High density QCD on a Lefschetz thimble?

Luigi Scorzato (ECT*, Trento)in collaboration with

Marco Cristoforetti (ECT*) and Francesco Di Renzo (U.Parma)

Oberwölz, 4 September 2012QCD – History and Prospects

The sign problem in QCD

But, when det[Q] oscillates strongly, reweighting is not effective:

the cost is expected to scales as ~ eV,and even lattices as tiny as V=44 are unfeasible.

∼ 0

0

The fermionic determinant is not real&positive, when the chemical pot. µB≠0.Without a positive measure, Monte Carlo methods are not applicable directly.

hOi = Tr [O e�(H�µBNq)/T ]

Tr [e�(H�µBNq)/T ]=

RdU O[U ] eSG[U ] det(Q[U ;µB ])Nf

RdUeSG[U ] det(Q[U ;µB ])Nf

�O� ∶= ∫ dU O[U] eSG[U] �det(Q)Nf �∫ dUeSG[U] �det(Q)Nf �A formal way out is reweighting:

�O� = �OeiNf arg det(Q)��eiNf arg det(Q)�which works well when the corrections are small

Saddle-point integration

A B

C

R t

I t

γ

i

√x

x

Ai(x) ∶= 1

2⇡ �∞−∞ e

i( t33 +xt)dt

Strongly oscillating, low dimensional integrals are treated very effectively with the

→ 1

2⇡ ��

ei( z33 +xz) dz

Steepest Descentof real part along γ

NOTE γ’(t) is not constant, but changes smoothly

-5 5

-1.0

-0.5

0.5

1.0 <ei( t3

3 +xt)

-5 5

0.1

0.2

0.3

0.4

0.5 <ei( t3

3 +xt)

Constant imaginarypart along γ

Saddle-point integration

• It is a classic and elementary tool that works extremely well for low dimensional oscillating integrals.

• It is usually combined with an asymptotic expansion around the stationary point.

‣ But, that would correspond to some version of Perturbation Theory, which is not what we want.

• However, the idea of finding a path with a stationary phase and where the important contributions are more localized, is independent of the series expansion and is very attractive from the point of view of the sign problem.

‣ A Monte Carlo integral along the curves of steepest descent (SD)?

comments

Higher dimensions

Under suitable conditions on f(x) and g(x), Morse theory (Pham ‘83, Vassiliev ‘02,

Nicolaescu ‘11, Witten ‘10) tells us that the Jσ are smooth manifolds of real dimension

n immersed in �n, and, for each cycle C, where the integral converges,:

The generalization of the paths of SD are called Lefschetz thimbles Jσ,

For each stationary point pσ of the complexified f(z), Jσ is the union of the paths

of SD that fall in pσ at ∞.

Z

Cdx g(x)ef(x) =

X

�

n

�

Z

J�

dz g(z)ef(z)i.e. the thimbles provide a basis of the relevant homology group, with integer coefficients.

C =X

�

n�J� (in the homological sense)

Z

Rn

dx

n

g(x)ef(x)

xpσ

R t

I tJ�

pσ

J�x

The path integral of a QFT?Can we use these ideas to compute the path integral of QFT?

hOi =RCQ

x

d�x

e�S[�]O[�]RCQ

x

d�x

e�S[�]

• On a finite lattice, the integration cycle is well defined (e.g. for QCD: C=SU(3)4V ), and for a generic choice of the parameters, the decomposition described above is actually valid:

• So, in principle, the original integral could be expressed as a sum of integrals over the thimbles Jσ, where the integral is rapidly convergent and

the phase of the integrand is constant (~stationary phase idea).

• Interesting, BUT, finding all the stationary points pσ of the complexified action, computing the integer coefficient nσ, performing a Monte Carlo on each Jσ, is not feasible.

C =X

�

n�J�

• But, do we really need to reproduce the original integral exactly?

• We may choose a different regularization, if it is more convenient.

• Consider the trivial vacuum (e.g.φ=0), which is a stationary point. We will see

that the associated thimble J0 defines a local QFT with the same degrees of freedom, the same symmetries and symmetry representations and also the same perturbative expansion as the original formulation.

• By universality, we expect that these properties essentially determine the behavior of physical quantities near a critical point (i.e. in the continuum limit), and hence the formulation in J0 is also an acceptable regularization of that QFT.

