Spectral gap in mean-field $\mathcal O(n)$-model
Simon Becker Angeliki Menegaki

Probing the evolution of heavy-ion collisions using direct photon interferometry
    Klaus Reygers^b
    December 30, 2023
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We study the dependence of the spectral gap for the generator of the Ginzburg-Landau dynamics for all 𝒪(n)-models with mean-field interaction and magnetic field, below and at the critical temperature on the number N of particles. For our analysis of the Gibbs measure, we use a one-step renormalization approach and semiclassical methods to study the eigenvalue-spacing of an auxiliary Schrödinger operator.



§ INTRODUCTION AND MAIN RESULTS




 §.§ 𝒪(n)-model
 The model we are concerned with in this article is the generator of the Ginzburg-Landau dynamics, or Langevin dynamics, of the mean-field 𝒪(n)-model in the critical and supercritical regime β≥ n, as defined precisely in Section <ref>. Our objective is to study the scaling of the spectral gap in terms of the system size N, for all the numbers of components n≥1, and including the cases with or without external magnetic field, in the low temperature and critical regime, extending the study of the subcritical regime β < n in <cit.>. When β<n, the spectral gap of the generator remains open uniformly in N and for any number of components n, in the full temperature range. 

The mean-field 𝒪(n)-model is defined by the energy function 

    H(σ)    = 12∑_x ∈ [N]σ(x)(-Δ_MFσ)(x) - 1/β∑_x ∈ [N]⟨ h, σ (x) ⟩

acting on spin configurations σ: {1,..,N}→𝕊^n-1 where Δ_MF is the mean-field Laplacian and h ∈ℝ^n an external magnetic field. For our study of spectral gaps, we consider the Ginzburg-Landau dynamics associated with the Gibbs measure dρ∝ e^-β H(σ) with Hamilton function (<ref>). The inverse temperature parameter β is such that lower temperatures (higher β) favors alignment of spins. The study of mean-field 𝒪(n)-models is motivated by the fact that their behavior approximates that of the full 𝒪(n)-model on high-dimensional tori <cit.>.



 §.§ State of the art and motivation
 The study of spectral gaps in 𝒪(n) models is a popular problem that has received a lot of attention over the last decades. The study of logarithmic Sobolev (and other functional) inequalities is a classical and very effective tool to study concentration of measures and to quantify the relaxation rates, i.e. the mixing properties, of the dynamics. In particular, the spectral gap (the speed of relaxation) is determined by the constant in the Log-Sobolev inequalities. We define the spectral gap to be the size of the gap between 0 and the rest of the spectrum of the associated generator L, defined in (<ref>). The gap then can be also characterized by 

    λ_S:= inf_f ∈ L^2(dρ)∖{0}-⟨ Lf,f⟩_L^2(dρ)/Var_ρ(f)

 where Var_ρ is the variance relative to the equilibrium measure ρ. All these quantities will be specified for our setting in the following section. 
For further background on functional inequalities see <cit.> and references therein.  

There are only few general approaches for the study of spectral gaps of spin systems, using log-Sobolev inequalities, available and many of them rely on an asymptotic study of log-Sobolev inequalities <cit.> or <cit.> for a more recent result in that direction. In the article <cit.>, a simpler proof for a log-Sobolev inequality was provided for bounded and unbounded spin systems and sufficiently high temperatures. The novelty of the approach in <cit.> is the combination of the study of log-Sobolev inequalities with a simple renormalization group approach to decompose the stationary measure in a way that makes it accessible to simple Bakry-Émery techniques. 

Inspired by the method in <cit.>, we invoke the same one-step renormalization group procedure to reduce the high-dimensional problem to the study of a low-dimensional renormalized measure and a fluctuation measure. In the subcritical regime β < n, which is the regime analyzed in <cit.>,  the renormalization of the equilibrium measure is particularly efficient, since the renormalized potential is strictly convex such that the Bakry-Émery criterion can be directly applied to this measure and implies that the spectral gap remains open. This renormalization group method has recently also been successfully applied in the study of the spectral gap for hierarchical spin models <cit.> and for a lattice discretization of a massive Sine-Gordon model <cit.>. 


The low temperature regime, which is the regime we are concerned about within this article, has a non-convex renormalized potential. In this regime, after a single renormalization step, the renormalized potential is not convex. This makes the asymptotic analysis much more difficult and requires new methods:


While we analyze the Ising model, n=1, without magnetic field, directly using explicit criteria for spectral gap and log-Sobolev inequalities <cit.>, we heavily use the equivalence between the generator of the Ginzburg-Landau dynamics and a Schrödinger operator to analyze multi-component,n ≥ 2, 𝒪(n)-models. This analysis builds heavily upon ideas by Simon <cit.> and Helffer–Sjöstrand <cit.> who developed effective semiclassical methods to study the low-lying spectrum of Schrödinger operators in the semiclassical limit (which in our case corresponds to N →∞). These results are discussed thoroughly in the final chapters of <cit.>. In this article however, we have to study the spectrum of Schrödinger operators beyond the harmonic approximation. In this case, the limiting operator is not explicitly diagonalizable anymore and the spacing between eigenvalues is no longer linear in the semiclassical parameter N, the number of spins. 

The mixing time of the Glauber dynamics of the mean-field Ising model (𝒪(1)) without magnetic field has been carefully analyzed in <cit.>. There it is shown-among others- that the mixing time in the subcritical regime β<1 is N log (N), the scaling at the critical point N^3/2 for β=1 and in the supercritical regime β>1 it is exponential growing in N. This is to be compared to a spectral gap that remains open for β<1, closes like N^-1/2 for β=1 and closes exponentially fast also for β>1. Thus, the mixing time for the Glauber dynamics are -up to a factor 1/N- comparable to our findings on the spectral gap, cf. Theorem <ref>.


Our main result on the mean-field Ising model in the supercritical regime β>n is stated in the following Theorem:
[Spectral gap–Supercritical Mean-field Ising models, β>1]

Let N be the number of spins and n the number of components. For the  supercritical mean-field Ising model (n=1, β>1), the spectral gap λ_N of the generator


  * for the case of small magnetic fields | h | < h_c, closes as N →∞ exponentially fast, λ_N =e^-NΔ_small(V)(1+(1)). In particular, for magnetic fields h ∈ [0,h_c) 

    Δ_small(V) = ∫_γ_1(β)^γ_2(β)β(φ-tanh(βφ+h) ) d φ
 
where γ_1(β) ≤γ_2(β) ∈ are the two smallest numbers satisfying the condition

    γ(β) = tanh(γ(β)  β+h).
 

  * For critical magnetic fields | h|=h_c, the spectral gap does not close faster than Θ(N^-1/3) anymore. 

  * Finally, for strong magnetic fields | h|> h_c, it is bounded away from zero uniformly in N.
 
where h_c=√(β(β-1))- arccosh(√(β)).


In the case of supercritical multi-component systems (n≥ 2, β>n) without magnetic fields, it is the rotational invariance of the model that leads to a closing spectral gap as N tends to infinity. To capture this property, we call a function f:(𝕊^n)^N → radial, if it is only a function of the norm of the mean spin |σ̅|. 
Our main results for all multi-component systems in the supercritical regime β>n are summarized in the following Theorem: 
[Spectral gap–Supercritical Mean-field 𝒪(n)-models, β>n≥ 2]

Let N be the number of spins and n the number of components. For the  supercritical mean-field 𝒪(n)-models(n ≥ 2, β>n), the spectral gap λ_N of the generator 
 

  * closes as λ_N=Θ(N^-1) if there is no external magnetic field h=0, but remains open λ_N= Θ(1) for radial functions.

  * is bounded away from zero uniformly in the number of spins for all h ∈ℝ^n\{0}. 
 

We also analyze the behavior of the spectral gap at the critical point β=n and h=0. Using a discrete Fourier analysis approach implemented in Section <ref> for the Ising case n=1 and a direct asymptotic analysis for all higher component systems n ≥ 2, we find a different asymptotic of the spectral gap from both the supercritical β>n (exponentially fast closing) and subcritical β<n (spectral gap remains open) regimes:
[Spectral gap–Critical Mean-field 𝒪(n) models, β=n]

For all critical, β=n, h=0 mean-field 𝒪(n)-models the spectral gap closes as λ_N=Θ(N^-1/2). In particular, the rate N^-1/2 is attained for the magnetization 

    M(σ) = N^-1/2∑_x ∈ [N]σ(x).



We emphasize that at the critical points (β=n, h=0), the gap does no longer close once a non-zero magnetic field is present: 
[Spectral gap–Mean-field 𝒪(n) models, β=n, h ≠ 0 ]

For all, β=n and h ≠ 0, the spectral gap of all mean-field 𝒪(n)-models remains open.

The proof of Theorem <ref> is along the lines of Theorem <ref> in the regime h > h_c and follows from Proposition <ref> in the Ising-case, n=1, and in the multi-component case, n ≥ 2, from Proposition <ref>.



 §.§ Organization of the article

The article is organized as follows:


  * In Section <ref> we introduce the mean-field 𝒪(n)-model.

  * In Section <ref> we introduce the renormalized methods.

  * In Section <ref> we analyze the mean-field Ising model in the supercritical regime β>1 and prove Theorem <ref>.

  * In Section <ref> we analyze the higher-component mean-field 𝒪(n)-models in the supercritical regime β>n and prove Theorem <ref>.

  * In Section <ref> we study the critical regime and prove both Theorems <ref> and <ref>.

  * Our article contains an appendix that contains technical details and further details on numerical methods.


Acknowledgements 
Both authors were supported by the EPSRC grant EP/L016516/1 for the University of Cambridge CDT, the CCA. The authors are grateful to Roland Bauerschmidt for many useful discussions and bringing this problem to our attention.


Notation
We write f(z) = 𝒪(g(z)) to indicate that there is C>0 such that | f(z) |≤ C | g(z) | and f(z)= (g(z)) for z → z_0 if there is for any ε>0 a neighbourhood U_ε of z_0 such that | f(z)|≤ε| g(z) |. We say that f(z) = Θ(g(z)) if there are k_1, k_2>0 and z_0 such that for all z ≥ z_0  we have k_1  g(z) ≤ f(z) ≤ k_2  g(z).
The expectation with respect to a measure μ is written denoted by 𝔼_μ(X).
The normalized surface measure on the n sphere is denoted as dS_𝕊^n. We write  to denote a vector or matrix whose entries are all equal to one and id for the identity map. Finally, we introduce the notation [N]:={1,...,N }. The eigenvalues of a self-adjoint matrix A shall be denoted by λ_1(A)≤ ... ≤λ_N(A). 




§ THE MEAN-FIELD 𝒪(N)-MODEL


We study the mean-field 𝒪(n)-model with spin configuration σ: [N] →𝕊^n-1 and introduce the mean-field Laplacian (Δ_MFσ)(x):= 1N∑_y ∈ [N]( σ(y)-σ(x)).

The mean spin is defined as σ:=1N∑_x ∈ [N]σ(x).
The energy of a spin configuration σ is given by the Curie-Weiss Hamiltonian

    H(σ)    = 12∑_x ∈ [N]σ(x)(-Δ_MFσ)(x) - 1/β∑_x ∈ [N]⟨ h, σ (x) ⟩
        = 14N∑_x,y ∈[N]|σ(x)-σ(y) |^2 - 1/β∑_x ∈ [N]⟨ h, σ (x) ⟩
       =N2(1-|σ|^2) - Nβ ⟨ h,  σ⟩.

where the constant vector h ∈ℝ^n represents an external magnetic field and β is the inverse temperature of the system. The critical temperature for the 𝒪(n)-models is β=n and we study both regimes: the supercritical regime β>n and the critical regime β=n.

