Supersymmetric 1+1d boundary field theory

I discuss recent work with Anatoly Konechny proving a gradient formula for the boundary beta function of the general supersymmetric one-dimensional quantum system with boundary that is critical in the bulk but not at the boundary. I concentrate on some unanswered questions about which Aliosha expressed curiousity. PACS numbers: 11.10.Hi, 11.10.Kk, 11.10.Wx, 11.25.Uv, 64.60.fd One-dimensional quantum systems with boundary that are critical in the bulk but not at the boundary are characterized by their boundary coupling constants λ . The renormalization group leaves the bulk properties fixed, while the effective boundary coupling constants flow according to the boundary renormalization group equation −T ∂λ a ∂T = β(λ) (1) (using the temperature T as the scale parameter). Anatoly Konechny and I have proved a couple of general gradient formulae for the boundary beta functions β(λ) [1, 2]. The gradient formula for the general boundary system is ∂s ∂λa = −gab(λ)β(λ), (2) where s is the so-called boundary entropy, and gab(λ) is a certain positive-definite metric on the space of boundary systems, constructed from the two-point correlation functions (response functions) of the operators localized in the boundary. As a corollary, −T ∂s ∂T = β ∂s ∂λa = −βgab(λ)β 0 (3) establishing the second law of boundary thermodynamics—that the boundary entropy decreases with temperature, as it does in an isolated system whose entropy is S = (1 + T ∂/∂T ) ln tr(e−H/T ),H being the Hamiltonian. * In memory of Aliosha Zamolodchikov. 1751-8113/09/304015+05$30.00 © 2009 IOP Publishing Ltd Printed in the UK 1 J. Phys. A: Math. Theor. 42 (2009) 304015 D Friedan The second gradient formula pertains to the general supersymmetric one-dimensional system with boundary, critical in the bulk. The Hamiltonian of a supersymmetric system is of the form H = Q̂2, where Q̂ is the supersymmetry generator. The supersymmetric boundary systems are characterized by the boundary coupling constants that preserve supersymmetry. We continue to write these as λ , in the context of supersymmetric systems. Changing scale preserves the supersymmetry, so the boundary beta functions are supersymmetric. The second gradient formula is ∂ ln z ∂λa = −gS ab(λ)βb(λ), (4) where z is the so-called boundary partition function, and g ab(λ) is a certain positive-definite metric on the space of supersymmetric boundary systems (not the same as the restriction of the first metric gab(λ) from the space of all boundary systems). As a corollary, −T ∂ ln z ∂T = β ∂ ln z ∂λa = −βgab(λ)β 0, (5) so the boundary energy T 2∂ ln z/∂T of the supersymmetric system is non-negative, as it is in an isolated supersymmetric system (whose thermodynamic energy is given by T 2∂ lnZ/ ln T = T 2∂ ln tr(e−Q̂/T )/∂T = 〈Q̂2〉 0). These gradient formulae should give some general control over the boundary rg flow, since they provide functions—s and ln z—that decrease under the flow, and whose critical manifolds are the fixed points of the rg flow. The gradient formulae are easily generalized to arbitrary quantum circuits that consist of bulk-critical quantum wires. A boundary is the simplest kind of junction in such a circuit. I have argued that such quantum circuits are ideal physical systems for asymptotically largescale quantum computers [3]. The gradient formulae might be useful for general analysis of the computational power of such circuits. The technical details of the proofs can be found in the papers [1, 2]. Here, I will only bring up some questions about the significance of the assumptions that the proofs rest on. The boundary partition function z was constructed by Affleck and Ludwig [4, 5] by taking the full partition function ZL = tr(e−HL/T ) of a one-dimensional system of length L, in the limit of large L, then dividing by the universal partition function of the bulk system: ZL ∼ e πc/6 zz′, (6) where c is the conformal central charge of the bulk-critical system. The remaining Lindependent factors z and z′ are the boundary partition functions of the two boundaries. They actually did this construction in the special circumstances where the boundary is also scale invariant, in which case the boundary partition function is a number independent of T, which they called g. The construction generalizes directly to boundaries that are not scale invariant. Affleck and Ludwig emphasized that z = g is not the partition function of an isolated quantum system, pointing out that a genuine partition function in the limit T → 0 goes to a positive integer, the ground state degeneracy, while, for critical boundaries, z = g is typically not an integer, and can be less than 1. The boundary entropy s is constructed from the entropy of the one-dimensional system by subtracting the universal bulk entropy per unit length: SL ∼ ( 1 + T ∂ ∂T ) lnZL = L 6 + s + s ′. (7) It is not obvious that this quantity s can be interpreted as the entropy of the boundary as a distinct sub-system. The second law of boundary thermodynamics, which follows from the gradient