One-thimble regularisation of lattice field theories: is it only a dream?
OOne-thimble regularisation of lattice field theories:is it only a dream?
Francesco Di Renzo ∗ , Simran Singh and Kevin Zambello Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma and INFN,Gruppo Collegato di Parma, I-43124 Parma, ItalyE-mail: [email protected] , [email protected] , [email protected] Lefschetz thimbles regularisation of (lattice) field theories was put forward as a possible solutionto the sign problem. Despite elegant and conceptually simple, it has many subtleties, a major oneboiling down to a plain question: how many thimbles should we take into account? In the originalformulation, a single thimble dominance hypothesis was put forward: in the thermodynamic limit,universality arguments could support a scenario in which the dominant thimble (associated to theglobal minimum of the action) captures the physical content of the field theory. We know by nowmany counterexamples and we have been pursuing multi-thimble simulations ourselves. Still, asingle thimble regularisation would be the real breakthrough. We report on ongoing work aimingat a single thimble formulation of lattice field theories, in particular putting forward the proposalof performing Taylor expansions on the dominant thimble. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ a r X i v : . [ h e p - l a t ] F e b ne-thimble regularisation of lattice field theories Francesco Di Renzo
1. Thimbles and single thimble dominance hypothesis in a nutshell
The sign problem is a major obstacle to lattice simulations of theories we would be interestedin, among which QCD at finite baryon density. The problem is quite easily described : we want tocompute < O > = Z (cid:90) dx e − S ( x ) O ( x ) with S ( x ) = S R ( x ) + iS I ( x ) but (with a complex action in place) e − S can not be regarded as a decent (positive) probability mea-sure and Monte Carlo simulations are not viable. Thimble regularisation [1, 2] is easily describedas well • One complexifies the degrees of freedom, i.e. x → z = x + iy and S ( x ) → S ( z ) . • One then looks for critical points p σ where ∂ z S = • The thimble J σ attached to each critical point is the union of all the Steepest Ascent paths(SA) which are the solutions of ddt z i = ∂ ¯ S ∂ ¯ z i stemming from the critical point (initial condition). • Due to the olomorphic nature of S , the original integral is convergent on the thimble, with S I staying constant (so that the sign problem is killed).Thimbles are manifolds of the same (real) dimension of the original manifold the theory was for-mulated on, but they are embedded in a manifold of twice that dimension. The integration measureon the thimble encodes the orientation of the latter with respect to the embedding manifold and thissadly reintroduces a residual sign problem due the so-called residual phase. We have no simplerecipe to compute the integration measure other than the following one, which is easy to under-stand, but heavy to implement • At the critical point one has to solve the Takagi problem for the Hessian of the action H ( S , p σ ) v ( i ) = λ i ¯ v ( i ) . • The Takagi values λ i fixe the rate at which the real part of the action increases along the SApaths. • The Takagi vectors v ( i ) provide a basis for the tangent space at the critical point. • The tangent space at each point on the thimble can be reconstructed by parallel transportingthe Takagi vectors along the SA paths.All in all, Lefschetz/Picard theory states that a thimble decomposition for the original integral holds < O > = ∑ σ n σ e − iS I ( p σ ) (cid:82) J σ dz e − S R O e i ω ∑ σ n σ e − iS I ( p σ ) (cid:82) J σ dz e − S R e i ω (1.1)In (1.1) the harmless constant phase factors due to S I are factored in front of the integrals, while theresidual sign problem is due to the residual phases e i ω . Both the numerator and the denominator In the following we adopt a light notation in which a field theory looks like an ordinary integral. ne-thimble regularisation of lattice field theories Francesco Di Renzo ( i.e. the partition function) receive contributions in principle by all the critical points, even if the intersection numbers n σ can be zero for possibly many critical points. Actually n σ = instable thimble does not intersect the original integration manifold .Collecting contributions from multiple