption for the transverse correlations.Basically, we use Bogoliubov inequalities, the bounds derived from the reflection positivity,and the Griffiths-type positivity of the correlations. The Hamiltonian of the quantum XY model is given by H (Λ) ( B ) = − X | x − y | =1 [ S (1) x S (1) y + S (3) x S (3) y ] − B X x S (3) x . This Hamiltonian is different from (3.2) in only the terms S (2) x S (2) y about the second com-ponent of the spins. Therefore, both of the reflection positivity [6] and the positivity ofthe transverse correlation hold also for the XY model. Clearly, the Bogoliubov inequalityholds as well. Thus, the same argument yields the exponent η = 0 in the case of non-zerotemperatures.For the case of zero temperature, we can obtain the lower bound η ′ ≥ β → ∞ in the same way. Unfortunately, we have not been able to find a usefulanalogue of the Kennedy-Lieb-Shastry inequality [7] for the ground state of the XY model.Therefore, the upper bound of η ′ is missing at zero temperature. The Hamiltonian of the classical Heisenberg model with N component spins is given by[5] H (Λ) ( B ) = − X | x − y | σ x · σ y − B X x σ ( N ) x , where σ x = ( σ (1) x , σ (2) x , . . . , σ ( N ) x ) ∈ R N with integer N ≥
2. The model satisfies thereflection positivity, and the positivity of the transverse correlations holds [5]. Using theseproperties, the lower bound η ≥ which they considered is different from ours. The upperbound η ≤ in the sameway as in [10]. Thus, η = 0 at non-zero temperatures. For the field theoretical case, see [14]. The first classical analogs of Bogoliubov inequalities were found by Mermin [15]. For a more sophis-ticated treatment, see, e.g., Sec. III.6 of the book [16]. eferences [1] Y. Nambu, Axial Vector Current Conservations in Weak Interactions , Phys. Rev.Lett. (1960) 380–382.[2] Y. Nambu, and G. Jona-Lasinio, Dynamical Model of Elementary Particles Based onan Analogy with Superconductivity. I , Phys. Rev. (1961) 345–358.[3] J. Goldstone,
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