The path integral of a QFT?Can we use these ideas to compute the path integral of a QFT?

hOi =RCQ

x

d�x

e�S[�]O[�]RCQ

x

d�x

e�S[�]

Regularize the QFT on J0 ?

Witten, Morse and Chern-Simon (CS)

This leads to a jump in the thimbles Jσ → Jτ + Jσ which is compensated by a jump in the intersection indices nσ in: C =

X

�

n�J�

(a) (c)p!

p"

(b)p!

p"

p!

p"

Witten:1001.2933

λ

Our work was largely inspired by the work of Witten arXiv:1001.2933.Witten uses Morse theory to extend the CS theory to some values of the

parameters where the functional integral is not manifestly convergent.He stresses that if we don’t want to reproduce the original integral exactly, we have in fact much freedom to choose a combination of thimbles as our

extension of CS, for an initial choice of the parameters.But then, Witten needs to compute how the partition function depends on the

parameters of the action (in presence of a knot background).In order to do that, he needs to take into account the fact that the thimbles may cross under a change of parameters (Stokes phenom.), and the indices nσ

must change in order to ensure a smooth dependence on the parameters.

Stokes phenomenon and Monte CarloComputing the partition function Z(k,λ) analytically and how it changes

depending on the parameters (k,λ) is certainly the standard way to compute analytically the observables associated to (k,λ).

For this, it is important to have a smooth dependence on the parameters, and the Stokes jumps must be taken into account.

But this is not the only possible way to study a system. In particular, in Monte Carlo calculations, the partition function itself is never computed.

In a MC, the normalization is fixed by computing a limited set of observables.And this can be done independently for each choice of the parameters.

From this point of view it is more natural to regularize the system always on that thimble J0 that ensures, at small coupling, the right perturbative limit.

J0C =X

�

n�J�

thimble attached to the perturbative vacuum

A Scalar QFT(which already contains most of the interesting aspects)

• Formulate on the thimble• Check Symmetries and Perturbation Theory• Formulate a Monte Carlo algorithm

to do:

Let me apply these ideas to a simple model first

A scalar field with a sign problemThe model that we consider is a complex scalar field with U(1) symm.

When μ≠0, the action is not real, Re[exp[-S]] is not positive and we have a sign problem.

S =

Zd

4x[|@�|2 + (m2 � µ

2)|�|2 + µj0 + �|�|4] j⌫ := �⇤ !@⌫ �

Formulation of the scalar QFT on a thimble

⇔ d

d⌧

�

a,x

(⌧) = − �S[�(⌧)]��

a,x

, ∀a, x,

Assume for now that φ=0 is the global minimum in C,

�O� = ∫J0∏

x,a

d�a,x

e−S[�]O[�]∫J0∏

a,x

d�a,x

e−S[�]

where the thimble J0 is defined as the union of the curves of steepest descent (SD) for SR, that end in φ=0 in the limit of τ→∞.

d

d⌧

�

(R)a,x

(⌧) = −�SR

[�(⌧)]��

(R)a,x

, ∀a, x,d

d⌧

�

(I)a,x

(⌧) = −�SR

[�(⌧)]��

(I)a,x

, ∀a, x,

�x

= �1,x + i�2,x → (�(R)1,x + i�(I)1,x) + i(�(R)2,x + i�(I)2,x)where the usual complexification of a complex scalar field is assumed:

U(1) Symmetry

�̂x e↵�2 �̂

x

⇒ The symmetry transformations are defined on the thimble,

⇒ The Ward Identities can be proved.

One can prove that the thimble attached to φ=0 is invariant under U(1).The reason is the (partial) covariance of the SD equation:

d

d⌧

�

a,x

(⌧) = − �S[�(⌧)]��

a,x

, ∀a, x,

Because of the conjugation, it turns out that the equation is covariant only under the physical subgroup of the whole complexified symmetry group.

Otherwise, the degrees of freedom would not be correct.

Perturbation Theory

dp

d�p

Z

J0(�,µ)d� e�S[�;�,µ]O�,µ[�]

!

|�=0

Z

J0(0,µ)d�

dp

d�p |�=0

⇣e�S[�;�,µ]O�,µ[�]

⌘

ordinary PT

It is a gaussian integral performed along the path of steepest descent. This coincides with the

original integral as long as the latter is convergent

d

d� |�=0

"Z

J0(�,µ)d� e�S[�;�=0,µ]O�=0,µ[�]P [�;µ]

#

0

The integral is constant under small variations of the path around the path of steepest descent.

