Sharing Experiments and their Provenance David Koop Juliana Freire Large-Scale Visualization and Data Analysis (VIDA) Center Polytechnic Institute of New York University www.vistrails.org NSF Community Codes 2012 Science Today 011100101 111001011 001001101 101010110 111000110 Collect/Generate/Obtain Filter/Analyze/Visualize Publish/Share Data Results Findings www.vistrails.org NSF Community Codes 2012 2 Science Today 011100101 111001011 001001101 101010110 111000110 Collect/Generate/Obtain Filter/Analyze/Visualize Publish/Share Data Results Findings • There’s more... - Revisit or extend the initial result - Share with a colleague who wants to reproduce an experiment - Investigate the effect of new techniques in the same framework - Determine how flawed data or algorithms impacted results www.vistrails.org NSF Community Codes 2012 2 Provenance, Reproducibility, and Sharing • Goals: - Capture necessary provenance - Support reproducibility - Improve sharing and collaboration Visualizations Results Source Code Workflows Libraries 011100101 111001011 001001101 101010110 111000110 Text Data www.vistrails.org NSF Community Codes 2012 3 5 a) honeycomb rung terms as 0.56 0.56 J =sin✓ and J = cos ✓ , 0.48 0.48 r p p ) / J 0.4 0.4 L where ✓ =0corresponds to the unperturbed Hamiltonian. ( ∆ The phase diagrams as a function of ✓ have been mapped out 0.32 0.32 for both the DFib model18 and the DYL model,4 respectively. 0.24 0.24 Directly probing the topological order in the DYL model 0.16 width W = 2 0.16 width W = 3 and its Hermitian counterpart we show the lifting of their re- finite-size gap 0.08 0.08 spective ground-state degeneracies in Figs. 6 and 7 when in- cluding a string tension. We find a striking qualitative dif- 0 0 ference between these two models: For the DYL model the 0 0.1 0.2 0.3 0.4 0.5 inverse system size 1/L lifting of the ground-state degeneracy is exponentially sup- pressed with increasing system size – characteristic of a topo- logical phase. For the Hermitian model, on the other hand, we b) ladder find a splitting of the ground-state degeneracy proportional to 0.32 0.32 JrL. The linear increase with both system size and coupling can be easily understood by the different matrix elements of p 0.24 0.24 the string tension term on a single rung for the two degener- ) / J L ( ate ground-states of the unperturbed model. Plotting the low- ∆ energy spectrum in Fig. 7 clearly shows that the two-fold de- 0.16 0.16 generacy of the unperturbed Hermitian model arises from a (fine-tuned) level crossing. Similar behavior is found in the honeycomb lattice model (not shown). 0.08 0.08 finite-size gap Considering the model in a wider range of couplings, as shown in Fig. 8, further striking differences between the non- 0 0 0 0.05 0.1 0.15 0.2 0.25 Hermitian DYL model and its Hermitian counterpart are re- inverse system size 1/L vealed: The DYL model exhibits two extended topological phases around ✓ =0and ✓ = ⇡/2 (with two and four de- DemoFIG. 4. (color online) Scaling of the finite-size gap ∆(L) (in units generate ground states, respectively), which are separated by of Jp) with linear system size for the Hermitian projector model ✓ = ⇡/4 herm a conformal critical point at precisely c as discussed H on two different lattice geometries: the honeycomb lattice extensively in Refs. 4 and 18. In contrast, the Hermitian model with L W plaquettes (top panel) and 2-leg ladder systems of length Galois Conjugates of Topological Phases Hherm exhibits no topological phase anywhere, and the inter- L (bottom⇥ panel). M. H. Freedman,1 J. Gukelberger,2 M. B. Hastings,1 S. Trebst,1 M. Troyer,2 and Z. Wang1 mediate coupling ✓ = ⇡/4 does not stand out. 1Microsoft Research, Station Q, University of California, Santa Barbara, CA 93106, USA 2Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland (Dated: July 6, 2011) Galois conjugation relates unitary conformal field theories (CFTs) and topological quantum field theories (TQFTs) to their non-unitary counterparts.a Here we investigate↵ Galois conjugatesb of quantum double models, such as the Levin-Wen model. While these Galois conjugated Hamiltonians are typically non-Hermitian, we find that their ground state wave functions still obey a generalized version of the usual code property (local operators non-Hermitian DYL model do not act on the ground state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such non-unitaryδ topological phasesβ can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states 3 3 lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this we rigorously prove that no local change of basis (IV.5) can transform the ground states of the Galois conjugated doubled Fibonacci theory into the groundd states of a topologicalγ model whosec Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasi-local Hamiltonians. A similar statement holds ) x 1000 for many other non-unitary TQFTs. One consequence is that the “Gaffnian” wave function cannot be the ground 0 E state of a gapped fractional quantum Hall state. - 1 PACSFIG. numbers: 5. 