The dynamics we consider is the continuous-time Ginzburg-Landau dynamics

    ∂_t f    = ∑_x ∈ [N]⟨∇^(x)_𝕊^n-1, β^-1∇^(x)_𝕊^n-1f+ f ∇^(x)_𝕊^n-1 H ⟩_ℝ^n

to the invariant distribution of the mean-field 𝒪(n)-model which is the Gibbs measure dρ(σ):=e^- β H(σ) /Z  dS^⊗ N_𝕊^n-1(σ) with normalizing constant Z. The operators Δ^(x)_𝕊^n-1 defined by ⟨ f,-Δ^(x)_𝕊^n-1f ⟩:=⟨∇^(x)_𝕊^n-1f,∇^(x)_𝕊^n-1f⟩ and ∇^(x)_𝕊^n-1 are the Laplace-Beltrami and gradient operator on 𝕊^n-1 acting on spin i, respectively. We recall that for the Ising model n=0 and a function F:𝕊^0 →ℝ, the gradient is given by (∇_𝕊^0 F)(σ)=F(σ)-F(-σ).
The L^2((𝕊^n-1)^N)-adjoint of the generator of the Kramers-Smoluchowski equation (<ref>) is the generator

    (Lζ)(σ)   := ∑_x ∈ [N]β^-1(Δ^(x)_𝕊^n-1ζ)(σ)-⟨ (∇^(x)_𝕊^n-1 H)(σ), (∇^(x)_𝕊^n-1ζ)(σ) ⟩_ℝ^n.

Studying the operator L on the weighted space L^2((𝕊^n-1)^N,  dρ) makes this generator self-adjoint.
The quadratic form of the generator (<ref>) is just a rescaled Dirichlet form

    -⟨ Lf,f⟩_L^2(dρ)    = β^-1∑_x ∈ [N]‖∇_𝕊^n-1^(x)f ‖^2_L^2(dρ).





 





§ RENORMALIZED MEASURE AND MATHEMATICAL PRELIMINARIES


We start with the definition of entropy with respect to probability measures:
[Entropy]
For a probability measure μ on some Borel set Ω the entropy Ent_μ(F) of a positive measurable function F:Ω→ℝ_≥ 0 with ∫_Ω F(x) log^+(F(x))  dμ(x)< ∞ is defined as 

    Ent_μ(F)    :=∫_Ω F(x) log(F(x) /  ∫_Ω F(y)  dμ(y) )  dμ(x).



Instead of studying the generator of the dynamics directly, we apply a one step renormalization first <cit.>:

[Renormalized quantities]
The renormalized single spin potential V_n associated with the mean-field 𝒪(n)-model for φ∈ℝ^n is defined as 

    V_n(φ)    = -log∫_𝕊^n-1 e^-β2‖φ-σ‖^2+ ⟨ h,  σ⟩ dS_𝕊^n-1(σ) 
       = β2(1+‖φ‖^2 ) - log( Γ(n2) (2‖βφ + h ‖)^n/2-1 I_n2-1( ‖βφ +h ‖) )

where I is the modified Bessel function of the first kind.
The N-particle renormalized measure is defined for a normalizing constant ν_N^(n) by 

    dν_N(φ) = ν_N^(n)e^-NV_n(φ) dφ on ℝ^n.


[Fluctuation measure]
For any φ∈ℝ^n, there is a probability measure μ_φ,  the fluctuation measure, on (𝕊^n-1)^N defined as 

    𝔼_μ_φ(F) = ∫_(𝕊^n-1)^N F(σ)  e^NV_n(φ)∏_x ∈ [N] e^-β2‖φ-σ(x) ‖_2^2+ ⟨ h, σ(x) ⟩ dS(σ(x)).

 
A straightforward calculation shows that the stationary measure dρ can be decomposed into the fluctuation and renormalized measure such that 𝔼_ρ(F)= 𝔼_ν_N(𝔼_μ_φ(F)).




In the case of the Ising model (n=1) the renormalized potential is 

    V_1(φ) = β2(1+  φ^2)  - log(cosh(βφ+h)).

For the XY model (n=2) the renormalized potential reads

    V_2(φ) = β2(1+  ‖φ‖^2)  - log(I_0(‖βφ+h ‖))

where I is the modified Bessel function of the first kind.

For the Heisenberg model (n=3) one finds

    V_3(φ)= β2(1+ ‖φ‖^2)- log( sinh‖βφ  +h‖/‖βφ+h  ‖).



For the N-asymptotic study of eigenvalues we observe that the renormalized potential grows quadratically at infinity such that Δ V_n ∈ L^∞(ℝ^n).

The Ginzburg-Landau dynamics dynamics for the renormalized measure is then given by the self-adjoint operator L_ren:D(L_ren) ⊂ L^2(ℝ^n, dν_N) → L^2(ℝ^n, dν_N), satisfying

    (L_renζ)(φ) =  (Δ_ℝ^nζ)(φ) - N⟨∇_ℝ^n V_n(φ), ∇_ℝ^nζ (φ)⟩.

The renormalized generator L_ren satisfies 

    -⟨ L_ren f,f ⟩_L^2(dν_N)=‖∇_ℝ^nf ‖^2_L^2(dν_N).


The renormalized Schrödinger operator with null space spanned by e^-NV_n is the operator defined by conjugation - Δ_ren = e^-NV_n/2 L_ren  e^NV_n/2

    Δ_ren =- Δ_ℝ^n+N^24|∇ V_n(φ)|^2- N2Δ V_n(φ).




[LSI and SGI]
Let μ be a Borel probability measure on ℝ^n. We say that μ satisfies a logarithmic Sobolev inequality LSI(k) iff

    Ent_μ(f^2) ≤2k‖∇ f ‖_L^2(d μ)^2

for all smooth functions f. 
The LSI(k) implies <cit.> that μ satisfies a spectral gap inequality SGI(k)

    Var_μ(f) ≤1k‖∇ f ‖_L^2(d μ)^2.


Thus, in light of the characterisation (<ref>), the spectral gap of L_ren is by (<ref>) precisely the constant in the SGI of the renormalized measure.

If f vanishes outside a set Ω of measure μ(Ω) <1 and if μ satisfies a SGI(k) then

    ‖ f ‖_L^2(d μ)^2≤1k(1-μ(Ω))‖∇ f ‖_L^2(d μ)^2.



For Borel probability measures μ on ℝ there is an explicit characterization of the measures satisfying a LSI <cit.>:


Any such measure μ satisfies a LSI(k) iff there exist absolute constants K_0=1/150 and K_1=468 such that the optimal value k in the LSI(k) satisfies K_0(D_0+D_1)≤ 1/k ≤ K_1(D_0+D_1) for finite D_0 and D_1. Let m be the median of μ and p(t)  dt the absolutely continuous part of μ with respect to Lebesgue measure. The constants D_0 and D_1 are given by

    D_0    :=sup_x<m( -μ((-∞,x])log(μ((-∞,x]))  ∫_x^m ds/p(s)) and 
     
    D_1    :=sup_x>m( -μ([x,∞))log(μ([x,∞))  ∫_m^x ds/p(s)).


For constants 

    B_0    :=sup_x<m( μ((-∞,x])  ∫_x^m ds/p(s))  and 
    B_1 :=sup_x>m( μ([x,∞)) ∫_m^x ds/p(s))

one defines the Muckenhoupt number <cit.> B:=max(B_0,B_1). The measure μ satisfies then a SGI with optimal constant c=1/k if and only if B is finite in which case 

    B/2  ≤ c ≤ 4 B

<cit.>. 



The proof given in <cit.> shows that the characterization of LSI constants holds true not only by splitting at the median: Instead, there is ε>0 such that for any ζ for which μ((-∞,ζ]), μ([ζ,∞)) ∈ (1/2-ε,1/2+ε) the above characterization (<ref>) holds true when the median m is replaced by ζ. The same is, up to an unimportant adaptation of the lower bound in (<ref>), for the SGI as well, cf. <cit.>. 

We continue by observing that the fluctuation measures satisfy a LSI(2γ_n) independent of h or φ. This follows for n=1 with γ_n = 4 from a simple application of the tensorization principle to the classical bound on the Bernoulli distribution <cit.>. For number of components n ≥ 2 one can use the results from <cit.>.


 Let the renormalized measure ν_N satisfy a LSI(λ), then the full equilibrium measure ρ satisfies a LSI 

    Ent_ρ(F^2) ≤2γ_n(1+8 N  β^2λ) ∑_x ∈ [N]‖∇_𝕊^n-1^(x)F ‖^2_L^2(dρ)

and if the renormalized measure ν_N satisfies a SGI(λ), then the equilibrium measure ρ satisfies a SGI

    Var_ρ(F) ≤1γ_n(1+4N β^2λ)∑_x ∈ [N]‖∇_𝕊^n-1^(x)F ‖^2_L^2(dρ).



The proof of the SGI is as follows:
For the SGI we obtain the decomposition

    Var_ρ(F)   =𝔼_ν_N( Var_μ_φ(F))+ Var_ν_N(𝔼_μ_φ(F)) 
       ≤1γ_n∑_x ∈ [N]‖∇_𝕊^n-1^(x)F ‖^2_L^2(dρ) + 1λ𝔼_ν_N(|∇_φ𝔼_μ_φ(F) |^2).

To bound the second term in the above estimate, we compute using the Cauchy-Schwarz inequality and the spectral gap inequality for fluctuation measures μ_φ on the sphere, defined by (<ref>) such that, see <cit.>,

    ∇_φ𝔼_μ_φ(F)    =N ∇ V(φ) 𝔼_μ_φ(F) -β∑_x ∈ [N]𝔼_μ_φ(F(φ-σ_x)).


We then use that by the explicit expression (<ref>)

    ∇ V(φ) =β∫_𝕊^n-1 e^-β/2‖φ-σ_x ‖^2 (φ-σ_x)dS_𝕊^n-1(σ_x)
    /∫_𝕊^n-1 e^-β/2‖φ-σ_x ‖^2 dS_𝕊^n-1(σ_x) =β( φ- 𝔼_μ_φ(σ_x)).

Inserting this into (<ref>) we find that 

    ∇_φ𝔼_μ_φ(F)  = β∑_x ∈ [N]𝔼_μ_φ(Fσ_x)- 𝔼_μ_φ(F) 𝔼_μ_φ(σ_x) = β∑_x ∈ [N]cov_μ_φ(F, σ_x).

Thus, we have using Cauchy-Schwarz that 

    |∇_φ𝔼_μ_φ(F) |^2  ≤ N  β^2 ∑_x ∈ [N]|cov_μ_φ(F, σ_x) |^2.

We can then use that by Cauchy-Schwarz again 

    cov_μ_φ(F, σ_x)   =𝔼_μ_φ((F-𝔼_μ_φ(F))(σ -𝔼_μ_φ(σ)) ) 
       ≤√(𝔼_μ_φ(F-𝔼_μ_φ(F))^2)√(𝔼_μ_φ(F-𝔼_μ_φ(F))^2)√(𝔼_μ_φ(σ-𝔼_μ_φ(σ))^2)
       ≤ 2√(Var_μ_φ(F)).

Finally, inserting this into 
(<ref>) and using the LSI for the fluctuation measure, we find 

    𝔼_ν_N(|∇_φ𝔼_μ_φ(F) |^2)    ≤ 4N β^2 𝔼_ν_NVar_μ_φ(F) 
       ≤4N β^2/γ_n𝔼_ρ∑_x ∈ [N]|∇^(x)_𝕊^n-1 F |^2

which after inserting this bound into (<ref>) implies the claim. To prove the LSI we follow <cit.> and write 
    Ent_ρ(F^2)    = 𝔼_ν_N(  Ent_μ_φ(F^2)  ) + Ent_ν_N(𝔼_μ_φ( F^2 ) )  
       ≤2γ_n∑_x ∈ [N]‖∇_𝕊^n-1^(x)F ‖^2_L^2(dρ) + 2λ𝔼_ν_N(  |∇_φ√(𝔼_μ_φ(F^2) )|^2 ).
 
