runcatedT Hilbert Space Approach for the 1+1D 4...

De�nition of modelFinite size physics

Numerics and data evaluationResults

Truncated Hilbert Space Approach for the 1+1D ϕ4

theory

Supervisor: Zoltán Bajnok

Márton Lájer

ELFT Particle Physics Seminar, 2016

Márton Lájer THSA for ϕ4 model



Outline

1 De�nition of model

2 Finite size physics

3 Numerics and data evaluation

4 Results




Recent works

A. Coser, G. Mussardo et al., Truncated Conformal Space

Approach for 2D Landau-Ginzburg Theories,arXiv:1409.1494v2 (2014)

S. Rychkov, L. G. Vitale, A Hamiltonian Truncation Study of

the φ4 Theory in Two Dimensions, arXiv:1412.3460 (2014)

S. Rychkov, L. G. Vitale, A Hamiltonian Truncation Study of

the φ4 Theory in Two Dimensions II: The Z_2-Broken Phase

and the Chang Duality , arXiv:1512.00493 (2015)

J. Elias-Miro, M. Montull, M. Riembau, The renormalized

Hamiltonian truncation method in the large ET expansion,arxiv:1512.05746 (2015)

Z. Bajnok, M. Lajer, Truncated Hilbert space approach

to the 2d ϕ4 theory , arXiv:1512.06901 (2015)




Classical action

S =∫d2xL (x), L (x) =

1

2∂µϕ(x)∂

µϕ(x)−V (ϕ(x))

V (ϕ) = m2

(−12

ϕ2 +

g

6ϕ4

)=−m

2

2ϕ2 +

λ

4!ϕ4

=−g2ϕ2 +g4ϕ

4

g =6g4m2

=λ

4m2

1+1 dimensions, h = c = 1

Symmetries of S : Poincaré, internal Z2, spatial parity

GS breaks Z2 parity




Fluctuations and kinks

Elementary �uctuations around minima:

m0 =√2m, g =

12g4m2

0

Classical kink/antikink (K/K )solution:

φ± =±

√3

2gtanh

(m0x

2

), M0 =

m0

g

(anti)kink nonperturbative, disjunct topological sectors

KK bound state (breather) classically not stable

Semiclassics: quantum �uctuations around cl. kink background

SC normal ordering: wrt. free boson with m0 mass, in�nitevolume




Semiclassics

Relevant quantities: masses and S-matrix

�rst SC correction to kink mass:

Msc = M0−m0c +O (g) , c =3

2π− 1

4√3

breathers:

mn = 2Msc sin

(n

ζ

2

), ζ =

g

1−gc

Stability against m1: mn < 2m1, SC: always true for m2, neverfor m3

Mussardo: there are at most two stable breathers




Semiclassics

B1 S-matrix below two-particle treshold: unitarity, crossing

S (θ) = S (−θ)−1 = S (iπ−θ)

where θ is the rapidity di�erence,

k1 =−k2 = m1 sinhθ

2

Simplest nontrivial solution:

S (θ , iα) =sinhθ + i sinα

sinhθ − i sinα

Pole at m2 = 2m1 cosα

2

Semiclassics:

m2 = 2m1 cosζ

2=⇒ α = ζ




Canonical Quantization

Hamiltonian for �nite volume L:

H =

L∫0

H dx , H =1

2(∂tϕ)2+

1

2

(∂xϕ

2)− 1

2m2

ϕ2+

g

6m2

ϕ4

Boundary conditions:

ϕ (x) =±ϕ (x +L)

Quantization of free boson:

ϕ (x , t) = ∑n

1√2ωnL

(ane−iωnt+iknx +a†

neiωnt−iknx

)[an,am] =

[a†n,a

†m

]= 0 ;

[an,a

†m

]= δn,m

ωn =√

µ2 +k2n ; kn =2π

Ln

PBC: n integer, ABC: n half-integerMárton Lájer THSA for ϕ4 model



Normal ordering

wrt a mass scale µ and volume L:

H = H µ,L0

+g0 (µ,L) +g2 (µ,L) : ϕ2 :µ,L +g4 (µ,L) : ϕ

4 :µ,L

vacuum is annihilated by all anFree Hamiltonian:

Hµ,L0

=

L∫0

: (∂tϕ)2+(∂xϕ)2+µ2ϕ2 : dx+E±

0(µ,L) = ∑

n

ωna†nan+E±

0

GS energy and energy density in�nite for non-renormalizedHamiltonianComparing renormalization schemes:

E±0

(µ,L) = µ

∞∫−∞

dθ

2πcoshθ log

(1∓ e−mLcoshθ

)Márton Lájer THSA for ϕ4 model



Normal ordering

Relate ::µ,L normal ordering to ::µ ′,L′ :

: ϕ2 :µ ′,L′ =: ϕ

2 :µ,L +∆Z

: ϕ4 :µ ′,L′ =: ϕ

4 :µ,L +6∆Z : ϕ2 :µ,L +3(∆Z )2 ,

∆Z = Z (µ,L)−Z(µ′,L′), Z (µ,L) = ∑

|n|<N

1

2ωnL

Semiclassical scheme: L′→ ∞, µ ′ = m0, perturbationparameters are de�ned here:

g0 (m0,∞) = 0, g2 (m0,∞) =−34m2

0, g4 (m0,∞) =gm2

0

12




Normal ordering

Changing the scheme from ::m0,∞ to ::m,L,

g0 (m,L) = m2

(g

2z2± (mL)− z± (mL)− ln2

8π

)g2 (m,L) = m2

(g z2± (mL)−1

)g4 (m,L) =

m2g

6

where z± = z±+ ln2

4πfor PBC (+) and ABC (−) with

z+ (mL) =

∞∫0

dθ

π

(emLcoshθ −1

)−1z− (mL) = 2z+ (2mL)− z+ (mL)




