A new method for fermionic singular-drift problem in the complex Langevin method Keitaro Nagata a), Jun Nishimura a,b) , Shinji Shimasaki a) a) KEK, b) Sokendai Shimasaki, Nishimura, PRD92,1,011501, (2015); 1504.08359 Nagata, Nishimura, Shimasaki, arXiv:1508.02377 Nagata, Nishimura, Shimasaki, in preparation Workshop Sign2015 Atomki, Debrecen, Hungary, September 29-October 2, 2015 2015/07/29 1 Purpose • CLM + gauge cooling opened a way to study – heavy dense QCD [Seiler, Sexty, Stamatescu(‘12)] – full QCD at high temperature [Sexty (’14), Sexty et al. (‘15)] • Can we study QCD at low temperature with light quarks in CLM ? – Dirac low-modes cause a problem [Mollgaard, Splittorff(’13)] • We develop new method to overcome the problem caused by the fermion drift term at low temepature with light quarks – generalization of gauge cooling 2015/07/29 2 Outline • Introduction – current status of CLM • Problem at low temperature with light quarks • A new method to overcome the fermionic singular drift problem • Discussion – comment on norm-dependence of Dirac eigenvalues – physical part of eigenvalues and BC-like relation 2015/07/29 3 Introduction : current status of CLM 2015/07/29 4 Stochastic quantization with Langevin eq. • Example : (real) LM for one dimensional integral [Parisi, Wu] η: Gaussian noise • Noise average of an observable • The probability distribution converges to the Boltzmann weight according to the Fokker-Planck equation 2015/07/29 5 Complex Langevin method(CLM) • Stochastic quantization does not rely on the probability interpretation of the weight. • This is in principle available even if action is complex [Parisi(’83), Klauder(‘83)] – originally real variables are extended to complex due to complex drift term: x -> z • The complexification sometimes causes the convergence to wrong limits. 2015/07/29 6 Justification of CLM • CLM is justified if some conditions are satisfied [Aarts, et. al. PRD81, 054508(’10), EPJC71,1756(’11)]. integra6on by parts – holomorphy of observables – fast fall-off of the probability distribution in the imaginary direction – regularity of drift terms [Seiler talk in Sign 2012, Nishimura, Shimasaki, PRD92 (2015) 1, 011501 arXiv:1504.08359 ] 2015/07/29 7 Excursion problem and “gauge cooling” • Large probability in the imaginary direction leads to wrong convergence (excursion problem). • Gauge cooling [Seiler, Sexty, Stamatescu(’12), Sexty (’14)] – complexification of link variable SU(3) -> SL(3,C) – “unitarity norm” : measure of the distance from SU(3) matrices – suppress this norm by using complexified gauge symmetry, i.e. SL(3,C). – heavy dense QCD [Seiler, Sexty, Stamatescu(‘12)]/full QCD at high temperature [Sexty (’14), Sexty et al. (‘15)] 2015/07/29 8 Justification of the gauge cooling • Justification of the gauge cooling [Nagata, Shimasaki, Nishimura, 1508.02377] – gauge cooling with infinitesimal small parameter is equal to adding a new term in the Langevin equation. – the new term does not affect the expectation value, if it is invariant under “gauge” transformation. 2015/07/29 9 Nagata-­‐J.N-­‐Shimasaki: arXiv:1508.02377 [hep-­‐lat] (discre6zed) complex Langevin eq. slide by Nishimura (2015) complex Langevin eq. including “gauge cooling” Choose appropriately as a func6on of the config. before gauge cooling e.g.) In order to solve the problem of insufficient fall off of P(x,y;t) in y-­‐direc6ons, minimize: complex Langevin eq. including “gauge cooling” (cont’d) Nagata-­‐J.N-­‐Shimasaki: arXiv:1508.02377 [hep-­‐lat] slide by Nishimura (2015) the effects of gauge cooling Jus6fica6on of “gauge cooling” Nagata-­‐J.N-­‐Shimasaki: arXiv:1508.02377 [hep-­‐lat] Evolu6on of P(x,y,;t) slide by Nishimura (2015) the effects of gauge cooling the effects of gauge cooling disappears as long as f(z) is O(N,C) invariant !!! Problem at low temperature with light quarks 2015/07/29 13 A difficulty in QCD at low-T and small mass • Nonzero µ violates the anti-hermiticity of the Dirac operator – the Dirac eigenvalues acquire the real part – at µ=mπ/2, the eigenvalue distribution starts to cover the origin (early onset problem) – the singularity violates the integration by parts used in the justification of CLM [Seiler(‘12), Nishimura, Shimasaki(‘15)] det (D+m) ~ 0 Schematic figure of eigenvalues of D+m 2015/07/29 14 Failure of CLM in Random matrix theory exact phase quench Chiral condensate for Nf=2, N = 30, µ=2/Sqrt(N), m~=m N [Mollgaard, Splittorff, PRD88 (2013), 11,116007] CL simula6on