For the second term we have from applying the Cauchy-Schwarz inequality: 
    |∇_φ√(𝔼_μ_φ(F^2) )|^2  = β^2 |∑_x ∈ [N]cov_μ_φ(F^2, σ_x) /√(𝔼_μ_φ(F^2) )|^2 ≤β^2N ∑_x ∈ [N]|cov_μ_φ(F^2, σ_x)  |^2/𝔼_μ_φ(F^2).
 

By  doubling the variables σ_x, σ_x', we write 

    |cov_μ_φ(F^2(σ_x), σ_x) |   = 1/2|𝔼_μ_φ( (F^2(σ_x)-F^2(σ_x')  )(σ_x-σ_x') )  |
       ≤√(Var_μ_φ(F))√(1/2𝔼_μ_φ⊗μ_φ(  ( F(σ_x)+F(σ_x'))^2(σ_x-σ_x')^2)   )
       ≤√(Var_μ_φ(F))√(8𝔼_μ_φ(F^2)   )
 
where in the last two lines we applied CS inequality and used that | σ_x - σ_x'|≤ 2. Then  
    |cov_μ_φ(F^2, σ)  |^2 ≤ 8 Var_μ_φ(F)  𝔼_μ_φ(F^2).
 
This gives

    |∇_φ√(𝔼_μ_φ(F^2) )|^2 ≤ 8 N β^2 Var_μ_φ(F).

Overall we have 
 
    Ent_ρ(F^2)    ≤2γ_n∑_x ∈ [N]‖∇_𝕊^n-1^(x)F ‖^2_L^2(dρ) + 16 β^2 N λγ_n∑_x ∈ [N]‖∇_𝕊^n-1^(x)F ‖^2_L^2(dρ)
        = 2γ_n(1+ 8 β^2 Nλ)  ∑_x ∈ [N]‖∇_𝕊^n-1^(x)F ‖^2_L^2(dρ).
 
 










§ THE MEAN-FIELD ISING MODEL


Without loss of generality, we assume h ≥ 0 when studying the Ising model.
We define the critical magnetic field strength in the Ising model 

    h_c(β):=√(β(β-1))- arccosh(√(β))
 for temperatures β≥ 1 as the supremum of all h>0 such that x = tanh(β x+h) has three distinct solutions for x ∈ [-1,1]. In particular h_c(β) is monotone with respect to the inverse temperature β. 



The critical magnetic field strength is chosen in such a way that for fields h<h_c(β) there are two potential wells in the renormalized potential landscape, see Figure <ref>, whereas for h ≥ h_c(β) there is only one, see Figure <ref> in subsection <ref> where this case is discussed.



 §.§ Lower bound on spectral gap in weak field h< h_c(β) regime

We start by showing that the inverse spectral gap in the Ising model in the case of subcritical magnetic fields, i.e. h<h_c(β), converges at most exponentially fast to zero as the number of spins, N, increases. 



We start by showing a LSI with exponential constant for the renormalized measure. This implies by Prop. <ref> that such an LSI must also hold for the full many-particle measure dρ.



Let β>1 and h<h_c(β) such that V_1 is a double well potential where the depth of the smaller well is denoted by Δ_small(V), cf. Fig. <ref>. The mean-field Ising model satisfies a LSI(e^-NΔ_small(V)(1+(1)))[If the magnetic field is zero, i.e. h=0, both wells are of equal size.]

    Ent_ν_N(F^2) ≲  e^NΔ_small(V)(1+(1))∫_ℝ| F' |^2  dν_N.



The renormalized potential V_1 has on [0,∞) a global minimum with positive second derivative at some φ_min satisfying φ_min = tanh(βφ_min+h). This follows since the renormalized potential (<ref>) reduces to

    V_1(φ) = β/2φ^2-log(cosh(βφ+h))
 
and the critical points of this potential are easily found to satisfy φ= tanh(βφ+h), see also <cit.>.
For small temperatures, i.e. β→∞, one has φ_min=1+(1).

We first consider h=0:
In this case, the median of the renormalized measure is located precisely at φ=0 and φ_min>0 is one of the two non-degenerate global minima of the renormalized potential (the other minimum is located at -φ_min by axisymmetry).


An application of Laplace's principle, see <cit.>, shows that for all x>0:

    lim_N →∞1Nlog(-ν_N( [x,∞)) log(ν_N( [x,∞))) ∫_0^x e^N V_1(φ)/ν_N^(1) dφ) 
       =lim_N →∞1N( log∫_x^∞ e^-NV_1(φ)  dφ  + log(- log(ν_N( [x,∞))))+ log∫_0^x e^N V_1(φ ) dφ) 
       = -inf_t ∈ [x,∞) V_1(t)  +sup_t ∈ [0,x] V_1(t).

The supremum of (<ref>) is attained at x=φ_min such that 

    -inf_t ∈ [x, ∞) V_1(t)  +sup_t ∈ [0,x] V_1(t)=Δ_small(V)>0.


Here, we used that for x> φ_min we get by Laplace's principle 

    -log(ν_N [x,∞) ) = N(V(x)-V(φ_min))(1+(1))
 
and thus lim_N →∞log(-log(ν_N [x,∞) ))N=0. 

 On the other hand, if x ∈ (0,φ_min) then, again by Laplace's principle, -log(ν_N [x,∞) ) = -log(12)+(1) and thus lim_N →∞log(-log(ν_N [x,∞) ))N=0 as well. The case x=φ_min can be treated analogously.
Hence, we obtain for the constant D_1 as in (<ref>) 

    D_1   :=sup_x>0(-ν_N( [x,∞)) log(ν_N( [x,∞))) ∫_0^x e^N V_1(φ)/ν_N^(1) dφ)  
       = e^NΔ_small(V)(1+(1)).

The symmetry of the distribution for h=0 implies then that D_0=D_1. 


We now consider h>0: The renormalized potential possesses a unique global minimum at some φ_min and the median of the renormalized measure converges to this point φ_min, see Fig. <ref>, as Laplace's principle implies 

    ∫_φ_min^∞ e^-N V_n(φ) dφ/∫_-∞^∞ e^-N V_n(φ) dφ=1/2+𝒪(1/N).

Hence, it suffices to verify the LSI bounds (<ref>)  for m=φ_min as argued in Remark <ref>.

Arguing as in (<ref>) yields for h>0 and x< φ_min:

    lim_N →∞1Nlog(-ν_N( (-∞,x] log(ν_N((-∞,x] )) ∫^m_x e^N V_1(φ)/ν_N^(1) dφ) 
       =lim_N →∞1N( log∫_-∞^x e^-NV_1(φ ) dφ   + log(- log(ν_N( [x,∞))) )+ log∫_x^m e^N V_1(φ ) dφ) 
       = -inf_t ∈ (-∞,x] V_1(t)  +sup_t ∈ [x,m] V_1(t)

which shows D_0= e^NΔ_small(V)(1+(1)) by taking x to be the minimum of the smaller well of the renormalized potential.
For the constant D_1 we get on the other hand for x>φ_min, since the renormalized potential is monotonically increasing on [φ_min,∞), 

    lim_N →∞1Nlog(-ν_N( [x,∞)) log(ν_N( [x,∞)) ∫_m^x e^N V_1(φ)/ν_N^(1) dφ) 
       =lim_N →∞1N( log∫_x^∞  e^-NV_1(φ ) dφ   + log∫_m^x e^N V_1(φ ) dφ  + log(- log(ν_N( [x,∞)))) ) 
       = -inf_t ∈ [x,∞) V_1(t)  +sup_t ∈ [m,x] V_1(t) = 0

such that D_1 is negligible compared with D_0.




 §.§ Upper bound on spectral gap in weak field h<h_c(β) regime.

The upper bound on the spectral gap is obtained by finding an explicit trial function saturating the SGI. For this construction, we use the notation and results of Lemma <ref>.



In order to fix ideas first, we assume h=0. We start by observing that the mean spin σ can only take values in the set ℳ:={-1, -1+2/N,...,1 }. The weights of the stationary measure dρ are given by functions η_N:ℳ→ℝ

    η_N(i) :=∑_σ;σ=i e^-β H(σ)=NN/2(1+i) e^-Nβ2(1-i^2).

where we used (<ref>).



We also introduce trial functions f_N: {± 1 }^N →ℝ for the spectral gap inequality given by

    f_N(σ) := ∑_i ∈ℳ; 0 ≤ i ≤σ_{i ≤γ_3(β)}/η_N(i)

with indicator function  and γ_3(β) is the largest solution to φ = tanh(βφ+h).
Since f_N depends only on the mean spin, we can identify them with functions g_N:ℳ→ℝ

    g_N(m) =∑_i ∈ℳ; 0 ≤ i ≤ m_{i ≤γ_3(β)}/η_N(i) such that f_N(σ)=g_N(σ).

For the L^2 norm of the f_N we find

    ‖ f_N ‖_L^2(dρ)^2
        = ∑_i ∈ℳη_N(i)/Z| g_N(i) |^2  ≥∑_i ∈ℳ; i > γ_3(β)η_N(i)/Z(∑_j ∈ℳ; 0 ≤ j ≤γ_3(β)1/η_N(j))^2

where Z is the normalization constant of the full measure dρ.
For the gradient of f_N we find 
    |∇_𝕊^0^(i) f_N(σ) |^2  = | g_N(σ)-g_N(σ± 2/N) |^2 ≲η_N(σ)^-2.

Hence, for some C>0

    ∑_i ∈ [N]‖∇_𝕊^0^(i) f_N‖_L^2(dρ)^2  ≤CN/Z∑_i ∈ℳ; 0 ≤ i ≤γ_3(β)1/η_N(i).

Using (<ref>) with μ(Ω) = 12 implies by comparing (<ref>) with (<ref>) that the constant γ in the SGI is bounded from below by 

    1/NC∑_i ∈ℳ; i > γ_3(β)η_N(i)  ∑_i ∈ℳ; 0≤ i ≤γ_3(β)1/η_N(i)≤1/2γ.

We recall from the discussion in Lemma <ref> that the continuous approximation η_N(i) attains its maximum in the limit at i= γ_3(β) and the summand 1/η_N(i) in the second sum attains its maximum in the limit at i= 0.

Thus it suffices to study the asymptotic of the logarithm of the leading order summands in (<ref>) using the asymptotic behaviour of ζ_N := ∂_s log(η_N(s)) given in (<ref>)

    log(η_N(γ_3(β))η_N(0))    = N ∫_0^γ_3(β)ζ_N(s)  ds= N ∫_0^γ_3(β) (β s -arctanh(s))  ds  (1+(1)) 
       = N (βγ_3(β)^2/2 - ∫_0^γ_3(β)βarctanh(x/β)/β dx )(1+(1))
       = -N ∫_0^γ_3(β)β  (x -tanh(β x))  dx  (1+(1)) 
       = N Δ_small(V)(1+(1)).

Here, we used integration of the inverse function to obtain the last line and (<ref>) in the last one.
In the case of a positive weak magnetic field h ∈ (0, h_c(β)) we choose a trial function f_N,h: {± 1 }^N →ℝ given by

    f_N,h(σ)    := ∑_i ∈ℳ;  σ<i<γ_3(β) _{i ≥γ_1(β)}/η_N,h(i)
    where for i ∈ℳ
    η_N,h(i)    := NN/2(1+i) e^-Nβ2(1-i^2)+h N i .

Proceeding as above in (<ref>) we obtain for the L^2 norm the lower bound

    ‖ f_N,h‖_L^2(dρ)^2 ≥1/Z∑_i ∈ℳ; i < γ_1(β)η_N,h(i) ( ∑_j ∈ℳ;γ_3(β)> j ≥γ_1(β)1/η_N,h(j))^2.

For the Dirichlet form we find, as for (<ref>), for some C>0

    ∑_x ∈ [N]‖∇_𝕊^0^(x) f_N,h‖_L^2(dρ)^2  ≤CN/Z∑_j ∈ℳ;γ_3(β)> j ≥γ_1(β)1/η_N,h(j).