05.30.Pr, Edge 73.43.-f labeling for a plaquette of the ladder lattice. E 2 2 I. INTRODUCTION Abelian Levin-Wen model.8 This model, which is also called “DFib”, is a topological quantum field theory (TQFT) whose states are string-nets on a surface labeled by either a triv- Galois conjugation, by definition, replaces a root of a poly- ial or “Fibonacci” anyon. From this starting point, we give nomial by another one with identical algebraic properties. For a rigorous argument that the “Gaffnian” ground state cannot example, i and i are Galois conjugate (consider z2 +1=0) be locally conjugated to the ground state of any topological The1+p−5 quasi-one1 1 p5 dimensional2 geometry allows to numerically as are φ = and = − (consider z z 1=0), 2 − φ 2 − − phase, within a Hermitian model satisfying Lieb-Robinson 3 3 2⇡i/3 3 2⇡i/3 3 9 as well as p2, p2e , and p2e− (consider z 2= (LR) bounds (which includes but is not limited to gapped diagonalize systems up− to linear system size L = 13. The 1 L = 4 1 0). In physics Galois conjugation can be used to convert non- local and quasi-local Hamiltonians).herm L unitaryfinite-size conformal field theories gap (CFTs) of theto unitary Hermitian ones, and Lieb-Robinson model boundsH are a technicalis again tool for local found lattice = 6 vice versa. One famous example is the non-unitary Yang-Lee models. In relativistically invariant field theories, the speed of L = 8 CFT,to which vanish is Galois conjugate in the to the Fibonacci thermodynamic CFT (G2)1, light is a limit, strict upper bound showing to the velocity a of linear propagation. de-In L = 10 the even (or integer-spin) subset of su(2)3. lattice theories, the LR bounds provide a similar upper bound Inpendence statistical mechanics on non-unitary the inverse conformal field system theo- by a velocity size called as the shown LR velocity, in but inFig. contrast 4b). to the rel- To ries have a venerable history.1,2 However, it has remained less ativistic case there can be some exponentially small “leakage” clearfurther if there exist demonstrate physical situations in which the non-unitary fragilityoutside of the these light-cone gapless in the lattice case. ground The Lieb-Robinson states bounds are a way of bounding the leakage outside the light- ground-state degeneracry splitting ( 0 0 models can provide a useful description of the low energy 18 physicsagainst of a quantum local mechanical perturbations system – after all, Galois wecone. add The a LR string velocity is tension set by microscopic details of the -0.1 0 0.1 conjugation typically destroys the Hermitian property of the Hamiltonian, such as the interaction strength and range. Com- -0.05 0.05 Hamiltonian. Some non-Hermitian Hamiltonians, which sur- bining the LR bounds with the spectral gap enables us to prove coupling parameter θ / π prisingly have totally real spectrum, have been found to arise locality of various correlation and response functions. We will in the study of PT-invariant one-particle systemspert3 and in call a Hamiltonian a Lieb-Robinson Hamiltonian if it satisfies some Galois conjugate many-body systemsH4 and might= beJrLR bounds. δl(r),⌧ (13) seen to open the door a crack to the physical use of such We work primarily with a single example, but it should be FIG. 6. (color online) Ground-state degeneracy splitting of the non- models. Another situation, which has recently attracted some clearrungs that ther concept of Galois conjugation can be widely ap- arXiv:1106.3267v3 [cond-mat.str-el] 5 Jul 2011 interest, is the question whether non-unitary models can de- pliedX to TQFTs. The essential idea is to retain the particle Hermitian doubled Yang-Lee model when perturbed by a string ten- scribe 1D edge states of certain 2D bulk states (the edge holo- types and fusion rules of a unitary theory but when one comes graphicfavoring for the bulk). In the particular, trivial there is currently label a discus-l(r)=to writing1 on down each the algebraic rung form of of the theF -matrices ladder. (also sion (✓ =0). sion on whether or not the “Gaffnian” wave function could be called 6j symbols), the entries are now Galois conjugated. A 6 the ground state for a gapped fractional quantum Hall (FQH) slight complication, which is actually an asset, is that writing stateWe albeit with parameterize a non-unitary “Yang-Lee” the CFT couplings describing its an ofF -matrix the requires competing a gauge choice and plaquette the most convenient and edge.5–7 We conclude that this is not possible, further restrict- choice may differ before and after Galois conjugation.