det(D+m) close to zero This problem will happen in QCD at low T and small mass 2015/07/29 15 A new method to overcome the fermionic singular drift problem 2015/07/29 16 Generalization of “gauge cooling” • The gauge cooling for the unitarity norm does not solve the fermionic singular drift problem. – eigenvalues of D(µ)+m can touch the origin even if the link variables are unitary. • Suitable choice of the repsentation (Polar coorinate) solves this problem for the RMT [Mollgaard, Splittorff(2014)] – not available in QCD • We extend the gauge cooling in two respects – not only uniratiry norm, but also other norms – not only gauge invariance, but also other symmetries 2015/07/29 17 Generalization of “gauge cooling” • We introduce new types of norms to overcome problems caused by Dirac low-modes. – sensitive to singular drift problem – (available for QCD) • We test the method in the RMT using SL(N,C) invariance (complexified SU(N)) – although it is not a gauge symmetry, the symmetry allows us to transform all the dynamical variables. 2015/07/29 18 Random matrix theory • Bosonic part • Fermionic part • Symmetry g and h originally belong to SU(N), then extended to SL(N,C) This extended symmetry allows us to perform cooling procedure. 2015/07/29 19 New norms to control logarithmic singularity Find such a norm that increases if there are eigenvalues near the origin ! x x : det (D+m)=0 cooling for N1 x cooling for N2, N3 x 2015/07/29 20 New norms to control logarithmic singularity Find such a norm that increases if there are eigenvalues near the origin ! • Type 1 : anti-hermiticity x x : det (D+m)=0 cooling for N1 x cooling for N2, N3 x 2015/07/29 21 New norms to control logarithmic singularity Find such a norm that increases if there are eigenvalues near the origin ! • Type 1 : anti-hermiticity x • Type 2 : M+M (M=D+m) x : det (D+m)=0 cooling for N1 x cooling for N2, N3 (eigenvalues of 2015/07/29 M+M smaller than 1/ξ are shifted up) x 22 Norms to control logarithmic singularity • New norms (M=D+m) • Hermiticity norm : an analog of the unitarity norm in lattice QCD • Total norm tunable parameter 2015/07/29 23 Setup for Random matrix theory • Physical setup (same as in Mollgaard & Splitorff(2013)) – N = 30, Nf = 2, µ = 2/Sqrt(N) • Setup for Langevin step – dtau = 1.d-4, # of steps = 50 000 – total Langevin time = 5.0 – measurement for each 200 Langevin steps. • Setup for gauge cooling – 10 times cooling after each Langevin step – ξ=300 – r=0 for N1 and r=0.01 for N3 2015/07/29 24 Result - chiral condensate and baryon number density 2.5 3 2 2.5 1.5 Re nB 3.5 Re 2 1.5 exact w/ cooling N1 w/ cooling N2 w/o cooling 1 0.5 1 exact w/ cooling N1 w/ cooling N2 w/o cooling 0.5 0 0 5 10 ~ m 15 0 -0.5 20 0 5 10 ~ m 15 20 Cooling for new types of norms works well even for quite small mass. 2015/07/29 25 Result – eigenvalues of D+m Eigenvalues of Dirac operator w/o cooling Eigenvalues of Dirac operator w/ cooling N1 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -2.5 -0.5 0 0.5 -2.5 -0.5 0 0.5 Eigenvalues of Dirac operator w/ cooling N2 2.5 2 The new method successfully removes problematic eigenvalues in a gauge invariant manner. 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -2.5 2015/07/29 -0.5 0 0.5 26 Discussion 2015/07/29 27 Discussion1 : norm-dependence of Dirac eigenvalues • As we anticipated, the present method helps to overcome the fermionic singular drift problem, where the distribution of the eigenvalues depends on the norms used. • Question – if such an unphysical dependence is allowed ? – how we extract physical part of the eigenvalue distribution in the CLM ? 2015/07/29 28 Discussion1 : norm-dependence of Dirac eigenvaludes • The eigenvalue density in the CLM is defined by - this is not holomorphic • Justification of CLM relies on the holomorphy of observables The eigenvalue density in the CLM has nothing to do with the physical one in the original theory. depend on norms 2015/07/29 do not depend on norms 29 Discussion2 : physical part of Dirac eigenvalues • Although the eigenvalue density in CLM is not physical ... – a part of the eigenvalue distribution is connected to chiral condensate (similar to Banks-Casher relation) – and therefore, they are physical • We derive a BC like relation to define the physical part of the Dirac eigenvalues accessible in CLM. – see also – BC-like equation at non zero µ for original (non-complexified) theory [Osborn, Splittorff, Verbaarschot, Phys. Rev. Lett. 94 (2005) 