We can apply (<ref>) with μ(Ω) = 11-ε for some ε>0 since the trial function (<ref>) vanishes to the right of the global maximum such that by comparing (<ref>) with (<ref>)  the constant γ in the SGI is bounded from below by 

    1/NC∑_i ∈ℳ; i < γ_1(β)η_N,h(i) ∑_j ∈ℳ;γ_3(β)> j ≥γ_1(β)1/η_N,h(j)≤1/εγ.

The weight η_N,h(i) in the first sum attain their maximum (in the limit) at i=γ_1(β) and the summands 1/η_N,h(i) in the second sum attain their maximum at i=γ_2(β).

To explicitly state an upper bound on the spectral gap it suffices to study the asymptotic of the logarithm of the leading order summands

    log(η_N,h(γ_1(β))η_N,h(γ_2(β)))    =N ∫_γ_2(β) ^γ_1(β)ζ_N,h(s)  ds=  ∫_γ_2(β)^γ_1(β) (β s -arctanh(s))  ds  (1+(1)) 
       = N(β (γ_1(β)^2-γ_2(β)^2)/2 - ∫_γ_2(β)β^γ_1(β)βarctanh(x/β)/β dx )(1+(1))
       = -N ∫_γ_2(β)^γ_1(β)β  (x -tanh(β x))  dx  (1+(1)) 
       = NΔ_small(V)(1+(1)).














 §.§ Spectral gap in strong magnetic field regime h> h_c(β).



Next, we study the case of strong magnetic fields for the Ising model, that is V_1' has at most one root, for β >1. We also include the case β=1 and h ≠ 0.
Unlike in the case of weak magnetic fields, in which case the constant in the LSI for the renormalized measure is exponentially increasing in the number of spins, the spectral gap of the renormalized measure is now linearly increasing in the number of spins. Responsible for this uniform gap is the local uniform convexity at the minimum of the renormalized potential. More precisely, we have

    V_1'(φ) =β (φ-tanh(βφ +h))  and  V_1”(φ) = β(1 - βsech(βφ +h)^2).

Thus, V_1”(φ)=0 yields φ_±= -h ±arccosh(√(β))/β. Inserting this into V_1'(φ_±)=0 implies that h_±=±arccosh(√(β)) ∓√(β(β-1)) with sign (h_±)=∓ 1 and thus φ_±= ±√(β-1/β).
In particular, in the subcritical regime β>1 all global minima φ_* have sign (φ_*)=(h), such that the renormalized potential satisfies V_1”(φ_*)>0. Moreover, for β =1 and h ≠ 0 there are no points at which both the first and second derivative vanish. The third derivative at this point however is always non-zero and given by 

    V_1^(3)(φ_±) = ∓ 2 √(β(β-1))β.



Let β≥ 1 and h> h_c(β), i.e. V_1 is a single well potential. We obtain for the Ising model a SGI(γ)

    Var_ν_N(F) ≤1γ∫_ℝ| F' |^2  dν_N
 
where 1/γ is uniformly bounded in N. 


Since the renormalized Schrödinger operator and renormalized generator are unitarily equivalent up to a factor, see (<ref>), the semiclassical eigenvalue distribution stated in <cit.> implies the statement of the Proposition: 



It follows immediately from the renormalized Schrödinger operator (<ref>)

    Δ_ren =- d^2dφ^2+N^24| V_1'(φ)|^2- N2 V_1”(φ).

that the low-lying eigenfunction of Δ_ren accumulate at the unique non-degenerate (the second derivative is non-zero) potential well and the spectral gap of the renormalized measure grows linearly in N. The result then follows from Prop. <ref>.





 §.§ Critical magnetic fields in n=1




Let h=h_± and β > 1. The spectral gap of the radial renormalized Schrödinger operator grows as Θ( N^2/3) and in particular, the spectral gap of the full measure does not close faster than Θ( N^-1/3).


Let λ:= N/2 and consider the Schrödinger operator, defined in (<ref>),

    ℋ   :=-∂_x^2 + λ^2 | V'_1(x) |^2-λ V_1”(x)

for the renormalized potential and auxiliary Schrödinger operators, which are obtained as the Taylor expansion of (<ref>)

    H_φ_*   =-∂_x^2 + λ^2 | V_1'(φ_*) |^2 (x-φ_*)^2 - λ V_1”(φ_*)  and 
    
    H_φ_±   =-∂_x^2 + λ^2 β^3(β-1) (x-φ_±)^4 ±λ  2√(β(β-1))β (x-φ_±)

on L^2() localized to the two critical points, the inflection point φ_± and the global minimum φ_*.
We then define j ∈ C_c^∞((-2,2);[0,1]) such that j(x)=1 for | x|≤ 1 and from this functions

    J_φ_*(x)    := j(λ^2/5| x-φ_*| ),  J_φ_±(x) := j(λ^3/10| x-φ_±| )   and 
       J(x) := √(1-J_φ_±(x)^2-J_φ_*(x)^2) with 
       ‖∇  J_φ_*‖_ℝ^n^2=𝒪(λ^4/5),‖∇ J_φ_±‖_^n^2 = 𝒪(λ^3/5).

Invoking then unitary maps U_φ_*, U_φ_±∈ℒ(L^2(ℝ)) defined as 

    (U_φ_*f)(x) := λ^-1/4f(λ^-1/2(x+φ_*))  and (U_φ_±f)(x) := λ^-1/6f(λ^-1/3(x+φ_±))

shows that the two Schrödinger operators in (<ref>) are in fact unitarily equivalent, up to multiplication by powers of λ, to the λ-independent Schrödinger operators

    S_φ_*   =- ∂_x^2+| V_1'(φ_*) |^2 x^2 -V_1”(φ_*) 
    
    S_φ_±   = - ∂_x^2+β^3(β-1) x^4 ± 2√(β(β-1))β x,

respectively. Both operators have discrete spectrum and that inf((S_φ_±))>0 is shown in Section <ref>. We illustrate the behaviour of the smallest eigenvalues of S_φ_± in Figure <ref>. More precisely, we have that

    λ  U_φ_*^-1 S_φ_* U_φ_*   = H_φ_* and λ^2/3  U_φ_±^-1 S_φ_± U_φ_±= H_φ_±.



Taylor expansion of the potential at the respective critical point and the estimate on the gradient (<ref>) imply that  

    | J_φ_*(ℋ-H_φ_*)J_φ_*|   = 𝒪(λ^4/5)  and also | J_φ_±(ℋ-H_φ_±)J_φ_±| = 𝒪(λ^3/5).

Let 0=e_1<e_2≤.. be the eigenvalues (counting multiplicities) of S_φ_* and 0<f_1≤ f_2≤... the ones of S_φ_±  and choose τ such that λ e_n+1>τ >λ e_n and λ^2/3 f_m+1>τ >λ^2/3 f_m with P_i being the projection onto the eigenspace to all eigenvalues of S_i below τ.

The IMS formula, see <cit.> for a version on manifolds, implies that  

    ℋ = J ℋ J -|∇ J |^2+∑_i ∈{φ_*, φ_±}(J_i H_i J_i + J_i (ℋ-H_i) J_i-  |∇ J_i|^2 ).

On the other hand, it follows that 

    J_φ_* H_φ_* J_φ_*   = J_φ_* H_φ_* P_φ_* J_φ_* + J_φ_* H_φ_* (id-P_φ_*) J_φ_*
       ≥ J_φ_* H_φ_*P_φ_*J_φ_* + λ^ e_n J_φ_*^2

and also 

    J_φ_± H_φ_± J_φ_±   = J_φ_± H_φ_± P_φ_± J_φ_± + J_φ_± H_φ_± (id-P_φ_±) J_φ_±
       ≥ J_φ_± H_φ_±P_φ_±J_φ_± + λ^2/3 f_m J_φ_±^2.

In particular, we find 

    ‖  V_1' ‖^2_ℝ^n≥ cλ^-6/5 on  J for some c>0.
 
and 

    ‖ V_1”‖_ℝ^n≥ c λ^-3/10 on  J for some c>0.
 
This implies for large λ that

    J H J ≥λ^2/3 f_m J^2.

From (<ref>) we then conclude that for some C>0

    ℋ≥λ^2/3 f_m J^2 - C λ^4/9 +∑_i ∈{φ_±,φ_*} J_i H_i P_i J_i = λ^2/3 f_m+ ∑_i ∈{φ_±,φ_*} J_i H_i P_i J_i -o(√(λ)).

This implies the claim of the Proposition, since 
    rank(J_0 H_±PJ_0) ≤ n.
 More precisely, for the eigenvalues E_1(λ)≤ E_2(λ)≤.. of ℋ we have shown that 

    lim inf_λ→∞λ^-2/3E_n(λ)≥ f_n-1>0  for n≥ 2 .
 
In particular, the lowest possible eigenvalue e_1=0 of the renormalized Schrödinger operator is of course attained as the nullspace of the renormalized Schrödinger operator H_φ_* is non-trivial.
This shows that the spectral gap of the renormalized Schrödinger operator grows at least proportional to λ^2/3. 







§ MULTI-COMPONENT 𝒪(N)-MODELS





 §.§ n≥ 2: Zero magnetic field, h=0


Let h=0 then the renormalized potential for n ≥ 2 is radially symmetric and possesses a critical point at φ=0. In the supercritical case, i.e. β>n, the renormalized potential possesses another critical radius r=‖φ‖∈ (0,1), see Figure <ref>. 
To see this, we differentiate the renormalized potential

    ∂_r V_n(r) = β r (1- I_n/2(β r)r I_n/2-1(β r)).

It is now obvious that r=0 is a critical point of the renormalized potential at which 

    lim_r ↓ 0I_n/2(β r)r I_n/2-1(β r)= 2nβ2>1  such that ∂_r^2 V_n(0)=β(1-βn)<0

where we used that β>n is supercritical.
To conclude the existence of precisely one other critical radius r_min at which the renormalized potential attains its global minimum it suffices therefore to show that I_n/2(β r)r I_n/2-1(β r) decays monotonically to zero. We prove this in Lemma <ref> in the appendix. This implies that also the factor (1- I_n/2(β r)r I_n/2-1(β r)) has precisely one root, i.e. the second critical radius.

In the next proposition we show that the radial part of the measure dν_N(φ) which we denote by dr_N:=ν_N^(n) r^n-1 e^-NV_n(r) dr in the sequel satisfies a SGI with a constant that is uniformly bounded in the number of spins.



 §.§ Zero magnetic field- A lower bound on the spectral gap



When h=0 and n ≥ 2, then the renormalized Schrödinger operator (<ref>) for λ:=N/2 is the self-adjoint operator 
    Δ_ren= - Δ_ℝ^n + λ^2 |∇_ℝ^n  V |^2 - λΔ_ℝ^n V.

This operator is also rotationally symmetric such that by separating (spherical coordinates) the angular part from the radial part, the remaining radial component Δ^rad,ℓ_ren of the renormalized Schrödinger operator on L^2((0,∞),r^n-1 dr )  for ℓ∈ℕ_0 reads

    Δ^rad,ℓ_ren = - (∂_r^2  + n-1r∂_r - ℓ(ℓ+n-2)r^2) +  λ^2 |∂_r V_n(r) |^2-λ∂_r^2 V_n(r).

Here, the term ℓ(ℓ+n-2) accounts for the eigenvalues of the angular part of the Laplacian.
The renormalized potential possesses, when h=0 and n ≥ 2, exactly two critical radii at which |∂_r V_n(r) |^2=0. The radii are r=0 and r=r_min, see the beginning of this Section <ref>.
However, V_n(r) is strictly concave at 0, i.e. ∂_r^2 V_n(0)<0, and by Lemma <ref> strictly convex at r_min such that ∂_r^2 V_n(r_min)>0. This follows from

    ∂_r^2 V_n(r_min) =β r_min∂_r |_r=r_min(1- I_n/2(β r)r I_n/2-1(β r))>0,

see the beginning of Section <ref>.

