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| -rw-r--r-- | bezout.bib | 53 | ||||
| -rw-r--r-- | bezout.tex | 50 |
2 files changed, 85 insertions, 18 deletions
@@ -283,3 +283,56 @@ } +@article{cristoforetti2012new, + title={New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble}, + author={Cristoforetti, Marco and Di Renzo, Francesco and Scorzato, Luigi and AuroraScience Collaboration and others}, + journal={Physical Review D}, + volume={86}, + number={7}, + pages={074506}, + year={2012}, + publisher={APS} +} +@article{tanizaki2017gradient, + title={Gradient flows without blow-up for Lefschetz thimbles}, + author={Tanizaki, Yuya and Nishimura, Hiromichi and Verbaarschot, Jacobus JM}, + journal={Journal of High Energy Physics}, + volume={2017}, + number={10}, + pages={100}, + year={2017}, + publisher={Springer} +} + + +@article{scorzato2015lefschetz, + title={The Lefschetz thimble and the sign problem}, + author={Scorzato, Luigi}, + journal={arXiv preprint arXiv:1512.08039}, + year={2015} +} + +@article{witten2010new, + title={A new look at the path integral of quantum mechanics}, + author={Witten, Edward}, + journal={arXiv preprint arXiv:1009.6032}, + year={2010} +} + +@article{witten2011analytic, + title={Analytic continuation of Chern-Simons theory}, + author={Witten, Edward}, + journal={AMS/IP Stud. Adv. Math}, + volume={50}, + pages={347}, + year={2011} +} + +@article{behtash2015toward, + title={Toward Picard-Lefschetz theory of path integrals, complex saddles and resurgence}, + author={Behtash, Alireza and Dunne, Gerald V and Sch{\"a}fer, Thomas and Sulejmanpasic, Tin and Unsal, Mithat}, + journal={arXiv preprint arXiv:1510.03435}, + year={2015} +} + + @@ -72,26 +72,40 @@ constraint remains $z^2=N$. The motivations for this paper are of two types. On the practical side, there are indeed situations in which complex variables appear naturally in disordered -problems: such is the case in which they are \emph{phases}, as in random laser +problems: such is the case in which the variables are \emph{phases}, as in random laser problems \cite{Antenucci_2015_Complex}. Quiver Hamiltonians---used to model black hole horizons in the zero-temperature limit---also have a Hamiltonian very close to ours \cite{Anninos_2016_Disordered}. - -There is however a more fundamental reason for this study: we know from +A second reason is that we know from experience that extending a real problem to the complex plane often uncovers -underlying simplicity that is otherwise hidden. Consider, for example, the -procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying -$\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ -to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ is a -polynomial of degree $p$ chosen to have simple, known saddles. Because we are -working in complex variables, and the saddles are simple all the way (we shall -confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, -while with real variables minima of functions appear and disappear, and this -procedure is not possible. The same idea may be implemented by performing -diffusion in the $J$s and following the roots, in complete analogy with Dyson's -stochastic dynamics \cite{Dyson_1962_A}. - -The spherical constraint is enforced using the method of Lagrange multipliers: +underlying simplicity that is otherwise hidden, and thus sheds light on the original real problem +(think, for example, in the radius of convergence of a series). + +Deforming a real integration in $N$ variables to a surface of dimension $N$ in +the $2N$ dimensional complex space has turned out to be necessary for correctly defining and analyzing path integrals with complex action (see \cite{witten2010new,witten2011analytic}), and as a useful palliative for the sign-problem \cite{cristoforetti2012new,tanizaki2017gradient,scorzato2015lefschetz}. +In order to do this correctly, the features of landscape of the action in complex space must be understood. Such landscapes are in general not random: here we propose to follow the strategy of Computer Science of understanding the generic features of random instances, expecting that this sheds light on the practical, nonrandom problems. + +%Consider, for example, the +%procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying +%$\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ +%There is however a more fundamental reason for this study: +%we know from experience that extending a real problem to +%the complex plane often uncovers underlying simplicity that +%is otherwise hidden. Consider, for example, the procedure of +% +%$\lambda H_{00} + (1-\lambda) H_0$ evolving adiabatically from $\lambda=1$ to $\lambda=0$, as +%is familiar from quantum annealing. The $H_{00}$ is a polynomial +%of degree N chosen to have simple, known saddles. Because we +%are +%working in complex variables, and the saddles are simple all the way (we shall +%confirm this), we may follow a single one from $\lambda=0$ to $\lambda=1$, +%while with real variables minima of functions appear and disappear, and this +%procedure is not possible. The same idea may be implemented by performing +%diffusion in the $J$s and following the roots, in complete analogy with Dyson's +%stochastic dynamics \cite{Dyson_1962_A}. + +Returning to our problem, +the spherical constraint is enforced using the method of Lagrange multipliers: introducing $\epsilon\in\mathbb C$, our energy is \begin{equation} \label{eq:constrained.hamiltonian} H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right). @@ -444,9 +458,9 @@ the complex case. The relationship between the threshold, i.e., where the gap appears, and the dynamics of, e.g., a minimization algorithm or physical dynamics, are a problem we hope to address in future work. -This paper provides a first step for the study of a complex landscape with + This paper provides a first step towards the study of a complex landscape with complex variables. The next obvious one is to study the topology of the -critical points and gradient lines of constant phase. We anticipate that the +critical points, their basins of attraction following gradient ascent (the Lefschetz thimbles), and descent (the anti-thimbles) \cite{witten2010new,witten2011analytic,cristoforetti2012new,behtash2015toward,scorzato2015lefschetz}, that act as constant-phase integrating `contours'. Locating and counting the saddles that are joined by gradient lines -- the Stokes points, that play an important role in the theory -- is also well within reach of the present-day spin-glass literature techniques. We anticipate that the threshold level, where the system develops a mid-spectrum gap, will play a crucial role as it does in the real case. |