Asymptotic FOEL for the Heisenberg model on boxes

Transkript

1 Asymptotic FOEL for the Heisenberg model on boxes Mathematical Many Body Theory, June BCAM, Bilbao, Spain Shannon Starr University of Alabama at Birmingham June 16, Bruno Nachtergaele, U. C. Davis, Wolfgang Spitzer, FernUniversität Hagen. 1 / 28

2 Outline 1) on graph G =, E) 2) Define the ferromagnetic ordering of energy levels condition FOEL) 3) Linear spin wave approximation for {1,..., L} d Z d at energies on the order of 1/L 2 2 / 28

3 Definition History Usual Ordering of Energy Levels 1. Define Heisenberg model Graph: G =, E), <, E { {x, y} : x, y, x y }. Let: = {x 1,..., x N } with = N. Hilbert space: H = C 2 ) N = C 2 C 2. S 1) = [ ] 0 1/2, S 2) = 1/2 0 Operators S a) x k : H H, S a) [ ] 0 i/2, S 3) = i/2 0 x k = 1 C 2 1 C 2 S a) 1 C 2 1 C 2 = 1 k 1) for a {1, 2, 3}, k {1,..., N}. C 2 [ ] 1/ /2 S a) 1 N k C 2, 3 / 28

4 Definition History Usual Ordering of Energy Levels [ ] S x 1), S y 2) = i 2 δ x,y S x 3), and cyclic permutations. Define S a) = x S a) x. For each {x, y} E, define h x,y : H H, h x,y = S x S y = Then the total Hamiltonian is [ This is a model such that H G = S a) {x,y} E h x,y. a=1 S x a) S y a). ], H G = 0 for a = 1, 2, 3. 4 / 28

5 Definition History Usual Ordering of Energy Levels Brief review: history for Heisenberg model Fröhlich, Simon and Spencer Comm. Math. Phys., 1976) classical Heisenberg ferromagnet proved LRO for d 3 and sufficiently small T > 0 invented reflection positivity technique to prove Gaussian domination for 2-point correlation functions. Dyson, Lieb, Simon J. Stat. Phys., 1978) proved the quantum antiferromagnet is reflection positivity, not ferromagnet. See, also, Eugene Speer, Lett. Math. Phys., 1985.) Physicists use the linear spin wave approximation. Mathematicians have worked to justify this: Correggi, Giuliani, Seiringer, Comm. Math. Phys. 2015). 5 / 28

6 Definition History Usual Ordering of Energy Levels Usual ordering of energy levels: Lieb-Mattis theorem Suppose G is bipartite: G =, E) with = A B and { } E {x, y} : x A, y B, and assume A = B, balanced. Then Lieb and Mattis J. Math. Phys., 1962) proved that, for the antiferromagnet H G, the ground state is a spin singlet. More generally, they allow antiferromagnetic couplings between the two parts and ferromagnetic couplings with single parts H J) = J x,y h x,y. {x,y} E J x,y 0 if x, y) A B) B A), J x,y 0 if x, y) A A) B B). 6 / 28

7 Definition History Usual Ordering of Energy Levels For each s {0, 1,..., N/2}, define the total-spin s subspace 3 ) ) 2 H totsp s) = ker ss + 1)1. a=1 S a) Then H G H totsp n) H totsp by SU2) symmetry. Lieb and Mattis also proved ) min spec H G H totsp 0)... min spec H G H totsp 1 ) 2 N). More generally if A and B are not balanced, then they proved the ground state has total spin 1 2 A B and A B )) min spec. H G H totsp 2... min spec H G H totsp N ) 2 ). 7 / 28

8 Spectral gap Digression: CLR theorem The FOEL property Denote E 0 G, s) = min spec H G H totsp ) s). For G =, E), N = the ground state energy is E 0 G, 1 2 N). For open chains = {1,..., L} Z, E = { } {k, k + 1} : 1 k L 1, Koma and Nachtergaele Lett. Math. Phys., 1997) proved that the spectral gap is E 0 [1, L], 1 2 L 1). So E 0 [1, L], 1 2 L ) E 0 [1, L], 1 2 L 1 ) min E 0 [1, L], s). s< 1 2 L 1 8 / 28

9 Spectral gap Digression: CLR theorem The FOEL property Induction argument Consider increasing chain from [1, L] to [1, L + 1]. The definition of each pair interaction 0 h x,y = 1 4 S x S y = 1/2 1/2 1/2 1/2 0 x,y 1 C 2) z, z {x,y} is positive semi-definite and has ground state energy 0. So E 0 [1, L], 1 2L) = 0. Koma and Nachtergaele calculated E 0 [1, L], 1 2 L 1) = 1 cosπ/l) = 2 sin2 π 2L), strictly decreasing. 9 / 28