By the tensorization principle we already know that the rotational invariance of the renormalized measure implies that the spectral gap inequality for the renormalized measure is at least uniform in N. In our next Proposition we therefore study the low-lying spectrum of the radial component, Δ_ren^rad,0, as λ→∞. 


Let h=0, n ≥ 2 and β >n. The spectral gap of the radial renormalized Schrödinger operator Δ_ren^rad,0 grows linearly in N.


To study the low-lying spectrum of the radial component of the renormalized Schrödinger operator, let λ:= N/2 and consider Schrödinger operators

    H^0_osc(λ)=  - (∂_r^2  + n-1r∂_r)+ λ^2 |∂_r^2 V_n(0) |^2 r^2 - λ ∂_r^2 V_n(0)  and
       H^r_min_osc(λ)= - ∂_x^2 + λ^2  |∂_r^2 V_n(r_min) |^2(x-r_min)^2-λ ∂_r^2 V_n(r_min)

where we use the variable x rather than r to emphasize that the last operator is defined on L^2(ℝ), unlike the first one which is an operator on L^2((0,∞),r^n-1 dr). Observe that in (<ref>) we replaced the gradient term of the Schrödinger operator by its Taylor approximation at the critical point. This explains the occurrence of the second derivative at the critical point in (<ref>).
Invoking the unitary maps U_0 ∈ℒ( L^2((0,∞),r^n-1 dr)) and U_r_min∈ℒ(L^2(ℝ)) defined as 

    (U_0f)(x) = λ^-n/4f(λ^-1/2x)   and  (U_r_minf)(x) = λ^-1/4f(λ^-1/2(x+r_min))

shows that the two Schrödinger operators in (<ref>) are in fact unitarily equivalent, up to multiplication by λ, to the λ-independent Schrödinger operators

    S^0_osc   = - (∂_r^2  + n-1r∂_r)+ |∂_r^2 V_n(0) |^2 r^2 -   ∂_r^2 V_n(0)
    
    S^r_min_osc   = - ∂_x^2 +   |∂_r^2 V_n(r_min) |^2x^2- ∂_r^2 V_n(r_min),

respectively. More precisely, we have that

    U_0^-1λ S^0_osc U_0   = H^0_osc(λ)   and   
    U_r_min^-1λ S^r_min_oscU_r_min= H^r_min_osc(λ).

Since the bottom of the spectrum of the operator S^0_osc is strictly positive S^0_osc≥ -∂_r^2 V_n(0)>0,
we conclude from (<ref>) that the bottom of the spectrum of H^0_osc(λ) increases linearly to infinity as λ→∞.


To connect the low-energy spectrum of the renormalized Schrödinger operator with the above auxiliary operators, take j ∈ C_c^∞(-∞,2) such that j(x)=1 for | x |≤ 1. Then, we define

    J_0(x)    = j(λ^2/5| x | ),  J_r_min(x) = j(λ^2/5| x-r_min| ) with ‖∇  J_i‖_ℝ^n = 𝒪(λ^2/5)

for i ∈{ 0,r_min} and J := √(1-J_r_min^2-J_0^2). 


Without loss of generality we can assume that λ is large enough such that J_0 and J_r_min are disjoint.



Taylor expansion of the potential at 0 and r_min respectively and the estimate on the gradient (<ref>) imply that  

    | J_i(Δ^rad,0_ren-H^i_osc)J_i | = 𝒪(λ^4/5)  for i ∈{ 0,r_min}.

Let 0=e_1<e_2≤.. be the eigenvalues (counting multiplicities) of S^0_osc⊕ S^r_osc and choose τ such that e_n+1>τ >e_n with P_i being the projection onto the eigenspace to all eigenvalues of H^i_osc below τλ.
The IMS (Ismagilov, Morgan, and Simon/Sigal) formula, see <cit.> for a version on manifolds, implies that  

    Δ^rad,0_ren = J Δ^rad,0_ren J  - |∂_r J |^2 +∑_i ∈{0,r_min}(J_iΔ^rad,0_ren J_i-  |∂_r J_i|^2 )

such that 

    Δ^rad,0_ren = J Δ^rad,0_ren J -|∂_r J |^2+∑_i ∈{0,r_min}(J_i H^i_osc J_i + J_i (Δ^rad,0_ren-H^i_osc) J_i-  |∂_r J_i|^2 ).

On the other hand, it follows that 

    J_i H^i_osc J_i   = J_i H^i_osc P_i J_i + J_i H^i_osc (id-P_i) J_i≥ J_i H^i_oscP_i J_i + λ e_n J_i^2.

By construction, since ∇ V_n vanishes linearly on the support of J_i, we have 


    ‖∇ V_n ‖^2_ℝ^n≥ c (λ^-2/5)^2 =cλ^-4/5 on  J for some c>0.
 
Since Δ V_n is globally bounded anyway, this implies for large λ that

    J Δ^rad,0_ren J ≥ J^2 (c λ^6/5-λ) ≥λ e_n J^2.

From (<ref>) we then conclude that for some C>0

    Δ^rad,0_ren≥λ e_n - C λ^4/5 + ∑_i ∈{0,r_min}J_i H^i_oscP_i J_i= λ e_n+ ∑_i ∈{0,r_min}J_i H^i_oscP_i J_i-o(λ).

This implies the claim of the Proposition, since 
    rank(∑_i ∈{0,r_min}J_i H^i_oscP_i J_i) ≤ n.
 More precisely, for the eigenvalues E_1(λ)≤ E_2(λ)≤.. of Δ^rad,0_ren we have shown that 

    lim inf_λ→∞λ^-1E_n(λ)≥ e_n.
 
In particular, the lowest possible eigenvalue e_1=0 of the renormalized Schrödinger operator is of course attained as the nullspace of the renormalized Schrödinger operator is non-trivial.
This shows that the spectral gap of the renormalized Schrödinger operator grows at least linearly in λ in the angular sector ℓ=0. 

 Let h=0, n ≥ 2 and β >n. The spectral gap of the full Gibbs measure ρ does not close faster than Θ(1/N). In particular, for radial functions, i.e. f only depends on |σ̅|, the spectral gap remains open.


Since the spectral gap of the radial component of the renormalized measure grows linearly in N and the spectral gap of the angular component is uniform in N, the tensorization principle implies that the full renormalized measure satisfies a SGI that is uniform in N. 
Due to Proposition <ref>, the spectral gap of the full measure does therefore not close faster than of order 1/N.

For radial functions f, the  ^n ∋φ↦ E_μ_φ(f) maps also into radial functions and therefore the spectral gap of the renormalized measure is only determined by the radial renormalized Schrödinger operator in Prop. <ref>. Using Proposition <ref> and (<ref>), this implies that for radial functions, the gap remains open.


In the next Proposition we show that the rate N^-1 in this case is in fact optimal: 
 Let h=0, n ≥ 2 and β >n. The spectral gap of the full measure ρ of the dynamics decays at least as fast as N^-1. 


We consider the mean-spin σ̅:(𝕊^n)^N →^n+1 defined by


    σ̅(σ) = 1/N∑_i=1^N σ_i  = (σ̅_1(σ),...,σ̅_n+1(σ)) ∈^n+1.
 

In analogy to the spherical harmonics which in cartesian coordinates reads x_1/‖ x ‖, we consider the function:  

    f(σ):= σ̅_1(σ)/‖σ̅(σ) ‖η( ‖σ̅(σ) ‖).
 
where η∈ C^∞(ℝ;[0,1]) is a cut-off function such that for fixed 1>δ>0: 
    η(t):={[               1,    when t> δ and;               0, when  t  ≤δ/2  . ].

 As we want to compute the covariant derivative ∇_σ_1 t(σ), we consider the parametrisation 
 γ_1(t) so that γ_1(0)= σ_1. Then we define γ(t):=(γ_1(t), σ_2, …, σ_N) and s(t):= σ̅(γ(t)). It is then clear that for v:=γ_1'(0) we have  s'(0)=v/N, the first coordinate of which is  s_1'(0)=⟨ e_1, v ⟩ /N. We define then 

    s_1(t):=σ̅_1 (γ(t))=⟨ e_1, s(t) ⟩.
 

Thus, since f(γ(t))=s_1(t)/‖ s(t) ‖,
we find for the derivative


    f'(0)   = 1/| s(0) |^2( | s(0) | s_1'(0)- s_1(0) s(0) · s'(0)/| s(0) |) η(| s(0)|)  
            + s_1(0)/| s(0) |η'(| s(0)| ) s(0) · s'(0)/| s(0) |
       = 1/N|σ̅(σ) |^2( |σ̅(σ) |⟨ e_1,v ⟩- ⟨ e_1, σ̅(σ) ⟩⟨σ̅(σ),v ⟩/|σ̅(σ) |) η(|σ̅(σ) | ) 
       +  σ̅_1(σ)/|σ̅(σ) |η'(|σ̅(σ ) | ) ⟨σ̅(σ),v ⟩/N |σ̅(σ) |   .

Therefore, we see that in terms of 
    μ(σ):= 1/N( 1/|σ̅(σ) |^2(|σ̅(σ) | e_1- σ̅_1(σ) σ̅(σ)/|σ̅(σ)|)η(|σ̅(σ) | ) +   
    σ̅_1(σ)/|σ̅(σ) |^2η'(|σ̅(σ ) | )   σ̅(σ) 
     ),

the derivative is just

    ∇_σ_1 f(σ) = μ(σ)-⟨μ(σ), σ_1 ⟩σ_1.

The cut-off function η ensures that |σ̅(σ) | is not small. Therefore we can bound | Z(σ)| =𝒪(1) which implies that 𝔼_ρ(| ∇_σ_1σ̅(σ)|^2) ≲ 1/N^2 or that 
    ∑_i ∈ [N]𝔼_ρ(| ∇_σ_iσ̅(σ)|^2) ≲ 1/N.

 
By rotational symmetry we also know that 𝔼_ρ(f)=0. For the second moment 𝔼_ρ(f(σ)^2), we have by rotational invariance again

    1   =∑_i=1^n+1𝔼_ρ( σ̅_i(σ)^2/|σ̅(σ) |^2)= ∑_i=1^n+1𝔼_ρ( σ̅_1(σ)^2/|σ̅(σ) |^2)
       = (n+1)𝔼_ρ( σ̅_1(σ)^2/|σ̅(σ) |^2η(|σ̅(σ)  |) ) + (n+1)ℛ_N
 
where ℛ_N is the error 

    ℛ_N:= 𝔼_ρ( σ̅_1(σ)^2/|σ̅(σ) |^2(1- η(  |σ̅(σ)  |) ) ).


Our aim is now to argue that ℛ_N is small as N is large.

For β>n we know that the renormalized potential attains its minimum at hyperspheres ∂ B_^n(0,r_min). This implies that the renormalized measure concentrates at such φ∈∂ B(0,r_min) with exponential tail bounds, i.e. the probability of φ away from ∂ B(0,r_min) is exponentially small in N. 
The fluctuation measure then enforces that also the mean spin σ̅ has to be outside of a ball of radius δ>0 with high probability.
To see this recall that the fluctuation measure can be rewritten as 

    𝔼_μ_φ(F) = ∫_(𝕊^n-1)^N F(σ)  e^β N ⟨φ, σ̅⟩ dS_𝕊^n-1^⊗^N(σ)/∫_(𝕊^n-1)^N e^β N ⟨φ, σ̅⟩ dS_𝕊^n-1^⊗^N(σ) =∫_(𝕊^n-1)^N F(σ)  e^β N ⟨φ, σ̅⟩ dS_𝕊^n-1^⊗^N(σ)/𝒩(φ)^N .

 Here, the radial normalizing function 
 
    𝒩(φ):=∫_𝕊^n-1 e^β⟨φ,  σ⟩ dS_𝕊^n-1(σ)=Γ(n2) (2‖βφ‖)^n/2-1 I_n2-1( ‖βφ‖)
 is a strictly monotonically increasing function of |φ| that satisfies 𝒩(φ) ≥ 1 and 𝒩(φ)=1 if and only if φ=0. This follows directly from the Taylor series of the modified Bessel function.
 