202001] – Conditions that Dirac eigenvalues should satisfy to generate chiral condensate [Splittorff, Phys. Rev. D91 (2015), no. 3 034507] – We identify the physical part of the Dirac eigenvalues using the BC like equation for CLM. 2015/07/29 30 BC like equation • Chiral condensate - this is dominated by eigenvalues smaller than mass because eigenvalues form a pair (γ5-hermiticity) - the eigenvalues near the origin dominates Σ in the chiral limit 2015/07/29 31 BC like equation • Chiral condensate • Physical part of the Dirac eigenvalue density 2015/07/29 32 Test of the BC-like relation ^ 4 r (r) • BC-like relation is obtained in thermodynamic and chiral limit 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 N=10 N=20 N=30 0 2015/07/29 our result Σ=3.75 for r-­‐>0, N-­‐>infinity (m=0.1) exact result = 4 consistent with discrepancy of O(m) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 r 33 Limitation of the method • It turns out that the new method breaks down if µ becomes larger, as long as we use the same parameter setup in the cooling procedure. – difficult to suppress the growth of the distribution ~ exp(µ) • The BC-like relation implies the accumulation of the Dirac eigenvalues near the origin is physical and inevitable in the chiral limit. • It is still unclear how large µ we can access in the present method. 2015/07/29 34 Limitation of the method • Increase µ with fixed cooling-parameters ~ =5, N=30, N =2 w/ cooling N m f 1 ~ =5, N=30, N =2 w/ cooling N m f 2 ~ µ=4 ~ µ=3 ~ µ=2 2 2 1 1 0 0 -1 -1 -2 -2 -2 2015/07/29 -1 0 1 ~ µ=4 ~ µ=3 ~ µ=2 2 -2 -1 0 1 2 35 Limitation of the method • Increasing number of cooling improves eigenvalue distribtuion ~ =5, ~ m µ=3, N=30, Nf=2 w/ cooling N1 no cooling 10 cooling 30 cooling 2 1 0 -1 -2 -2 2015/07/29 -1 0 1 2 36 Summary • We developed the method to overcome the problem at low temperature with light quarks – the cooling procedure for new types of norms – it is tested for the chiral random matrix theory. • The new method excludes eigenvalues near the origin, and reproduced exacts results even for quite small quark mass. – it extends the applicable range of CLM – application to QCD is straightforward • We discussed the validity of the method and physical meaning – non-holomorphy of the eigenvalue density and BC-like relation • study of QCD is also in progress 2015/07/29 37 Buckup 2015/07/29 38 How cooling affects Dirac eigenvalues – The eigenvalues are invariant under SL(N,C) transformation. – However, it changes the dynamical variables in the random matrices. – Then, the new dynamical variables affect the drift terms in the next Langevin step. – Performing the cooling after every Langevin step is important to suppress the norm appropriately. 2015/07/29 39 QCD at low temperature with light quarks • Physical setup – Nx = Ny = Nz = 2, Nt=20, beta = 5.7 – staggered fermion with Nf = 4 • Setup for Langevin step – dtau = 5.d-5, # of steps = 10 000 – measurement for each 10 Langevin steps. • Setup for gauge cooling – 10 times cooling after each Langevin step – ξ=100 – r=0.99 (r=0.999 for some parameters) for N2 2015/07/29 40 Baryon number density • m = 0.1 1 0.8 preliminary chiral*10(RHMC) chiral*10(CLM) baryon(RHMC) baryon(CLM) phase quench 0.6 0.4 0.2 0 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 µ 1.15 CLM with cooling • Silver Blaze phenomena seems to be observed. • This can be achieved only with the cooling for the unitarity norm. 2015/07/29 41 Snapshot of eigenvalues 5 4 3 2 1 0 -1 -2 -3 -4 -5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 • In the present setup, there is a void near the origin. • The new types of cooling will be needed when eigenvalues are distributed near the origin (work in progress). 2015/07/29 42 Buck up slides 2015/07/29 43 Langevin time history (RMT) • Left : Baryon Number, Right : chiral condensate ~ =, ~ history of baryon number; m µ=2, N=30, Nf=2 ~ =, ~ history of condensate; m µ=2, N=30, Nf=2 10 5.5 baryon condensate 5 5 4.5 4 0 3.5 -5 3 2.5 -10 2 1.5 -15 1 -20 0.5 0 50 100 2015/07/29 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 350 400 450 500 44 History • solid : CLM with cooling • dashed : RHMC for phase quenched model mass=0.4, mu=1.0 mass=0.1, mu=0.9 0.4 baryon(RHMC) baryon(CLM) baryon(RHMC) baryon(CLM) 0.4 0.3 0.2 0.2 0.1 0 0 -0.1 -0.2 -0.2 -0.3 -0.4 -0.4 2000 4000 6000 2015/07/29 8000 10000 12000 14000 16000 18000 20000 22000 24000 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 22000 24000 45