Ground States for Mean Field Models with a Transverse Component

Ioffe, Dmitry; Levit, Anna

doi:10.1007/s10955-013-0745-5

Cited by 13 publications

(17 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…in its stochastic representation via Feynman-Kac, [1], [3] and [7]. We are indebted to D. Ioffe for pointing out the connection and for useful comments.…”

Section: Introductionmentioning

confidence: 99%

Layered Systems at the Mean Field Critical Temperature

et al. 2015

View full text Add to dashboard Cite

We consider the Ising model on Z × Z where on each horizontal line {(x, i), x ∈ Z}, the interaction is given by a ferromagnetic Kac potential with coupling strength J γ (x, y) ∼ γJ(γ(x − y)) at the mean field critical temperature. We then add a nearest neighbor ferromagnetic vertical interaction of strength ǫ and prove that for every ǫ > 0 the systems exhibits phase transition provided γ > 0 is small enough.

show abstract

“…in its stochastic representation via Feynman-Kac, [1], [3] and [7]. We are indebted to D. Ioffe for pointing out the connection and for useful comments.…”

Section: Introductionmentioning

confidence: 99%

Layered Systems at the Mean Field Critical Temperature

et al. 2015

View full text Add to dashboard Cite

show abstract

“…60 See Rieckers (1981) and Gerisch (1993) for the analogue of (4.78) in the quantum Curie-Weisz model. 61 This seems well known, but the first rigorous proof we are aware of is very recent (Ioffe & Levit, 2013). 62 As points of B 3 , for 0 ≤ B < 1 these are given by ψ ± 0 = (B, 0, ± sin(arccos(B))).…”

Section: Ground Statesmentioning

confidence: 98%

Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

Landsman¹

2013

Studies in History and Philosophy of Science Part B: Studies in

View full text Add to dashboard Cite

Beginning with Anderson (1972), spontaneous symmetry breaking (ssb) in infinite quantum systems is often put forward as an example of (asymptotic) emergence in physics, since in theory no finite system should display it. Even the correspondence between theory and reality is at stake here, since numerous real materials show ssb in their ground states (or equilibrium states at low temperature), although they are finite. Thus against what is sometimes called 'Earman's Principle', a genuine physical effect (viz. ssb) seems theoretically recovered only in some idealization (namely the thermodynamic limit), disappearing as soon as the idealization is removed.We review the well-known arguments that (at first sight) no finite system can exhibit ssb, using the formalism of algebraic quantum theory in order to control the thermodynamic limit and unify the description of finite-and infinite-volume systems. Using the striking mathematical analogy between the thermodynamic limit and the classical limit, we show that a similar situation obtains in quantum mechanics (which typically forbids ssb) versus classical mechanics (which allows it).This discrepancy between formalism and reality is quite similar to the measurement problem (now regarded as an instance of asymptotic emergence), and hence we address it in the same way, adapting an argument of the author and Reuvers (2013) that was originally intended to explain the collapse of the wave-function within conventional quantum mechanics. Namely, exponential sensitivity to (asymmetric) perturbations of the (symmetric) dynamics as the system size increases causes symmetry breaking already in finite but very large quantum systems. This provides continuity between finite-and infinite-volume descriptions of quantum systems featuring ssb and hence restores Earman's Principle (at least in this particularly threatening case). Motto"The characteristic behaviour of the whole could not, even in theory, be deduced from the most complete knowledge of the behaviour of its components, taken separately or in other combinations, and of their proportions and arrangements in this whole. This is what I understand by the 'Theory of Emergence'. I cannot give a conclusive example of it, since it is a matter of controversy whether it actually applies to anything." (Broad, 1925, p. 59) * Final version, to be published in Studies in History and Philosophy of Modern Physics.

show abstract

“…The former are the ones traditionally studied for quantum spin systems, but the latter relate these systems to strict deformation quantization, since macroscopic observables are precisely defined by (quasi-)symmetric sequences which form the continuous cross sections of a continuous bundle of C * -algebras. This continuous bundle of C * -algebras is defined over base space I given by (1.9) with fibers 11) and continuity structure specified by continuous cross sections which are thus given by all quasi-symmetric sequences [15] [14, Ch.10]. 3 We refer to the appendix for some useful definitions or to [15] for a more comprehensive explanation.…”