Hence, we can pick δ such that e^β⟨φ, σ̅⟩ < (1+𝒩(φ))/2 for all φ∈∂ B(0,r_min) and |σ̅|≤δ. Hence, we see that for such φ 

    𝔼_μ_φ(_|σ̅|≤δ) = 𝒪(( 1/2𝒩(φ)+1/2)^N ).

This shows that 𝔼_ρ(_|σ̅|≤δ) = 𝒪(( 1/2𝒩(φ)+1/2)^N ) and hence that ℛ_N tends exponentially fast to zero as well, as N tends to infinity, under the condition that β>n. 






 §.§ Nonzero magnetic fields for n≥ 2



The situation h≠ 0 and n ≥ 2 cannot be reduced to a one-dimensional model due to lack of symmetries. Yet, the renormalized Schrödinger operator provides a very elegant tool to show that the spectral gap of the full generator of the Ginzburg-Landau dynamics dynamics remains open as N →∞. 



In fact, whereas the global minimum for h=0 of the renormalized potential is attained on a hypersphere, the global minimum for h ≠ 0 is attained at a single point, only. This allows us to identify the asymptotic of the low-energy spectrum of the renormalized Schrödinger operator directly with the spectrum of a quantum harmonic oscillator.



Let φ_c∈ℝ^n be a critical point of the renormalized potential (<ref>). We define the set 

    Σ:={∑_i=1^n(n_i|λ_i | + 12(|λ_i | - λ_i)) ,  with  n_i ∈ℕ_0, λ_i ∈σ(D^2V_n(φ_c)) }

where λ_1,...,λ_n comprise the entire spectrum of D^2V_n(φ_c).



Let e_k be the k-th smallest element counting multiplicity in Σ we then have the following Proposition:


Let h ≠ 0, β≥ n, and n ≥ 2.  Let E_k(λ) denote the k-th lowest eigenvalue of the renormalized generator then this eigenvalue satisfies the asymptotic law lim_λ→∞E_k(λ)/λ = e_k.
In particular, the ground state of the renormalized generator in the limit as λ→∞ is unique and the spectral gap of the renormalized Schrödinger operator remains open and linearly in λ.


When h ≠ 0 then the renormalized potential has a unique non-degenerate minimum. To see this recall that the renormalized potential reads

    V_n(φ,h)
        = β2(1+‖φ‖^2 ) - log( Γ(n/2) (2‖βφ +h‖)^n/2-1 I_n2-1(‖βφ +h‖) ).

Introducing the new variable ζ:=βφ + h implies that 

    V_n(φ(ζ),h)
        = 12β(β^2+‖ζ-h ‖^2 ) - log( Γ(n/2) (2‖ζ‖)^n/2-1 I_n2-1(‖ζ‖) ) 
       = 12β(β^2+‖ζ‖^2+‖ h‖^2 -2  ⟨ζ,h ⟩) - log( Γ(n/2) (2‖ζ‖)^n/2-1 I_n2-1(‖ζ‖) ).

Computing the gradient of that expression yields 

    ∇_ζ V_n(φ(ζ),h)= -1/βh + g_β(‖ζ‖) e_ζ

where we introduced the auxiliary function g_β(r):= (r/β -  I_n/2 (r)/I_n/2-1( r)). Thus for the gradient to vanish the vectors h and ζ have to be linearly dependent. 

Assuming thus that e_h = ±e_ζ we obtain from setting the gradient to zero the following equation 

    β I_n/2 (‖ζ‖) / I_n/2-1( ‖ζ‖)=( ‖ζ‖∓‖ h ‖).

Thus, when h and ζ are aligned, there is precisely one solution, the global minimum of the renormalized potential, satisfying 

    β I_n/2 (‖ζ‖) / I_n/2-1( ‖ζ‖)=( ‖ζ‖- ‖ h ‖)

with g_β(‖ζ‖)= β^-1‖ h ‖>0. That the aligned scenario corresponds to the global minimum is evident from the expression of the renormalized potential (<ref>).

The simplicity of the solution follows since the left hand side β I_n/2 (‖ζ‖)  I_n/2-1( ‖ζ‖) is a concave, monotonically increasing function from 0 to β as ‖ζ‖→∞.


When h and ζ point in opposite directions, there can, by concavity of the left-hand side, be between zero and two solutions to the equation

    β I_n/2 (‖ζ‖) / I_n/2-1( ‖ζ‖)=( ‖ζ‖+ ‖ h ‖)

with g_β(‖ζ‖)= -β^-1‖ h ‖<0.
In particular, for sufficiently low temperatures there exists a local maximum and a saddle point of the renormalized potential as shown in Figure <ref>. 




From differentiating (<ref>), the Hessian is given by

    D^2_ζ V_n(φ(ζ),h)= g'(‖ζ‖)ζζ^T/‖ζ‖^2+g_β(‖ζ‖)(id/‖ζ‖-ζζ^T/‖ζ‖^3).

We note that the Hessian has full rank unless at critical points unless g_β'(‖ζ‖)+g_β(‖ζ‖)(‖ζ‖ - ‖ζ‖^-1)=0, since g_β(‖ζ‖)≠ 0 by (<ref>) for non-zero magnetic fields.






In addition, there can be only a saddle point which can only happen at one fixed temperature depending on n. 



Finally, if the temperature is sufficiently high, yet still such that β>n, there may be no critical point if h and ζ point in opposite directions. This is in particular the case when β=n and h ≠ 0: Taylor expansion at zero yields

    β I_n/2 (‖ζ‖) / I_n/2-1( ‖ζ‖) = βΓ(n/2)/2Γ(1+n/2)‖ζ‖ + 𝒪(‖ζ‖^2)
 
where for β=n we find n Γ(n/2)/2Γ(1+n/2)=1 and concavity of the function ‖ζ‖↦β I_n/2 (‖ζ‖) / I_n/2-1( ‖ζ‖) show.


Thus ‖∇ V_n ‖^2 vanishes at not more than three critical points φ_c on the span of h. In particular, all eigenvalues of D^2V_n are non-negative only at the global minimum of V_n by (<ref>), since we already established that g'(‖ζ‖)<0 at the other two. To see that they are strictly positive there, it suffices to analyze for r=‖ζ‖

    g_β'(‖ζ‖)    = (β^-1 -ℐ(‖ζ‖)^-1) + ‖ζ‖ℐ'(‖ζ‖)/ℐ(‖ζ‖)^2
       =g_β(‖ζ‖)/‖ζ‖ + ‖ζ‖ℐ'(‖ζ‖)/ℐ(‖ζ‖)^2>0.


Hence, we find that

    g_β'(‖ζ‖)+g_β(‖ζ‖)(‖ζ‖ - ‖ζ‖^-1) = g_β(‖ζ‖) ‖ζ‖ +  ‖ζ‖ℐ'(‖ζ‖)/ℐ(‖ζ‖)^2.

In particular, this expression is strictly positive at the global minimum, since g_β(‖ζ‖) >0 and ℐ'(‖ζ‖)>0 by general principles, see Lemma <ref>.

The asymptotic behaviour of the spectrum of the renormalized Schrödinger operator has been computed in <cit.> and our above representation of Σ follows by noticing that 12 D^2  |∇ V_n |^2(φ_c) = (D^2V_n(φ_c))^2>0. 

Since the renormalized Schrödinger operator and renormalized generator are unitarily equivalent up to a factor, the semiclassical eigenvalue distribution stated in <cit.> implies the statement of the Proposition. 




§ THE CRITICAL REGIME, PROOF OF THEO. <REF>


We conclude our analysis by investigating the  critical case β=n and prove Theorem <ref>. As before, we distinguish between n=1 and multi-component systems n ≥ 2:


 §.§ Critical Ising model

It follows from (<ref>), which always vanishes for all x>0, that the spectral gap, at the critical point β=n=1, does not close exponentially fast in the number of spins. We want to show in this subsection that it closes at least polynomially, though. For a refined analysis in dimension n=1, we recall some basic ideas from discrete Fourier analysis: 


Let f: {± 1 }^N →ℂ be an arbitrary function on the hypercube. The L^2({± 1}^N,2^-Ndμ_count) inner product on the hypercube is defined as 

    ⟨ f,g ⟩_{± 1}^N := ∑_x ∈{± 1}^N 2^-N f(x_1,..,x_N)g(x_1,..,x_N).

The characteristic function χ_S for S ⊂ [N] is defined as χ_S(x):= ∏_i ∈ Sσ_i
and the family (χ_S)_S ⊂ [N] forms an orthonormal basis of L^2({± 1}^N). In particular, χ_∅=1.
We also define indicator vectors _S∈ℝ^N such that _S(x)=1 if x ∈ S and 0 otherwise. 

Every function f ∈ L^2({± 1}^N) admits a unique Fourier decomposition


    f = ∑_ S ⊂ [N]f(S) χ_S

where f(S) := ⟨ f, χ_S ⟩. The variance of the stationary measure is given as the sum of

    Var_ρ(f)=𝔼_ν_N( Var_μ_φ(f))+ Var_ν_N(𝔼_μ_φ(f)).

Since the first term on the right-hand side of this equation is always uniformly bounded by the Dirichlet form, as shown in the proof of Proposition <ref>, it suffices to study the behaviour of the second term.
Thus, applying the expectation with respect to the fluctuation measure yields by the Fourier decomposition (<ref>)

    𝔼_μ_φ(f)  = ∑_ S  ⊂ [N]f(S) 𝔼_μ_φ( χ_S).

In particular, using the explicit  form of V_1(ϕ), a direct computation yields for all x ∈ [N]

    𝔼_μ_φ( σ(x) )= e^V_1(φ)(e^-β/2|φ-1 |^2+h -e^-β/2|φ+1 |^2-h ) /2=  tanh(βφ+h).


Using that μ_φ is a product measure, this implies that the full expression for (<ref>) is given by 

    𝔼_μ_φ(f)   = ∑_ S  ⊂ [N]f(S)( tanh(βφ+h))^| S |.

Hence, we find for the variance 

    Var_ν_N(𝔼_μ_φ(f))    =  ∑_S_1,S_2 ⊂[N]^Nf(S_1)f(S_2)( 𝔼_ν_N( tanh(βφ+h)^| S_1|+| S_2|)
            -𝔼_ν_N( tanh(βφ+h)^| S_1|)𝔼_ν_N( tanh(βφ+h)^| S_2|) )

For the Dirichlet form, we find, with S_1  S_2 denoting the symmetric difference of sets S_1 and S_2,

    ∑_x ∈ [N]𝔼_ρ|∇_𝕊^0^(x) f |^2   = ∑_x ∈ [N]∑_ S_1,S_2  ⊂ [N]f(S_1)f(S_2)𝔼_ρ(∇_𝕊^0^(x)χ_S_1∇_𝕊^0^(x)χ_S_2) 
       = 4∑_x ∈ [N]∑_ S_1,S_2  ⊂ [N]δ_x ∈ S_1δ_x ∈ S_2f(S_1)f(S_2)𝔼_ρ(χ_S_1χ_S_2) 
       = 4∑_ S_1,S_2  ⊂ [N]∑_x ∈ S_1 ∩ S_2f(S_1)f(S_2)𝔼_ρ(χ_S_1 S_2) 
       = 4∑_ S_1,S_2  ⊂ [N]⟨_S_1, _S_2⟩_ℝ^Nf(S_1)f(S_2)𝔼_ρ(χ_S_1 S_2)
       = 𝔼_ρ| 2∑_ S  ⊂ [N]_Sf(S) χ_S |^2_ℝ^N.