thimbles can indeed make thimble regularisation a hardproblem. Not only one has to deal with the computation of many contributions; the combinationof many terms in the numerator (and denominator) of (1.1) can actually result in a renewed signproblem, given the multiple phase factors e − iS I ( p σ ) . In the original proposal a single thimble domi-nance hypothesis was put forward. There could be situations in which the dominant thimble alone( i.e. the one associated to the absolute minimum of the real action) could encode the result one isinterested in. A first argument is a very simple one: from semiclassical arguments, the contributionfrom the global minimum of the action is more and more enhanced in the thermodynamic limit.Moreover, universality arguments can be taken into account: the dominant thimble regularisationdefines a local QFT with exactly the same symmetries, the same number of degrees of freedom(belonging to the same representations of the symmetry groups) and the same local interactions asthe original theory. Moreover, the perturbative expansion is the same as in the original formulation.While these arguments are not enough to draw a definite conclusion, it was reassuring that in thefirst application of thimble regularisation (the relativistic Bose gas) the approximation proved towork very well [3].However, it did not take that long for counterexamples to show up. In the case of the Thirring modelit was shown that the dominant thimble could not capture the (complete) correct result [4, 5]. Thiswas one of the motivations for exploring alternative formulations inspired by thimbles. The idea ofcomplexifing the degrees of freedom (in general, of deforming the original domain of integration)is in fact a very general one. Alternatives to thimbles appeared, e.g. the holomorfic flow [5] orvarious approached to the query for sign-optimised manifods, possibly enforced by deep-learningtechniques [6, 7, 8] Defining < O > σ = (cid:82) J σ dz e − S R O (cid:82) J σ dz e − S R = (cid:82) J σ dz e − S R OZ σ (1.2)we can rewrite (1.1) as < O > = ∑ σ n σ e − iS I ( p σ ) Z σ < O e i ω > σ ∑ σ n σ e − iS I ( p σ ) Z σ < e i ω > σ . (1.3)(1.2) has the obvious interpretation of an expectation value ( VEV ) on a single thimble, with themeasure given by the real part of the action. (1.3) can in turns be intepreted as a weighted sum ofVEV contributions , the weights being given by coefficients involving the Z σ . Stated in this way, thetask of performing multiple thimbles simulations can be reduced to (a) performing single thimblesimulations; (b) computing the relative weights in (1.3). As one could expect, it turns out that (b) isa harder task than (a) . Nevertheless there were cases in which we could perform multiple thimblessimulations, according to two different strategies. The instable thimble is the union of the Steepest Descent (SD) paths stemming from a critical point. ne-thimble regularisation of lattice field theories Francesco Di Renzo • There are cases in which it turns out that only a limited number of thimbles contributes,maybe also in presence of symmetries ensuring that a few contributions are equal. Thisturned out to be the case for QCD in 0 + < O > = < O e i ω > σ + α < O e i ω > σ < e i ω > σ + α < e i ω > σ . (1.4)Here the idea is to rewrite (1.3) putting all our ignorance into a single parameter and givingup hope of a first principles derivation of relative weights. One should instead fix the valueof α assuming one known measurement as a normalization point and then predicting thevalue of other observables. Something similar can be put at work in the playground that firstrevealed the failure of single thimble simulations, i.e. the Thirring model [10]. • Relative weights can in turns be computed in a semiclassical approximation (according towhat is also referred to as the gaussian approximation ). A possible strategy is to start withthis semiclassical computation and then compute corrections to it. It turned out that thisworks pretty well in the context of a minimal version of the so-called Heavy Dense approxi-mation for QCD [11].All in all, we were able to show that some steps can be taken in the direction of multiple thimblessimulations. The path to success is nevertheless a difficult one and in the end the real breakthroughwould be going back to the idea of (some form of) one thimble simulation.