10 Spectral gap Digression: CLR theorem The FOEL property Suppose ψ H totsp [1,L+1] 1 2 L + 1) 2) is an eigenvector of H [1,L+1]. Then ) ) ψ 1, ψ 2 H totsp 1 [1,L] 2 L 2 + H totsp 1 [1,L] 2 L 1, such that ψ = ψ 1 + ψ 2, where and denote the standard basis of C 2. So, since h L,L+1 psd 0, this means H [1,L] psd H [1,L+1]. ψ, H [1,L+1] ψ ψ, H [1,L] ψ = ψ 1, H [1,L] ψ 1 + ψ 2, H [1,L] ψ 2. But for k {1, 2}, ψ k, H [1,L] ψ k ψ k 2 min {E 0 [1, L], 12 ) L 2, E 0 [1, L], 1 )}) 2 L 1 10 / 28

11 Spectral gap Digression: CLR theorem The FOEL property So this proves E 0 [1, L + 1], 1 2 L + 1) 2 ) min {E 0 [1, L], 12 ) L 2, E 0 [1, L], 1 )}) 2 L 1 If, as an induction hypothesis, we assume E 0 [1, L], 1 ) 2 L 1 E 0 [1, L], 1 ) 2 L 2, then this means E 0 [1, L + 1], 1 2 L + 1) 2 ) E 0 [1, L], 1 2 L 1 ). 11 / 28

12 Spectral gap Digression: CLR theorem The FOEL property But Koma and Nachtergaele showed E 0 [1, L + 1], 1 ) 2 L + 1) 1 E 0 [1, L], 1 ) 2 L 1, π i.e., sin 2 2L+1) ) sin2 π 2L ). So we deduce the induction step E 0 [1, L + 1], 1 ) 2 L + 1) 2 E 0 [1, L + 1], 1 ) 2 L + 1) / 28

13 Spectral gap Digression: CLR theorem The FOEL property Aldous s conjec; Caputo, Liggett, Richthammer s thm Handjani and Jungreis rediscovered this argument J. Theor. Prob., 1996). The Heisenberg ferromagnet is unitarily equivalent to the SEP. Aldous conjectured, for any graph G =, E), with N =, E 0 G, 1 ) 2 N 1 min E 0 G, s). s< 1 2 N 1 Caputo, Liggett and Richthammer proved this J. A. M. S., 2010). 13 / 28

14 Spectral gap Digression: CLR theorem The FOEL property CLR s network reduction Caputo, Liggett and Richthammer consider weighted graphs. Rates c xy coupling constants J xy. Network reduction G =, c) G z = \ {z}, c z) ) x, y \ {z}, c z) xy = c xy + c xz c yz w \{z} c wz Example: x x x 3 x 2 1 x 1 1/2 1 x 3 x 4 14 / 28

15 Spectral gap Digression: CLR theorem The FOEL property The definition of property FOEL-n We say a graph G =, E), with N =, satisfies FOEL-n if E 0 G, 1 ) 2 N n min E 0 G, s). s< 1 2 N n The FOEL-n property is a property that G may or may not satisfy, for each n {0,..., 1 2 N }. By the same reasoning as Koma and Nachtergaele s argument, if we can grow our graph one vertex at a time G 2n, G 2n+1,..., G N = G, and if E 0 G 2n, 0) E 0 G 2n+1, 1 2 ) E 0G, 1 2 N n), then G satisfies FOEL-n. 15 / 28

16 Spectral gap Digression: CLR theorem The FOEL property Example [1, L] á la Koma and Nachtergaele Using this argument, the graphs Koma and Nachtergaele considered, [1, L] Z, satisfy FOEL-n for each n 1 2 L. Note, by the CLR theorem, all graphs satisfy FOEL-1. By Lieb-Mattis all graphs also satisfy FOEL-0.) Koma and Nachtergaele actually considered quantum group SU q 2) = U q sl 2 ) symmetric XXZ model for q 0, ). This representation comes with dual canonical basis, that Temperley and Lieb called Hulthèn bracket basis. In this basis one has tree-like 1?) behavior of E 0 [1, L], 1 2L n) for all n. See Nachtergale, Spitzer, S J. Stat. Phys., 2004). 1 Wielandt minimax for Perron-Frobenius roots 16 / 28

17 Spectral gap Digression: CLR theorem The FOEL property Counterexamples for n 2 Since FOEL-1 as well as FOEL-0) is true for all graphs, is FOEL-2? No, not for the hexagon. Let C{ N be G =, E) for = {1,..., N} and } E = {1, 2}, {2, 3},..., {N 1, N}, {1, N}. Then C 2n violates FOEL-n 1) for n > 2: true for n = 3, numerically verified by Lanczos iteration for n = 4, 5, 6, 7 Tran,Spitzer,S; J. Math. Phys.,2012), and n = 8 Dhar and Shastry, Phys. Rev. Lett., 2000). Bethe s ansatz question Sutherland, Phys. Rev. Lett., 1995). 17 / 28