For zero magnetic fields, i.e. h=0, and β≥ 1,all functions with Fourier support on sets of fixed cardinality k ∈ℕ, i.e. for f given as 

    f = ∑_S ⊂ [N]; | S | = kf(S) χ_S.

satisfy the inequality Var_ν_N(𝔼_μ_φ(f)) ≤N4k∑_x ∈ [N]|∇_𝕊^n-1^(x)f |^2_L^2(dρ).

In particular, for the magnetization

    M = 1√(N)∑_x ∈ [N]σ(x)

we obtain an inequality 


    Var_ν_N(𝔼_μ_φ(M)) =  N 𝔼_ν_N(tanh( βφ)^2)4∑_x ∈ [N]|∇_𝕊^n-1^(x)M |^2_L^2(dρ).
 

Moreover, the spectral gap for critical β=1 closes at least like 𝒪(N^-1/2).


When h=0, it suffices to estimate the variance by Jensen's inequality as 

    Var_ν_N(𝔼_μ_φ(f))    ≤𝔼_ν_N∑_S_1,S_2 ⊂ [N] ; | S_1 |=| S_2 |=kf(S_1)f(S_2)𝔼_μ_φ(χ_S_1)𝔼_μ_φ(χ_S_2)  
       ≤𝔼_ν_N𝔼_μ_φ|∑_S ⊂[N] ; | S_1 |=| S_2 |=kf(S) χ_S |^2  
       = 1/k^2𝔼_ρ|⟨∑_ S ⊂[N] ; | S_1 |=| S_2 |=kf(S)   _Sχ_S,_[N]⟩_ℝ^N|^2 
       ≤N/k^2 𝔼_ρ|∑_ S ⊂ [N] ;  | S_1 |=| S_2 |=kf(S)_Sχ_S |_ℝ^N^2.

Using (<ref>) we then obtain the spectral gap inequality

    Var_ν_N(𝔼_μ_φ(f))    ≤N/k^2 𝔼_ρ|∑_ S ⊂[N] ;| S_1 |=| S_2 |=kf(S)  _Sχ_S |_ℝ^N^2 
        =  N/4k^2∑_x ∈ [N]|∇_𝕊^0^(x) f |_L^2(d ρ)^2.

Turning to the magnetization (<ref>), we can write the variance of the magnetization M in terms of the expectation value 𝔼_ν_N(tanh( βφ)^2)

    Var_ν_N(𝔼_μ_φ(M)) = 1N∑_ x,y ∈ [N]𝔼_ν_N( tanh( βφ)^2) = N 𝔼_ν_N(tanh( βφ)^2).

We now recall that tanh(βφ)^2 = β^2 φ^2 +𝒪(φ^4) and for β =1 

    V_1(φ) = 1/2+φ^4/12+𝒪(φ^6)
 by Taylor expanding around 0.
It therefore follows from Laplace's principle <cit.> that 

    𝔼_ν_N(tanh( βφ)^2) ∼ N^1/4N^-3/4=N^-1/2.




On the other hand, we can compute the Dirichlet form of the magnetization using (<ref>)

    ∑_x ∈ [N]𝔼_ρ|∇_𝕊^0^(x) M |^2
       = 4N𝔼_ρ|∑_ x  ∈ [N] _Sχ_S |^2 =4  𝔼_ρ(1)=4.

Thus, comparing (<ref>) with (<ref>) implies the claim together with the asymptotic (<ref>).

While Proposition <ref> shows that the magnetization leads for critical β=1 to a spectral gap that closes at least like ∼ N^-1/2, when h=0,  the next Proposition shows that the magnetization does not imply a vanishing spectral gap when h > 0.

Let h>0, β≥ 1, and f a function with Fourier transform supported on sets of cardinality ≤ k for some fixed k ∈ℕ_0 independent of N, i.e. 

    f = ∑_S ⊂[N]; | S |≤ k f(S) χ_S.
 
Then such functions satisfy an improved inequality with φ_min=argmin_φ V_1(φ)  

    Var_ν_N(𝔼_μ_φ(f))    ≤β^2 csch^2(2(βφ_min+h))/2V”_1(φ_min)∑_x ∈ [N]𝔼_ρ|∇_𝕊^0^(x) f |_ℝ^N^2(1+o(1))

with a constant β^2 csch^2(2(βφ_min+h))/2V”_1(φ_min) (1+o(1)) that strictly bounded away from zero in the limit N →∞. In particular, V_1”(φ_min) >0 by the discussion in the beginning of Section <ref>.


Using (<ref>), which applies since V”_1(φ_min)>0 by the discussion in Subsection <ref>, we conclude that 

    𝔼_μ_φ(f)   = ∑_ S  ⊂ [N]f(S)( tanh(βφ+h))^| S |

implies since 

    d/dφtanh(βφ+h) = βsech^2(βφ+h)= βcsch(βφ+h)^2 tanh(βφ+h)^2

that

    Var_ν_N(𝔼_μ_φ(f))
       =1/2N V”_1(φ_min)|∑_S ⊂[N] f(S)  β| S |tanh(βφ_min +h)^| S |+1csch(βφ_min +h)^2 |^2(1+o(1))
       =β^2tanh(βφ_min +h)^2csch(βφ_min +h)^4/2N V”_1(φ_min)|∑_S ⊂[N] f(S)  tanh(βφ_min +h)^| S |⟨_S, _[N]⟩_^N|^2(1+o(1))
       ≤2 β^2 csch^2(2(βφ_min+h))/V”_1(φ_min)|∑_S ⊂[N] f(S)  tanh(βφ_min +h)^| S |_S|^2(1+o(1))
       =  2 β^2 csch^2(2(βφ_min+h))/V”_1(φ_min)|𝔼_ρ∑_S ⊂[N] f(S)  χ_S _S|^2(1+o(1))
       ≤2 β^2 csch^2(2(βφ_min+h))/V”_1(φ_min)𝔼_ρ|∑_S ⊂[N] f(S)  χ_S _S|^2(1+o(1))
       =  β^2 csch^2(2(βφ_min+h))/2V”_1(φ_min)∑_x ∈ [N]𝔼_ρ|∇_𝕊^0^(x) f |_ℝ^N^2(1+o(1))

where we used (<ref>) in the first line, | S |=⟨_S,_[N]⟩ in the second line, Cauchy-Schwarz and csch(x)^4tanh(x)^2=4csch(2x)^2 in the third line, (<ref>) and (<ref>) in the fourth line, Jensen's inequality in the fifth line and finally (<ref>) in the last line.



 §.§ Critical multi-component systems

In this subsection, we prove the multi-component part of Theorem <ref>: 




The magnetization M =N^-1/2∑_x ∈ [N]σ(x) has in the multi-component case always unit Dirichlet norm

    ∑_x ∈ [N]𝔼_ρ|∇_𝕊^n-1^(x) M |^2=∑_x ∈ [N]𝔼_ρ(N^-1) =1.

On the other hand, we can explicitly calculate using the derivative of the modified Bessel function of the first kind, ∂_z I_ν(z) = ν/z I_ν(z)+I_ν+1(z), and (<ref>), the expectation value 𝔼_μ_φ(σ(x)) that is independent of x ∈[N] for φ≠ 0

    𝔼_μ_φ(σ(x)) 
       = e^NV_n(φ) e^-β/2(1+ ‖φ‖^2)∏_y ∈ [N]∫_𝕊^n-1 e^-β⟨φ, σ(y)⟩σ(x)  dS(σ(y)) 
       = I_n/2(‖βφ‖)/I_n/2-1(‖βφ‖)φ/‖φ‖.

Taylor expansion at zero then yields 

    ( 𝔼_μ_φ(σ(x))  )^2=(I_n/2(‖βφ‖)/I_n/2-1(‖βφ‖))^2 = Γ(n/2)^2 /4Γ(1+n/2)^2‖βφ‖^2+ 𝒪(‖βφ‖^4).

For the renormalized potential we find by Taylor expansion, which we shall already specialize to critical temperatures β=n, at zero

    V_n(φ) = n/2 + n^3/8+4n‖φ‖^4+ 𝒪(‖φ‖^5).

For the magnetization M (<ref>), we can write

    Var_ν_N(𝔼_μ_φ(M))    =  1N∑_ x,y ∈ [N]𝔼_ν_N(( I_n/2(‖βφ‖)/I_n/2-1(‖βφ‖))^2) 
       = N 𝔼_ν_N(( I_n/2(‖βφ‖)/I_n/2-1(‖βφ‖))^2).

We then have by radial symmetry of both the renormalized potential and the integrand that at critical temperatures β=n

    𝔼_ν_N(( I_n/2(‖ n φ‖)/I_n/2-1(‖ n φ‖))^2)=∫_0^∞ e^-NV_n(r) r^n-1( I_n/2(n r)/I_n/2-1(n r))^2 dr /∫_0^∞ e^-NV_n(r) r^n-1 dr .

Applying Laplace's principle, cf. <cit.>, with constants μ=4 and α=3+(n-1) implies that 

    𝔼_ν_N(( I_n/2(‖ n φ‖)/I_n/2-1(‖ n φ‖))^2) ∼ N^n/4N^-(n+2)/4 =N^-1/2.

Combining this asymptotic behavior with (<ref>) and (<ref>) then yields the multi-component claim of Theorem <ref>, i.e. the rate N^1/2 is caught for the trial (mean spin) function M and thus the spectral gap is decaying at least with speed N^-1/2. 
 

The following Proposition shows that the upper bound N^-1/2 on the spectral gap in the critical regime β=n for all dimensions n≥1, is in fact sharp:

Let h=0 and β=n ≥ 1. The spectral gap of the radial renormalized Schrödinger operator grows as Θ(N^1/2) and in particular, the spectral gap of the full measure does not close faster than Θ( N^-1/2).


Let λ:= N/2, we then consider the equivalent Schrödinger operators to the renormalized generator 

    ℋ_1   :=-∂_x^2 + λ^2 | V'_1(x) |^2-λ V_1”(x)  and for n ≥ 2   
    ℋ_n^ℓ   :=-(∂_r^2 + n-1r∂_r)+ℓ(ℓ+n-2)r^2 +λ^2 |∇ V_n |^2 -λΔ  V_n,   ℓ∈ℕ_0,

where we used that by rotational symmetry of the renormalized potential, for n ≥ 2, we can decompose the Schrödinger operator into individual angular sectors parametrized by ℓ∈ℕ_0. We then introduce auxiliary Schrödinger operators

    H_1   =-∂_x^2 + λ^2 x^69 - λ x^2  and for  n ≥ 2
    
    H_n^ℓ   =-(∂_r^2 + n-1r∂_r) +ℓ(ℓ+n-2)r^2 + λ^2n^6(2+n)^2 r^6 -λ3n^32+n  r^2,   ℓ∈ℕ_0

on L^2() and L^2((0,∞), r^n-1 dr), respectively. The five first eigenvalues of H_1 are shown in Fig. <ref>.
 
We then define j ∈ C_c^∞(-2,2) such that j(x)=1 for | x |≤ 1 and from this 

    J_0(x)    = j(λ^2/9| x | ) and J := √(1-J_0^2) with ‖∇  J_0‖_ℝ^n = 𝒪(λ^4/9).

Invoking the unitary maps U_1∈ℒ(L^2(ℝ)) and U_n ∈ℒ( L^2((0,∞),r^n-1 dr)) defined as 

    (U_1f)(x) := λ^-1/8f(λ^-1/4(x))  and (U_nf)(r) := λ^-n/8f(λ^-1/4r)

shows that the two Schrödinger operators in (<ref>) are in fact unitarily equivalent, up to multiplication by √(λ), to the λ-independent Schrödinger operators

    S_1   = - ∂_x^2+   19  x^6-x^2  and for  n ≥ 2 
    
    S_n^ℓ   = - (∂_r^2  + n-1r∂_r)+ℓ(ℓ+n-2)r^2+ n^6(2+n)^2  r^6 -3n^32+n  r^2,   ℓ∈ℕ_0

respectively. That inf((S_n^0))=0 is shown in Section <ref>. Since ℓ(ℓ+n-2)r^2>0 we have consequently that for ℓ>0 by monotonicity inf((S_n^ℓ))≥inf((S_n^1))>0.
More precisely, we have that

    λ^1/2  U_n^-1 S_n^ℓ U_n   = H_n^ℓ.