2. Thimble decomposition and Stokes phenomena
In order to gain some understanding on the thimble decomposition, one should consider thesituations in which it fails. This is when a Stokes phenomenon occurs. A Stokes phenomenon takesplace when two different critical points are connected by a SA/SD path. This simply means thatthe stable thimble of one critical point sits on top of the unstable thimble of another one. Underthis conditions there is no thimble decomposition. It turns out that changes in the thimble decom-position can be traced back to the occurence of Stokes phenomena: a very effective description ofall this can be found in [12] (just in the case of the Thirring model). A semplified, intuitive pictureof the relationship between thimble decomposition and Stokes phenomena can be given as in thefollowing. • A thimble decomposition is in place when the union of a given number of thimbles is essen-tially a deformation of the original integration contour, just like in applications of Cauchytheorem (this was just the spirit of [5]). • As they are solutions of the same differential equation subject to different initial conditions,different thimbles can not cross each other. This in turns means that they act as barriers toeach other in the thimble decomposition: when the union of a given number of thimbles is a correct deformation of the original integration contour, other thimbles are simply kept out.3 ne-thimble regularisation of lattice field theories
Francesco Di Renzo • Moving around in the parameter space of a given theory, thimbles do move around in themanifold embedding the original one, but they do it smoothly, i.e. they are always subject tothe constraint of not crossing each other. Thus the thimbles that contribute to the decompo-sition of the original domain of integration keep on keeping the others out. • There is only one way thimbles can cross each other and this is just when two thimbles sit ontop of each other (two different critical points are connected by a SA/SD path and the stablethimble of one sits on top of the unstable thimble of the other). When this occurs we are inpresence of a Stokes phenomenon and thimble decomposition fails. • After a Stokes phenomenon has occured, the relative arrangement of thimbles can changeand a different thimble decomposition is in place.
3. Taylor expansions on Lefschetz thimbles
A very important point we want to stress is that Stokes phenomena mark discontinuities in thethimble decomposition, i.e. in the coefficients n σ . This does not mean that physical observablesare discontinuous. Continuity of physical observables across discontinuities of the n σ coefficients( i.e. of the thimble decomposition) is indeed a possible handle on fixing the values of the n σ themselves . This is not the end of the story, and the continuity of physical observables acrossStokes phenomena has yet another interesting application.The idea is to Taylor expand an observable around a point µ where one single thimble is enoughto reconstruct the correct result (cid:104) O (cid:105) ( µ ) = (cid:104) O (cid:105) ( µ ) + ∂ (cid:104) O (cid:105) ∂ µ (cid:12)(cid:12)(cid:12)(cid:12) µ ( µ − µ ) + ∂ (cid:104) O (cid:105) ∂ µ (cid:12)(cid:12)(cid:12)(cid:12) µ ( µ − µ ) + . . . (3.1) n fermion number density Figure 1:
The fermion number density for the L = β = m = See the discussion of the simple φ toy model in [13]. ne-thimble regularisation of lattice field theories Francesco Di Renzo
All in all, we compute Taylor expansion coefficients on a single thimble in a region in whichthis is enough and then we reconstruct values of the observable in a region in which multiplethimbles would be needed to reconstruct the correct result.A toy model application is displayed in Figure 1: this is the computation of the fermion numberdensity for the L = β = m = i.e. a few stepstaken into the bad region by computing Taylor coefficients in the good one). A few more detailscan be found in [10].This is not the end of the story. After the conference we were able to show that by bridging differentregions in which one single thimble is enough to evaluate Taylor coefficients, one can in some casescircumvent the necessity of performing multiple thimbles simulations.
4. Conclusions
The ideas that were discussed at the conference were admittedly very prototypal ones and onlya very basic example of performing Taylor expansions on Lefschetz thimbles was provided. Afterthe conference we made some progress, which will be the subject of a paper which will be issuedsoon. All in all, we are confident that the idea of having Taylor expansions bridging regions inwhich single thimble simulations are correct can indeed (in some cases) circumvent the necessityof multiple thimble simulations.
Acknowledgements
This work has received funding from the European Union’s Horizon 2020 research and inno-vation programme under the Marie Skłodowska-Curie grant agreement No. 813942 (EuroPLEx).We also acknowledge support from I.N.F.N. under the research project i.s. QCDLAT . References [1] M. Cristoforetti et al. [AuroraScience Collaboration],
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