Hyperbolic Hubbard-Stratonovich transformations

Transcription

1 Hyperbolic Hubbard-Stratonovich transformations J. Müller-Hill 1 and M.R. Zirnbauer 1 1 Department of Physics, University of Cologne August 21, / 20

2 Outline Motivation Hubbard-Stratonovich transformation Results Sketch of the proof 2 / 20

3 Motivation Derivation of the supersymmetric non-linear sigma model (nlσm) Main steps: represent product of Green s functions as Gauss integral over supervector field Φ perform disorder average introduce supermatrix Q conjugate to N a=1 Φ a Φ a s perform integration over Φ saddle-point approximation 3 / 20

4 Hubbard-Stratonovich transformation (HSt) introduce composite (super)matrix Q conjugate to A = P N a=1 Φa Φ a s Restrict to Boson-Boson block Restrict to zero-dimensional case Starting point: e Tr AB e Tr A2 dφ, with A ij = Aim: Perform integral over Φ N Φ a,i Φa,j s jj a=1 Use Hubbard-Stratonovich transformation: e Tr A2 = c e Tr Q2 2i Tr AQ dq D 4 / 20

5 Hubbard-Stratonovich transformation (HSt) Why not simple Gauss integrals? HSt: e Tr A2 = c D e Tr Q2 e 2i Tr AQ dq Trivial generalization of e b = 1 π e x2 2ibx dx? Φ and Q integrations might not interchange! Topic for the rest of the talk: Discuss different choices of D for which e 2i Tr AQ is bounded. 5 / 20

6 Hubbard-Stratonovich transformation (HSt) Why not simple Gauss integrals? HSt: e Tr AB e Tr A2 dφ = e Tr AB c D e Tr Q2 e 2i Tr AQ dq dφ Trivial generalization of e b = 1 π e x2 2ibx dx? Φ and Q integrations might not interchange! Topic for the rest of the talk: Discuss different choices of D for which e 2i Tr AQ is bounded. 5 / 20

7 Hubbard-Stratonovich transformation (HSt) Example with pseudounitary symmetry Example: A ij = N a=1 Φ a,i Φ a,j s jj with s = A = sa s and thus A iu(p, q) ( ) k p g = u(p, q) = k p = p k ( 1p q ) and As > 0 with k := {X g X = X } and p := {X g X = X } 6 / 20

8 Hubbard-Stratonovich transformation (HSt) different domains of integration Standard Euclidean domain of integration Euclid : k p g C (X, Y ) ix + Y ik p with g = u(p, q), g C = g ig does not work: take A ip (off-diagonal and anti-hermitean) take Q p (off-diagonal and hermitean) e 2i Tr AQ is not bounded 7 / 20

9 Hubbard-Stratonovich transformation (HSt) different domains of integration Schäfer & Wegner domain (1980) SW : k p g C (X, Y ) ix + ibe Y se Y k ik with b > 0 p Pro: rigorous proof Con: k is not stable under conjugation by T U(p, q) has not been used much 8 / 20

10 Hubbard-Stratonovich transformation (HSt) different domains of integration Pruisken & Schäfer domain (1982) PS : k p g C (X, Y ) Q = ie Y Xe Y ik ip Pro: full invariance property Con: some results (using PS-domain in the original formulation) are only correct in the large N-limit domain has a boundary 9 / 20

11 Hubbard-Stratonovich transformation (HSt) different domains of integration Different parametrization of Pruisken & Schäfer domain: PS : U(p, q) R p+q g C (T, P) Q = TPT 1 which can be seen by: Q = ie Y Xe Y = e Y kpk 1 e Y = TPT 1 ix = kpk 1, k U(p) U(q), P real diagonal T := e Y k U(p, q) 10 / 20

12 Hubbard-Stratonovich transformation (HSt) recent developments Rigorous HSt using Pruisken & Schäfer domain U(p, q) symmetry, Fyodorov (2005) O(1, 1) and O(2, 1) symmetry, Wei and Fyodorov (2007) O(p, q) symmetry, Fyodorov, Wei, and Zirnbauer (2008) 11 / 20

13 Hubbard-Stratonovich transformation (HSt) recent developments Rigorous HSt using Pruisken & Schäfer domain U(p, q) symmetry, Fyodorov (2005) O(1, 1) and O(2, 1) symmetry, Wei and Fyodorov (2007) O(p, q) symmetry, Fyodorov, Wei, and Zirnbauer (2008) Essentially: e Tr A2 = c PS e Tr Q2 2i Tr AQ dq, Volume form dq = i e i, where {e i} basis of g dq is seen as dim g form on g C 11 / 20

14 Hubbard-Stratonovich transformation (HSt) recent developments Rigorous HSt using Pruisken & Schäfer domain U(p, q) symmetry, Fyodorov (2005) O(1, 1) and O(2, 1) symmetry, Wei and Fyodorov (2007) O(p, q) symmetry, Fyodorov, Wei, and Zirnbauer (2008) Essentially: e Tr A2 = c PS e Tr Q2 2i Tr AQ dq, Volume form dq = i e i, where {e i} basis of g dq is seen as dim g form on g C 11 / 20