18 Spectral gap Digression: CLR theorem The FOEL property Asymptotic FOEL-n If you have a sequence of graphs G 2n, G 2n+1,..., and if E 0 G N, 1 ) 2 N n min E 0 G M, 12 ) M n, M {2n,...,N} then G N satisfies FOEL-n. Lemma Suppose there are constants p > 0 and 0 = C 0 < C 1 <... such that the sequence of graphs above satisfy E 0 G N, 1 2 N n ) C n N p, as N, for each n {0, 1,... }. Then, for each n {0, 1,... }, there is N n such that G N satisfies FOEL-n for each N N n. 18 / 28

19 Energies of O1/L 2 ) on boxes B d L) Toth representation Filling in LSW approximation for energies of order 1/L 2 Consider a box B d L) = {1,..., L} d Z d. Then the energies of order 1/L 2 in H totsp B d L), 1 2 Ld n) are asymptotically π 2 2L 2 κ κ n 2) ranging over ordered n-tuples κ 1,..., κ n ) Z d \ {0}) n, modulo the action of S n. In particular, E 0 B d L), 1 ) 2 Ld n nπ2 2L 2. This means E 0 G N, 1 2 N n) C nn p for C n = nπ 2 /2, p = 2/d. 19 / 28

20 Energies of O1/L 2 ) on boxes B d L) Toth representation Filling in Disclaimer If you prove the claim, you do not actually need the inductive argument for FOEL. Correggi, Giuliani and Seiringer Comm. Math. Phys., 2015), alidity of the Spin-Wave Approximation for the Free Energy of the Heisenberg Ferromagnet, proved c > 0 such that for all L and all n E 0 B d L), ) 1 2 Ld n cn L / 28

21 Energies of O1/L 2 ) on boxes B d L) Toth representation Filling in Graph Laplacian of the configuration graph Given a graph G =, E), the graph Laplacian is the operator G : l 2 ) l 2 ) G f x) = E {x, y})[f x) f y)], y so that f, G f = 1 2 {x,y} E f x) f y) 2. One may define a new graph Φ n G) = n, Φ n) E)), where Φ n) E) is the set of all {x 1,..., x n ), y 1,..., y n )}, k {1,..., n}, s.t. {x k, y k } E, and x j = y j for j k. 21 / 28

22 Energies of O1/L 2 ) on boxes B d L) Toth representation Filling in The configuration graph The n particle configuration graph is Θ n) G) = Θ n) ), Θ n) E)), where } Θ n) ) = {x 1,..., x n ) n : j k x j x k, and Θ n) E) is the set of edges {x 1,..., x n ), y 1,..., y n )} Φ n) E), such that x 1,..., x n ), y 1,..., y n ) Θ n) ). 22 / 28

23 Energies of O1/L 2 ) on boxes B d L) Toth representation Filling in Then, for any function f l 2 Θ n) )) which is symmetric π S n, f x π1),..., x πn) ) = f x 1,..., x n ), we can map T n) : l2 symθ n) )) H kers 3) 1 2N + n) by ) T n) f = 1 f x 1,..., x n )Sx n! 1 Sx n x, x x 1,...,x n) Θ n) ) where S ± x = S 1) x ± is 2) x. Then T n) is a unitary transformation, and H G T n) ) f = T n) Θ n) G) f. 23 / 28

24 Energies of O1/L 2 ) on boxes B d L) Toth representation Filling in Another tool from CGS, and filling in If we know the spectrum of G then it is trivial to determine the spectrum of Φ n) G). Given a f l 2 Θ n) )) with low energy we could extend by zero to l 2 Φ n) )). Then we have to worry about 1 Φ n) E) {x, y}) f x) f y) 2. x Θ n) ) y Φ n) )\Θ n) ) For G = B d L), with d 3, Correggi, Giuliani and Seiringer have another powerful result: for any eigenfunction Ψ of H G with eigenvalue λ, ρ Cλ d ρ 1, where ρx, y) = Ψ, S x S + x S y S + y Ψ is the 2-particle density. 24 / 28

25 Energies of O1/L 2 ) on boxes B d L) Toth representation Filling in If d 2, then we can fill in. For any x Φ n) ) \ Θ n) ), just define f x) to be an average of f y) for nearby y Θ n) ). 25 / 28

26 Energies of O1/L 2 ) on boxes B d L) Toth representation Filling in 26 / 28

27 Energies of O1/L 2 ) on boxes B d L) Toth representation Filling in We raise the energy by a constant factor depending on n). But we just use the energy bound with the Chebyshev inequality to restrict the number of eigenmodes we must keep to have a good approximation to f in the eigenvector decomposition relative to Φ n) B d L)). We already have uniform bounds for eigenfunctions of Φ n) B d L)) and their norms restricted to Φ n) B d L)) \ Θ n) B d L)) are small. Then eigenfunctions f of Θ n) B d L)) with energy C/L2 must be close in norm to linear combinations of of eigenfunctions of Φ n) B d L)). So the min-max theorems prove that the restrictions of the spectra to [0, C/L 2 ] are close. 27 / 28

28 Energies of O1/L2 ) on boxes Bd L) Toth representation Filling in Thanks for your attention! 28 / 28