It is not difficult to compare the PT of the two formulations.Here there are more terms.

Spontaneous Symmetry Breakingwith Mexican Hat Potential

Consider now a nontrivial global minimum.

In this case, we want to do PT around the true minimum.And also regularize the QFT on the thimble attached to it.

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x

The Hessian must be non degenerate. The correct way to study SSB is by introducing an explicit SB term. This produces a single global minimum.

Much of what I said can be carried over (for the reduced symm.), but

Global Minima and Morse Theory

Remember the decomposition: C =X

�

n�J�

A nice argument by Witten, based on Morse theory, actually gives further support to this choice of regularization.

The integer nσ are defined as the intersection number nσ =〈 C, Kσ〉between the original integration domain

C and the dual thimbles Kσ , defined as the union of

the curves of steepest ascent.

JσKσ

I have proposed to regularize the QFT on the thimble attached to the global minimum, on the basis of symmetries and perturbation theory.

SR

p0SR(p0) = smin

C Cn

Cn

hC,K�i = 0

hC,K⇢i 6= 0?

but suppressed by

e�SR(p⇢)+smin

This ‘suggests’ that only those critical points in C that correspond

to global minima do contribute.SR(p�) smin

p� K�

J�

SR(p⇢) > smin

p⇢

K⇢

J⇢

A Monte Carlo algorithm on a thimble?

Thimble Rock, Tonto National Forest, Arizona, USA.

Algorithm1

Z0e�iSI

Z

J0

Y

x

d�x

e�SR[�]O[�]I want to compute

d

d⌧�(R)a,x

= � �SR

��(R)a,x

+ ⌘(R)a,x

d

d⌧�(I)a,x

= � �SR

��(I)a,x

+ ⌘(I)a,x

I could use the Langevin algorithm

How can I stay in J0 ? Preserve J0 by construction Need to be projected on

the tangent space to J0

Bounded from below on J0

How can I compute the tangent space Tφ(J0) at φ?

not easy...

(Not Complex Langevin!)d

d⌧�(R)a,x

(⌧) = −�SR

[�(⌧)]��(R)

a,x

+ ⌘(R)a,x

d

d⌧�(I)a,x

(⌧) = +�SR

[�(⌧)]��(I)

a,x

+ ⌘(I)a,x

Tφ=0(J0)

φ Tφ(J0)η

Projection on the tangent spaceAlthough the tangent space at φ is not

directly accessible, the tangent space at φ =0 is easy to compute.

So, I need a way to transport a vector η along the grad. flow ∂SR, so that it remains tangent to J0. This amounts to

require that:

Which also leads to a prescription to compute η:

, d

d⌧⌘j(⌧) =

X

k

⌘k(⌧)@k@jSR,

0 = [@SR, ⌘(⌧)]k =X

j

@jSR@j⌘k(⌧)�X

j

⌘j(⌧)@j@kSR

L@SR(⌘) = 0 , [@SR, ⌘] = 0

We can prevent the stiffness

We rather need an evolution that preserves the tangent space, possibly by orthogonal transformations

The Iwasawa decomposition of an algebra (here gl(2n,�)) is the infinitesimal version of the Gram-Schmidt orthogonalization process:

The skew-symmetric part defines an orthogonal evolution that generates the same vector space as the original evolution ⇒ This is what we want!

⌘̇ = @2SR⌘We do not really need to solve the equation:

⌘̇ = A⌘ Ai,j =

8<

:

[@2SR]i,j (i < j)�Aj,i (i > j)0 (i = j)

[@2SR]i,j = Ai,j +Di,j +Ni,j

compact (skew-symmetric)

abelian(diagonal)

nilpotent(upper triang.)

This orthogonal flow can be preserved also numerically (e.g. Implicit Midpoint rule ⇒ solve a linear system of size O(NτV))

(NEW w.r.t. arXiv:1205.3996, preliminary)

A graphical summary

φ(t+dt,τ)

φ(t,τ)

φ=0

pdt

d

d⌧⌘j(⌧) =

X

k

⌘k(⌧)@k@jSR[�(⌧)],

d

d⌧�j(⌧) = �@jSR[�(⌧)],

✓(@2S(� = 0) · ⌘) = 0

Numerically stable?