15 Results Exact statement: Setting I e Tr A2 = lim ɛ 0 c PS e Tr Q2 2i Tr AQ χ ɛ (Q)dQ, PS : k p g C, (X, Y ) Q = ie Y Xe Y g = ( ) k p p k g a real semisimple matrix Lie algebra, closed under θ(a) = sas 1 Cartan involution on g g = k p, k = {X g θ(x ) = X } and p = {Y g θ(y ) = Y } Y = Y for all Y p, A ig and is k 12 / 20

16 Results Setting II k + k p + p PS : p p + g C = k C p C, g = k k + p p + (X, Y ) Q = e Y Xe Y p + p k + k p p + k k + additional involution τ on g with τθ = θτ τ θ Cartan involution g = k p = k + k p+ p. s p and A [p +, s] p with As > 0. Contains O(p, q) case. More examples in review: M. Zirnbauer (1996), Riemannian symmetric superspaces and their origin in random-matrix theory. 13 / 20

17 Cartoon of the proof attach 14 / 20

18 Cartoon of the proof attach 14 / 20

19 Main ideas of the proof 1) Root decomposition Use root decomposition to apply intuition from 2d cartoons: choose maximal abelian subalgebra a of p diagonalize [a, ] action on g = g 0 α g α g ±α = span(x α ± Y α ), where X α k and Y α p [H, X α ] = α(h)y α [H, Y α ] = α(h)x α Essentially: Q = ie Y Xe Y = i α k [ cosh α(h)x α + sinh α(h)y α ] k 1 where Y = khk 1 and H a 15 / 20

20 Main ideas of the proof 1) Root decomposition Use root decomposition to apply intuition from 2d cartoons: choose maximal abelian subalgebra a of p diagonalize [a, ] action on g = g 0 α g α g ±α = span(x α ± Y α ), where X α k and Y α p [H, X α ] = α(h)y α ig α ix α iy α ig α [H, Y α ] = α(h)x α Essentially: Q = ie Y Xe Y = i α k [ cosh α(h)x α + sinh α(h)y α ] k 1 where Y = khk 1 and H a 15 / 20

21 Main ideas of the proof 2) Extending PS domain Nullsurfaces: Example: (t, s) te 1 + ise 1 is a 2d Nullsurface: dq(e 1, ie 1 ) = e 1 e 2 (e 1, ie 1 ) = 0 16 / 20

22 Main ideas of the proof 2) Extending PS domain Nullsurfaces: Example: (t, s) te 1 + ise 1 is a 2d Nullsurface: dq(e 1, ie 1 ) = e 1 e 2 (e 1, ie 1 ) = 0 attach ig α ig α? 16 / 20

23 Main ideas of the proof 2) Extending PS domain Nullsurfaces: Example: (t, s) te 1 + ise 1 is a 2d Nullsurface: dq(e 1, ie 1 ) = e 1 e 2 (e 1, ie 1 ) = 0 attach ig α ig α g α Extension essentially choose g α = X α + Y α Remark: halfplanes can be chosen such that e 2i Tr AQ guarantees convergence, As > 0 is needed 16 / 20

24 Main ideas of the proof 3) Deformation Deformation Φ t of the eps domain Φ 0 (eps) = eps and Φ 1 (eps) = Euclid Use Cauchy/Stokes: γ 1 f (z)dz = γ 2 f (z)dz γ 1 γ 2 f (z)dz = Φ(γ 1 ) d(f (z)dz) = 0 γ 1 γ 2 g(q)dq = Φ t1 (eps) Φ t2 (eps) Φ(ePS) d(g(q)dq) = 0 17 / 20

25 Main ideas of the proof 3) Cartoon of the deformation attach ig α ig α g α ix α ik Y α p It can be checked that: boundary terms at infinity vanish deformation does not degenerate 18 / 20

26 Conclusions Pruisken & Schäfer HSt is obtained through a deformation of standard Euclidean HSt using Stokes. Pruisken & Schäfer HSt can be used rigorously if volume form dq is used. Schäfer & Wegner HSt is also a deformation of standard Euclidean HSt using Stokes. 19 / 20

27 F. Wegner, The mobility edge problem: continuous symmetry and a conjecture, Z. Phys B 36, 207 (1979). L. Schäfer and F. Wegner, Disordered systems with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes, Z. Phys. B 38, 113 (1980). A.M.M. Pruisken and L. Schäfer, The Anderson model for electron localization nonlinear sigma model, asymptotoic gauge invariance, Nucl. Phys. B 200, 20 (1982). K.B. Efetov, Supersymmetry and the theory of disordered metals, Andv. Phys. 32, 53 (1983). Y.V. Fyodorov, On Hubbard-Stratonovich transformations over hyperbolic domains, J. Phys.: Condensed Matter 17, S1915 (2005). Y.V. Fyodorov, Y. Wei and M.R. Zirnbauer, Hyperbolic Hubbard-Stratonovich Transformation Made Rigorous, J. Math. Phys 9, (2008). 20 / 20