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
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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
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Introduction : current status of CLM
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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
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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.
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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 ]
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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)]
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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.
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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
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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
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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
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A new method to overcome the fermionic
singular drift problem
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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Discussion
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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 ?
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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
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do not depend on norms
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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.
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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
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BC like equation
•  Chiral condensate
•  Physical part of the Dirac eigenvalue density
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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
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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
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-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
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-1
0
1
2
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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
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Buckup
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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.
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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
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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.
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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
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Buck up slides
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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
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150
200
250
300
350
400
450
500
0
50
100
150
200
250
300
350
400
450
500
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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
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