The most delicate point is to check that the newly proposed configuration φ(t+dt)

belongs to J0 by integrating back the DS to the origin φ=0, and possibly correct.

But it is a typical case that should be treated as a boundary value problem (BVP),by using the condition limτ→∞ φ(t+dt,τ) =0.

This leads to the representation of the BVP ODE, as a finite difference problem, which is reduced to solve (NNewRaph) linear systems of size (NτV).The key is to have a good starting point, which we should have because φ(t+dt,τ) is already good to O(dt2) (thanks to the properties of η, we have φ(t+dt,τ) ∀τ).

Tφ=0(J0)

Tφ(J0)φ(t)xφ(t+dt)

x

Numerical stability of the SD step

This is hopeless, if treated as an ODE with an initial value problem (IVP), because any perturbation outside J0 leads to a divergent trajectory.

(NEW w.r.t. arXiv:1205.3996, preliminary)

I find it very unlikely, because the phase is constant, where the integrand is large (in fact, it is neglected in the saddle point method).Can the “perturbative” contribution become negligible?

hY

x

d�x

i ⌧ 1Is it a sign problem? This is so when

We have good reasons to hope that it can be treated with reweighting, but no justification to ignore it!

Residual phase(NEW w.r.t. arXiv:1205.3996, preliminary)

As noticed at the beginning, there is still a phase

1

Z0

Z

J0

Y

x

d�x

e�SR[�]O[�]

det(Tφ)Tφ is the tangent space to J0 in φ.

The first tests

Tiny lattice V=24, very preliminary (Euler method for ODE).

0

5

10

15

20

25

0 100 200 300 400 500 600 700 800

mu=0.97 lambda=1.0 mass=1

Real part of the action

First thermalization on a thimble!!

-140

-120

-100

-80

-60

-40

-20

0

20

40

0 200 400 600 800 1000 1200

mu=0.97 lambda=1.0 mass=1

Real part of the action

As expected, Euler integrator is not stable. But it resists long

enough to see a thermalization!

-5e-05

-4e-05

-3e-05

-2e-05

-1e-05

0

1e-05

2e-05

3e-05

0 50 100 150 200 250 300 350 400

mu=0.97 lambda=1.0 mass=1

Imaginary part of the action

(Should be constant on the thimble)

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 200 400 600 800 1000 1200

mu=0.97 lambda=1.0 mass=1

Imaginary part of the action

(Should be constant on the thimble)

QCD

QCD Action

S[U ] = �

X

x,µ<⌫

1� 1

3

< Tr U

µ,⌫

(x)

�+N

f

Tr logQ[U ]

Q[U ]xy

= (m+ 4)�xy

� 1

2

X

⌫=0,3

(1� �

⌫

)�x+⌫̂,y

e

µ �0,⌫U

⌫

(x)� 1

2

X

⌫=0,3

(1 + �

⌫

)�x�⌫̂,y

e

�µ �0,⌫U

⌫

(x� ⌫̂)�1

Uµ,⌫(x) = Uµ(x)U⌫(x+ µ̂)Uµ(x+ ⌫̂)�1U⌫(x)

�1 Wilson Plaquette

Wilson Fermion Matrix

Complexification

A

a⌫(x) ! A

a,R⌫ (x) + iA

a,I⌫ (x) a = 1 . . . N2

c � 1.

SU(3)4V ! SL(3,C)4V

Covariant Derivatives

rR

x,⌫,a

, rI

x,⌫,a

, rx,⌫,a

.and similar definitions for:

rx,⌫,a

= rR

x,⌫,a

� irI

x,⌫,a

,

rx,⌫,a

= rR

x,⌫,a

+ irI

x,⌫,a

Such that: And Cauchy-Riemann hold.

rx,⌫,a

F [U ] :=@

@↵

F

⇥e

i↵TaU

⌫

(x)⇤|↵=0

Note that the covariant derivatives do not commute:

[rx,⌫,a

,ry,�,b

] = �x,y

�⌫,�

fabc

rx,⌫,c

, [Ta, Tb] = ifabcTcwhere:

Note that the Hessian is still well defined and symmetric in the stationary points.

Defining the thimbles for gauge theoriesHow does the gauge invariance affects the construction of the thimble J0?Discussed by Atiyah-Bott (1982) and reviewd by Witten (2010).

Strictly speaking no stationary point can be non-degenerate(the Hessian has zero eigenvalues corresponding to the gauge d.o.f.)