More precisely, since (U_nf)(x) := λ^-n/8f(λ^-1/4x), it follows that 

    (S_n^ℓ U_nf)(r)=   -λ^-n/8((λ^-1/2f”(λ^-1/4r)  + λ^-1/4n-1r f'(λ^-1/4r))+ℓ(ℓ+n-2)r^2f(λ^-1/4r)
       + n^6(2+n)^2  r^6f(λ^-1/4r) -3n^32+n  r^2f(λ^-1/4r)).

Then, applying (U_n^-1f)(r) = λ^n/8 f(λ^1/4r) shows that 

    (U_n^-1S_n^ℓ U_nf)(r)=   λ^-1/2(-(f”(r)  + n-1r f'(r))+ℓ(ℓ+n-2)r^2f(r)
       + λ^2 n^6(2+n)^2  r^6f(r) -λ3n^32+n  r^2f(r)).


Taylor expansion of the potential at 0 and the estimate on the gradient (<ref>) imply that  

    | J_0(ℋ_n^ℓ-H_n^ℓ)J_0 | = 𝒪(λ^4/9).

Let 0=e_1<e_2≤.. be the eigenvalues (counting multiplicities) of S_n (over all angular sectors ℓ) and choose τ such that e_n+1>τ >e_n with P being the projection onto the eigenspace to all eigenvalues of H below τ√(λ).
The IMS formula, see <cit.> for a version on manifolds, implies that  

    ℋ_n = Jℋ_nJ  - |∇ J |^2 + (J_0ℋ_nJ_0-  |∇ J_0|^2 )

such that 

    ℋ_n = J ℋ_n J -|∇ J |^2+(J_0 H_n J_0 + J_0 (ℋ_n-H_n) J_0-  |∇ J_0|^2 ).

On the other hand, it follows that 

    J_0 H_n J_0   = J_0 H_n P J_0 + J_0 H_n (id-P) J_0≥ J_0 H_nP_n J_0 + √(λ) e_n J_0^2.

By construction, since ∇ V_n vanishes to third order on the support of J_0, we have 

    ‖∇ V_n ‖^2_ℝ^n≥ c (λ^-2/9)^6 =cλ^-4/3 on  J for some c>0.
 
Since Δ V_n vanishes to second order 

    ‖Δ V_n ‖_ℝ^n≥ c λ^-4/9 on  J for some c>0.
 
This implies for large λ that

    J H J ≥√(λ) e_n J^2.

From (<ref>) we then conclude that for some C>0

    ℋ_n ≥√(λ) e_n J^2 - C λ^4/9 +J_0 H_nP J_0= √(λ) e_n+ J_0 H_nP J_0-o(√(λ)).

This implies the claim of the Proposition, since 
    rank(J_0 H_nPJ_0) ≤ n.
 More precisely, for the eigenvalues E_1(λ)≤ E_2(λ)≤.. of ℋ_n we have shown that 

    lim inf_λ→∞λ^-1/2E_n(λ)≥ e_n.
 
In particular, the lowest possible eigenvalue e_1=0 of the renormalized Schrödinger operator is of course attained as the nullspace of the renormalized Schrödinger operator is non-trivial.
This shows that the spectral gap of the renormalized Schrödinger operator grows at least proportional to √(λ). 






§ NUMERICAL RESULTS



Recall that the eigenfunctions of the operator

    H_osc:=-ħ^2/2μd^2/dx^2+μω^2/2 x^2

are given for n ∈ℕ_0 by 

    ψ_n(x):=1/√(2^n n!)( μω/πħ)^1/4 e^-μω x^2/2 ħ H_n( √(μω/ħ)x ).

Then, it follows that 


    ⟨ψ_n, -ħ^2 ψ_m”⟩_L^2()=    ħμω/2(2n+1),  if  n=m 
       -ħμω/2√(n(n-1)),  if  n=m+2 
       -ħμω/2√((n+1)(n+2)),  if  n=m-2. 
    .


and 


    ⟨ψ_n,x^2 ψ_m ⟩_L^2()=    ħ/2 μω(2n+1),  if  n=m 
       ħ/2 μω√(n(n-1)),  if  n=m+2 
       ħ/2 μω√((n+1)(n+2)),  if  n=m-2. 
    .


Using the annihilation operator a=2^-1/2(∂_q+q) where q=√(μω/ħ) x and its adjoint we can explicitly compute the matrix elements of all (⟨ψ_n, x^n ψ_m ⟩) by writing q^n=√(2)(a+a^*)^n and using the well-known action of the annihilation operator on eigenstates of (<ref>). Using a finite-basis truncation of the above matrices allowed us then to obtain Figures <ref> and <ref>.




§ ASYMPTOTIC PROPERTIES OF THE ISING MODEL



Let β>1 and h ∈ [0,h_c). The three critical points of η_N,h:(-1,1)→ℝ

    η_N,h(s) =Γ(N+1)/Γ(N/2(1+s)+1)Γ(N/2(1-s)+1) e^-Nβ2(1-s^2)+Nhs

are given by s_N^c := γ(β)  (1+(1)) where γ(β) satisfies the critical equation for the continuous renormalized potential

    γ(β) = tanh(βγ(β)+h).

Let us order the solutions γ(β) to that equation by γ_1(β)<γ_2(β)<γ_3(β).
For h=0 the function η_N,0 attains (in the limit N →∞) its maximum at γ_1(β)=-γ_3(β)<0 and minimum at γ_2(β)=0. 



Let h>0, then the function η_N,h attains (in the limit N →∞) its unique global maximum at γ_3(β)>0 whereas both γ_1(β),γ_2(β)<0 and γ_1(β),γ_2(β) are local maxima and minima respectively.



The logarithmic derivative ζ_N,h(s) = ∂_s log(η_N,h(s)) satisfies 

    ζ_N,h(s)=    N(β s-arctanh(s )+  h)(1+(1)).



For h=0 we note that η_N,0 is even and for h>0 the global maxima of η_N,h must be attained at some s ≥ 0.
Direct computations show by the logarithmic scaling of the digamma function ψ_2(s) = log(Γ)'(s)= log(s)+ 𝒪(1/s) that the logarithmic derivative ζ_N,h(s) = ∂_s log(η_N,h(s)) is given by (<ref>).
Thus, for all critical values s_N^c of η_N,h, i.e. those values that satisfy ζ_N,h(s_N^c)=0, there exists γ(β) ∈ [-1,1] such that γ(β):=  lim_N →∞s_N^c
and γ(β) is any solution to γ(β) = tanh(γ(β) β+ h).



We then obtain (<ref>) directly by differentiating log(η_N,h) and using the identity

    -∂_s log(Γ(N(1+s)2+1)Γ(N(1-s)2+1)) 
       = -N/2(log(1+N/2(1+s)/1+N/2(1-s)) )(1+(1)) 
       = -N/2(log( 1 + s N/2/1+N/2/1 - sN/2/1+N/2) )(1+(1)) 
       = -N/2(log( 1 + s /1 - s) )(1+(1)) 
       = -N artanh(s)(1+(1)).




Moreover, we read off from (<ref>) that 

    lim_k ↑ 1ζ_N,h(k) = -∞ and lim_k ↓ -1ζ_N,h(k) = ∞.

In particular, γ(β) solves the implicit equation γ(β) = tanh(βγ(β)+h). For the second derivative of ζ_N,h which is h-independent, we find the closed-form expression using the derivative of the trigamma function ψ_3

    ζ_N,h”(s)    = -N^3/8(ψ_3'(1+N/2(1+s))-ψ_3'(1+N/2(1-s))) 
       = N^3/8∫_0^∞ z^2 e^-z(1+N/2(1+s))-e^-z(1+N/2(1-s))/1-e^-z dz.

This implies that ζ_N is strictly convex on [-1,0) and strictly concave on (0,1]. Using the asymptotic of the trigamma function ψ_3(s)= 1/s + 1/(2s^2)+ 𝒪(1/s^3) we find that ζ_N,h is strictly monotone increasing at zero, independent of h,

    ζ_N'(0) =Nβ -  N^2 ψ_3(1+N/2)/2 = N(β -   N/2/1+N/2)(1+(1))>0,

since β>1.



The function ℐ(r):= r I_n/2-1(r)/I_n(r) is strictly monotonically increasing on (0,∞). In particular ℐ'(r)>0.


By differentiating and using that I_ν'(r) = ν/r I_ν(r) + I_ν+1(r) we find

    ℐ'(r) = r(1-I_n/2-1(r)I_n/2+1(r)/I_n(r)^2).

Thus, it suffices to record that the product of Bessel functions satisfies I_n/2(z)^2 > (I_n/2-1I_n/2+1)(z):

    (I_n/2-1I_n/2+1)(z)    = (z/2 )^n∑_k=0^∞(n+k+1)_k(z^2/4)^k/k! Γ(n/2-1+k+1) Γ(n/2+1+k+1)
    
    (I_n/2I_n/2)(z)    = (z/2 )^n∑_k=0^∞(n+k+1)_k(z^2/4)^k/k! Γ(n/2+k+1) Γ(n/2+k+1)

Hence, the identity follows from 

    Γ(n/2+k+1)^2<Γ(n/2-1+k+1) Γ(n/2+1+k+1)
 
which follows itself from logarithmic convexity of the gamma function.



§ SUSY QUANTUM MECHANICS


We use ideas from supersymmetric quantum mechanics, to show positivity and analyze the ground state of several Schrödinger operators appearing in this article:

In one dimension, we recall that using operators 

    A=∂_x+W(x)  and  A^* = -∂_x+W(x)

with real-valued and smooth superpotential W,
we can write 

    A^*A = -∂_x^2 -W'(x)+W(x)^2  and  AA^*=-∂_x^2+W'(x)+W(x)^2.

In particular, W(x):=√(β(β-1))β x^2 yields operator S_φ_± defined in (<ref>).

However, solving Aψ=0 or A^*ψ = 0 shows that ψ=C e^±√(β(β-1))β x^3∉ L^2(). This shows that inf((AA^*)),inf((A^*A))>0.

We now analyze operators in (<ref>).
Choosing W(x):=x^3/3, yields A^*A=S_1 in (<ref>), and we find by solving 
Aψ(x)=0 that ψ(x)∝ e^-x^4/12 which implies that inf((A^*A))=0.

For radial operators on L^2((0,∞),r^n-1 dr), a similar argument applies: 

We define operators 

    A=∂_r + W(r)  and  A^*= -∂_r +n-1/r + W(r).



Choosing then W(r):=n^3/(2+n)r^3, such that A^*A =S_n with S_n in (<ref>), we find by solving 

    Aψ=0 ⇒ψ(r)∝ e^-n^3/4(n+2) r^4∈ L^2((0,∞),r^n-1 dr).




§ ASYMPTOTIC PROPERTIES



<cit.>

Let V: ℝ→ℝ be smooth with unique global minimum at φ_min∈ℝ and V”(φ_min)>0. Assume that ∫_ℝ e^-V(φ) dφ is finite and that {φ∈ℝ; V(φ) ≤ V(φ_min)+1 } is compact. We also define the probability measure dζ_N(φ) = e^-N V(φ) dφ / ∫_ℝe^-N V(φ) d φ. Then for any bounded smooth function g: ℝ→ℝ 

    𝔼_ζ_N(g)   = ∫_ℝ g(φ)e^-NV(φ) d φ/∫_ℝ e^-NV(φ) d φ
        = g(φ_min)+ g”(φ_min)/2N V”(φ_min) +3 V”'(φ_min) g'(φ_min)/4N V”(φ_min)^3 + 𝒪(1/N^2)

and for the variance

    Var_ζ_N (g) = g'(φ_min)^2/NV”(φ_min) + 𝒪(1/N^2)










alpha



