operators on Cn Toeplitz operators on compact manifolds Plan 1 Toeplitz operators Quantization Toeplitz operators on Cn Toeplitz operators on compact manifolds 2 Spin systems in the semiclassical limit Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds What is quantization? Classical mechanics Quantum mechanics Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds What is quantization? Classical mechanics Quantum mechanics Symplectic manifold M Hilbert Space H Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds What is quantization? Classical mechanics Quantum mechanics Symplectic manifold M Hilbert Space H Function a ∈ C∞(M, R) Self-Adjoint operator A ∈ L(H) Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds What is quantization? Classical mechanics Quantum mechanics Symplectic manifold M Hilbert Space H Function a ∈ C∞(M, R) Self-Adjoint operator A ∈ L(H) Hamiltonien flow of a Flow of eitA/h Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds What is quantization? Classical mechanics Quantum mechanics Symplectic manifold M Hilbert Space H Function a ∈ C∞(M, R) Self-Adjoint operator A ∈ L(H) Hamiltonien flow of a Flow of eitA/h Poisson Bracket Lie Bracket Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds What is quantization? Classical mechanics Quantum mechanics Symplectic manifold M Hilbert Space H Function a ∈ C∞(M, R) Self-Adjoint operator A ∈ L(H) Hamiltonien flow of a Flow of eitA/h Poisson Bracket Lie Bracket Quantization : for a given classical model, how to construct an associated quantum model ? Semiclassics : the quantum model is h-dependent. What can be said in the h → 0 limit ? Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Bargmann spaces Original idea: express Quantum Mechanics directly in phase space. The standard L2(Rn) is replaced with the Bargmann space, with parameter N > 0: BN = f ∈ L2(Cn), eN 2 |·|2 f is holomorphic on Cn . [1] Bargmann, V. Comm. Pure Appl. Math. 14, no. 3 (1961): 187–214. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Bargmann spaces Original idea: express Quantum Mechanics directly in phase space. The standard L2(Rn) is replaced with the Bargmann space, with parameter N > 0: BN = f ∈ L2(Cn), eN 2 |·|2 f is holomorphic on Cn . This is a closed subspace of L2(Cn), with reproducing kernel ΠN(x, y) = N π n exp − N 2 |x − y|2 + iN (x · y) . [1] Bargmann, V. Comm. Pure Appl. Math. 14, no. 3 (1961): 187–214. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Szegő kernel Hilbert basis indexed by Nn. eν = N|ν| ν1!ν2! . . . νn! zνe−N|z|2 2 . From there one recovers ΠN with ΠN(x, y) = ν∈Nn eν(x)eν(y). The Szegő kernel decays exponentially fast away from the diagonal. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Toeplitz quantization Let f ∈ C∞(Cn, C) bounded. The Toeplitz operator associated with f is the bounded operator TN(f) : BN(Cn) → BN(Cn) u → fu . Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Toeplitz quantization Let f ∈ C∞(Cn, C) bounded. The Toeplitz operator associated with f is the bounded operator TN(f) : BN(Cn) → BN(Cn) u → ΠN(fu). Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Toeplitz quantization Let f ∈ C∞(Cn, C) bounded. The Toeplitz operator associated with f is the bounded operator TN(f) : BN(Cn) → BN(Cn) u → ΠN(fu). If f has polynomial growth then TN(f) is an unbounded operator with dense domain. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Toeplitz quantization Let f ∈ C∞(Cn, C) bounded. The Toeplitz operator associated with f is the bounded operator TN(f) : BN(Cn) → BN(Cn) u → ΠN(fu). If f has polynomial growth then TN(f) is an unbounded operator with dense domain. If f is real-valued then TN(f) is ess. self-adjoint. If moreover f 0 then TN(f) 0. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Examples If f is holomorphic then TN(f) = f. If f is anti-holomorphic then TN(f) = f(N−1∂ + z 2 ). Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Examples If f is holomorphic then TN(f) = f. If f is anti-holomorphic then TN(f) = f(N−1∂ + z 2 ). The Toepliz quantization is anti-Wick: if f is anti-holomorphic and h is holomorphic then TN(fgh) = TN(f)TN(g)TN(h). Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Examples If f is holomorphic then TN(f) = f. If f is anti-holomorphic then TN(f) = f(N−1∂ + z 2 ). The Toepliz quantization is anti-Wick: if f is anti-holomorphic and h is holomorphic then TN(fgh) = TN(f)TN(g)TN(h). On C, TN(|z|2) = (N−1∂ + z 2 )z, with spectrum N−1 N and eigenvectors the (ek)k∈N . Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Composition of Toeplitz operators In general, composition yields a formal series: TN(f)TN(g) = TN fg + N−1C1(f, g) + N−2C2(f, g) + · · · ). Here Cj is bilinear and bidifferential of degree 2j. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Composition of Toeplitz operators In general, composition yields a formal series: TN(f)TN(g) = TN fg + N−1C1(f, g) + N−2C2(f, g) + · · · ). Here Cj is bilinear and bidifferential of degree 2j. In particular, [TN(f), TN(g)] = iN−1TN({f, g} + N−1C2(f, g) + · · · ). Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Hardy spaces and Szegő kernel Geometrical setting: compact Kähler manifold M. Symplectic form Complex structure [2] Woodhouse, N. Geometric Quantization. Oxford University Press, 1997. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Hardy spaces and Szegő kernel Geometrical setting: compact Kähler manifold M. Symplectic form Complex structure Compatibility condition Complex line bundle L → M with curvature −iω. [2] Woodhouse, N. Geometric Quantization. Oxford University Press, 1997. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Hardy spaces and Szegő kernel Geometrical setting: compact Kähler manifold M. Symplectic form Complex structure Compatibility condition Complex line bundle L → M with curvature −iω. Hardy space HN(M, L) of holomorphic sections of L⊗N. [2] Woodhouse, N. Geometric Quantization. Oxford University Press, 1997. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Hardy spaces and Szegő kernel Geometrical setting: compact Kähler manifold M. Symplectic form Complex structure Compatibility condition Complex line bundle L → M with curvature −iω. Hardy space HN(M, L) of holomorphic sections of L⊗N. Szegő projector SN : L2(M, L⊗N) → HN(M, L). [2] Woodhouse, N. Geometric Quantization. Oxford University Press, 1997. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Hardy spaces and Szegő kernel Geometrical setting: compact Kähler manifold M. Symplectic form Complex structure Compatibility condition Complex line bundle L → M with curvature −iω. Hardy space HN(M, L) of holomorphic sections of L⊗N. Szegő projector SN : L2(M, L⊗N) → HN(M, L). The spaces HN(M, L) are finite-dimensional in that case. The line bundles L⊗N correspond to the weights e−N 2 |·|2 in the flat case. [2] Woodhouse, N. Geometric Quantization. Oxford University Press, 1997. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Algebra of Toeplitz operators The Szegő kernel SN has a full expansion. [3] Boutet de Monvel, L, Sjöstrand, J. Journées EDP 34–35 (1975): 123–64. [4] Charles, L. Comm. Math. Phys. 239, no. 1–2 (2003): 1–28. [5] Berman, R., Berndtsson, B., Sjöstrand, J., Arkiv För Matematik 46, no. 2 (2008). [6] Kordyukov, Y. ArXiv Preprint ArXiv:1703.04107, 2017. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Algebra of Toeplitz operators The Szegő kernel SN has a full expansion. The dominant term is always ΠN. [3] Boutet de Monvel, L, Sjöstrand, J. Journées EDP 34–35 (1975): 123–64. [4] Charles, L. Comm. Math. Phys. 239, no. 1–2 (2003): 1–28. [5] Berman, R., Berndtsson, B., Sjöstrand, J., Arkiv För Matematik 46, no. 2 (2008). [6] Kordyukov, Y. ArXiv Preprint ArXiv:1703.04107, 2017. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds Algebra of Toeplitz operators The Szegő kernel SN has a full expansion. The dominant term is always ΠN. Toeplitz operators form a C∗-algebra as previously. [3] Boutet de Monvel, L, Sjöstrand, J. Journées EDP 34–35 (1975): 123–64. [4] Charles, L. Comm. Math. Phys. 239, no. 1–2 (2003): 1–28. [5] Berman, R., Berndtsson, B., Sjöstrand, J., Arkiv För Matematik 46, no. 2 (2008). [6] Kordyukov, Y. ArXiv Preprint ArXiv:1703.04107, 2017. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds An example: the 2D sphere Here M = S2. In the stereographic projection, L corresponds to the weight z → 1 1+|z|2 , so that HN(M, L) f holomorphic in C, C |f|2 (1 + |z|2)N+2 < ∞ = CN[X]. Alix Deleporte Low-energy spectrum of Toeplitz Operators
operators on Cn Toeplitz operators on compact manifolds An example: the 2D sphere Here M = S2. In the stereographic projection, L corresponds to the weight z → 1 1+|z|2 , so that HN(M, L) f holomorphic in C, C |f|2 (1 + |z|2)N+2 < ∞ = CN[X]. In the canonical basis N k −1 2 Xk, the Toeplitz quantization of the three base coordinates on S2 are the Spin matrices with spin S = N−1 2 . Alix Deleporte Low-energy spectrum of Toeplitz Operators
Toeplitz operators Quantization Toeplitz operators on Cn Toeplitz operators on compact manifolds 2 Spin systems in the semiclassical limit Alix Deleporte Low-energy spectrum of Toeplitz Operators
systems Systems with n spins correspond to the Kähler manifold (S2)n. We are interested in antiferromagnetic systems. Let G = (V, E) a finite graph, the antiferromagnetic symbol on (S2)|V| is set to hAF = (i,j)∈E xixj + yiyj + zizj. Alix Deleporte Low-energy spectrum of Toeplitz Operators
systems Systems with n spins correspond to the Kähler manifold (S2)n. We are interested in antiferromagnetic systems. Let G = (V, E) a finite graph, the antiferromagnetic symbol on (S2)|V| is set to hAF = (i,j)∈E xixj + yiyj + zizj. If the graph is bipartite, then the minimum is reached when two neighbours always have opposite values. Alix Deleporte Low-energy spectrum of Toeplitz Operators
In frustrated systems, the previous solution is not possible. [7] Douçot, B., and Simon, P., J. Physics A: Math and General 31, no. 28 (1998) Alix Deleporte Low-energy spectrum of Toeplitz Operators
In frustrated systems, the previous solution is not possible. If the graph is “made with triangles”, then on each triangle the sum of the vectors should be zero. [7] Douçot, B., and Simon, P., J. Physics A: Math and General 31, no. 28 (1998) Alix Deleporte Low-energy spectrum of Toeplitz Operators
In frustrated systems, the previous solution is not possible. If the graph is “made with triangles”, then on each triangle the sum of the vectors should be zero. This yields a degenerate minimal set, which is not a manifold. What behaviour should one expect for the eigenvectors with minimal eigenvalue of TN(hAF)? [7] Douçot, B., and Simon, P., J. Physics A: Math and General 31, no. 28 (1998) Alix Deleporte Low-energy spectrum of Toeplitz Operators
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