Solution: Substitute the concept of non-degenerate critical pointwith the one of non-degenerate critical manifold (Bott 1956)

A manifold N ⊂ D is a non-degenerate critical sub-manifold of D for the function F : D → � if:

1. dF = 0 , along N.

2. The Hessian ∂²F is non-degenerate on the normal bundle ν(N).

Under these conditions the normal bundle decomposes in ν(N) = ν+(N)+ν-(N)

associated to positive and negative eigenvalues of ∂²F .

In the case of QCD the manifold N corresponds to a gauge orbit.

In the case of the thimble J0, it is the gauge orbit generated from A=0.

nG := dimRN = (V � 1)⇥ (N2c � 1)

Equations of Steepest Descent

d

d⌧

U

⌫

(x; ⌧) = (�iT

a

rx,⌫,a

S[U ])U⌫

(x; ⌧)

with covariant derivatives, they take the form:

d

d⌧SR/I =

1

2

d

d⌧(S ± S) = �1

2rjS ·rjS ⌥ 1

2rjS ·rjS =

⇢� k rS k2

0

Note that this implies the following essential relations:

The thimble J0 can now be defined as:

J0 :=

n

U 2 (SL(3,C))4V | 9U(⌧) solution of SD Eq | U(0) = U & lim

⌧!1U(⌧) 2 N (0)

o

which has dimension: (n-nG)+nG=n.

Gauge Symmetry of the thimble

d

d⌧

U

⌫

(x; ⌧) = (�iT

a

rx,⌫,a

S[U ])U⌫

(x; ⌧)

Consider the SD equation:

(Ta

rx,⌫,a

S[U ]) !�⇤(x)�1

�†(T

a

rx,⌫,a

S[U ])⇤(x)†

Under gauge transformations it changes as:

U⌫(x) ! ⇤(x)U⌫(x)⇤(x+ ⌫̂)�1

Note that the full SD equation is covariant only under the SU(3) subgroup of SL(3,�). ⇤(x)† = ⇤(x)�1

Proof of gauge invariance is now essentially identical to the proof of U(1) global symmetry for the scalar case.

(This means that also Ward Identities are fulfilled).

Perturbation Theory

dp

dgp

Z

J0(g;µ)dA e�S2[A]+gSint[A] det(Q[A = 0]) F [A; g, µ] Q[A = 0;µ]�1 . . . Q[A = 0;µ]�1

!

|g=0

We need to compute:

In this expression, the fermion field is integrated out.

This leaves the determinant and the inverse fermion matrices (free propagators).

The integrand has the form of a gaussian times polynomials

Proof of equivalence is essentially identical to the scalar case.

Algorithm

Transport equation:

d

d⌧⌘j(⌧) = ⌘j0(⌧)rj0rjSR,

Only few difference w.r.t. the scalar case.

Langevin Eq:

d

d⌧

U

⌫

(x; ⌧) = �iT

a

(rx,⌫,a

S[U ] + ⌘

a,x,⌫

)U⌫

(x; ⌧),

(The fermion force needs to be computed precisely.How many noise vectors are enough?)

Algorithm

On the lattice (with Wilson fermions) there are no disconnected topological sectors.

There are gauge configurations with some topological content (approx. instantons),

By smoothing the configuration (i.e. cooling, which is essentially our steepest descent

evolution), the topological content is initially revealed (smooth instantons appear), but

eventually canceled since all the configurations fall in the zero topological charge.

In our case, we want to reach the zero charge sector as fast as possible.

If this is feasible at all, some Fourier acceleration will be presumably important.

Just one remark specific to QCD: topological sectors.

Conclusions• I have illustrated a new approach to deal with the sign problem that

afflicts a class of QFTs.

• It consists in regularizing the QFT on a Lefschetz thimble. Although it does not coincide with the usual regularization, it is a legitimate one on the basis of universality. In fact, we could prove that QCD on the thimble has the same symmetries, d.o.f. and PT as the usual formulation.

• I have also introduced an Monte Carlo algorithm to achieve an importance sampling of the configurations on the thimble. Its numerical implementation will be presumably challenging, and certainly expensive, but all the steps of the algorithm are, a priori, feasible and have acceptable scaling.

• A sign problem remains. It is expected to be mild and hence manageable with reweighting, but this must be checked.

• We are presently testing the method for a scalar QFT on tiny lattices with encouraging results.