Section: Spin Systems and Generalizationsmentioning

confidence: 99%

“…A typical example of a mean-field quantum spin system is the Curie-Weiss model (see, for example,[1,6,11,[25][26][27] and references therein) 3. The same result holds for an arbitary unital C * -algebra B playing the role of the matrix algebra M k (C) in the above setting[14].…”

mentioning

confidence: 94%

See 1 more Smart Citation

Bulk-boundary asymptotic equivalence of two strict deformation quantizations

Moretti

Ven

2020

Lett Math Phys

View full text Add to dashboard Cite

The existence of a strict deformation quantization of $$X_k=S(M_k({\mathbb {C}}))$$ X k = S ( M k ( C ) ) , the state space of the $$k\times k$$ k × k matrices $$M_k({\mathbb {C}})$$ M k ( C ) which is canonically a compact Poisson manifold (with stratified boundary), has recently been proved by both authors and Landsman (Rev Math Phys 32:2050031, 2020. 10.1142/S0129055X20500312). In fact, since increasing tensor powers of the $$k\times k$$ k × k matrices $$M_k({\mathbb {C}})$$ M k ( C ) are known to give rise to a continuous bundle of $$C^*$$ C ∗ -algebras over $$I=\{0\}\cup 1/\mathbb {N}\subset [0,1]$$ I = { 0 } ∪ 1 / N ⊂ [ 0 , 1 ] with fibers $$A_{1/N}=M_k({\mathbb {C}})^{\otimes N}$$ A 1 / N = M k ( C ) ⊗ N and $$A_0=C(X_k)$$ A 0 = C ( X k ) , we were able to define a strict deformation quantization of $$X_k$$ X k à la Rieffel, specified by quantization maps $$Q_{1/N}:/ \tilde{A}_0\rightarrow A_{1/N}$$ Q 1 / N : / A ~ 0 → A 1 / N , with $$\tilde{A}_0$$ A ~ 0 a dense Poisson subalgebra of $$A_0$$ A 0 . A similar result is known for the symplectic manifold $$S^2\subset \mathbb {R}^3$$ S 2 ⊂ R 3 , for which in this case the fibers $$A'_{1/N}=M_{N+1}(\mathbb {C})\cong B(\text {Sym}^N(\mathbb {C}^2))$$ A 1 / N ′ = M N + 1 ( C ) ≅ B ( Sym N ( C 2 ) ) and $$A_0'=C(S^2)$$ A 0 ′ = C ( S 2 ) form a continuous bundle of $$C^*$$ C ∗ -algebras over the same base space I, and where quantization is specified by (a priori different) quantization maps $$Q_{1/N}': \tilde{A}_0' \rightarrow A_{1/N}'$$ Q 1 / N ′ : A ~ 0 ′ → A 1 / N ′ . In this paper, we focus on the particular case $$X_2\cong B^3$$ X 2 ≅ B 3 (i.e., the unit three-ball in $$\mathbb {R}^3$$ R 3 ) and show that for any function $$f\in \tilde{A}_0$$ f ∈ A ~ 0 one has $$\lim _{N\rightarrow \infty }||(Q_{1/N}(f))|_{\text {Sym}^N(\mathbb {C}^2)}-Q_{1/N}'(f|_{_{S^2}})||_N=0$$ lim N → ∞ | | ( Q 1 / N ( f ) ) | Sym N ( C 2 ) - Q 1 / N ′ ( f | S 2 ) | | N = 0 , where $$\text {Sym}^N(\mathbb {C}^2)$$ Sym N ( C 2 ) denotes the symmetric subspace of $$(\mathbb {C}^2)^{N \otimes }$$ ( C 2 ) N ⊗ . Finally, we give an application regarding the (quantum) Curie–Weiss model.

show abstract

Ground States for Mean Field Models with a Transverse Component

Cited by 13 publications

References 30 publications

Layered Systems at the Mean Field Critical Temperature

Layered Systems at the Mean Field Critical Temperature

Spontaneous symmetry breaking in quantum systems: Emergence or reduction?

Bulk-boundary asymptotic equivalence of two strict deformation quantizations

Contact Info

Product

Resources

About