Finite size e�ects

�Finite volume�: space is compacti�ed via ϕ (x +L) =±ϕ (x)

Lüscher:

�nite volume spectrum of H⇐⇒ in�nite volume physical quantities

Leading order: �Bethe-Yang� corrections

Polynomial in L−1

due to momentum quantization (BC)

�Lüscher� corrections

exponentially smalldue to vacuum polarization (virtual particles)




In�nite volume data

Parameters:

masses: Ei =√m2

i +p2

Ei (θ) = mi cosh(θ) , pi (θ) = mi sinh(θ)

below a kinematical treshold θt only elastic processes →integrable framework applicabletwo-particle S-matrices: S11, S12, S22, Sk1, Sk2, Skk

general form restricted by unitarity, analicity, crossing, physical

poles and branch cuts

e.g. S11(θ): B1

(θ

2

)B1

(− θ

2

)→ B1

(− θ

2

)B1

(θ

2

)S11(θ) must have a pole corresponding to B2 if B2 is in the

spectrum




2-particle S-matrices

General form:

Saa(θ) = S(θ , iα)exp

{−∫

∞

θt

dθ ′

2π iρ(θ

′) logS(θ′,θ)

}θt : kinematic treshold rapidity for inelastic processes (e.g.B1B1→ B1B2)

bound states appear as poles, e.g. for S11 (θ):

S(θ ,x) =sinhθ + i sinα

sinhθ − i sinα

m2 = 2m1 cosα

2

Measuring ρ (θ) would be very interesting but di�cult





kink and antikink: mass M degenerate due to Z2 symmetry

Mussardo: Skk also diagonal, unitarity and crossing belowtreshold:

Skk(θ) = Skk(iπ−θ) = Skk(−θ)−1 = Skk(θ)

B1, B2 are kk bound states with fusion angles β1, β2

m1,2 = 2M cosβ1,2

2

if M <m1, and both B1 and B2 present:

Skk(θ) =−S(θ , iβ1)S(θ , iβ2)Sexp(θ)





kink-B sector has topological charge ⇒unitarity/crossingbelow kB2 treshold:

Sk1(θ) = S1k(iπ−θ) = S1k(−θ)−1 = S1k(θ) = S

1k(θ)

pole at i β

2:

Sk1(θ) = S

(θ , i

β1

2

)Fusion provides β1:

Sk1(θ + iβ1

2)Sk1(θ − i

β1

2) = S11(θ)

β1 = π−α m1 = 2M sinα

2

we know that m2 = 2m1 cosα

2=⇒ β2 = π−2α




Polynomial corrections

It should be decided whether the previous bootstrap-basedapproximations are useful

Momentum quantization (Bethe-Yang):

Volume large compared to particle numberParticles move freely most of the timeParticle wave functions pick up phase upon interacting

e ipjL ∏n:n 6=j

Sjn(θj −θn) =±1, E (L) =N

∑n=1

mn coshθn(L)

± depends on periodic/antiperiodic nature of wave function

exact for integrable models; here approximate for certainkinematical domains




Exponential corrections

virtual processes (Lüscher)

Leading order for vacuum:

lightest (massm) particle-antiparticle pair appears, travelsaround the world and annihilate

E0(L) =−m∫

∞

−∞

dθ

2πcoshθ e−mLcoshθ

Leading order for 1-particle state a:

F-term: virtual pair bb from vacuum → scattering ona→annihilate:

∆Fma =−mb

∫∞

−∞

dθ

2πcoshθ (Sba(iπ/2+ θ)−1)e−mbLcoshθ

µ-term: residue of F-term at pole bound state c:

∆µma =−∑b,c

θ(m2

a−|m2

b−m2

c |)µcab(−i)ResSab(θ)e−µc

abL




THSA

We choose a free massive Fock basis at volume mL0 = 1, massµ

m= 1

We truncate the H -space at some energy cuto� Emax wrt tofree theorydimensionless parameters:

H

m= h ; mL = l ; ωn = mωn = m

√1+

4π2n2

l2

Hamiltonian:

h = h0 +∫ L

0

mdx[g2(l) :φ

2 : +g4 :φ4 : +g0(l)

]h0 = ∑

n

ωn(l)a+n an + e±

0(l), e±

0(l) = E±

0(mL)/m

2 dimensionless parameters: g , l . m sets energy scaleWe solve hψ = Eψ numerically in truncated basis




Mini Hilbert Space

Periodic boundary condition: �mini superspace�

H = H mini⊗H

IR (zero momentum) mode has special role in low-lyingspectrum

φ(x ,0) = φ(x ,0) + φ0 ; φ0 =1√2l

(a0 +a+0

)

Mini Hamiltonian: anharmonic oscillator (QM)

hmini = ω0 a+0a0 +

1

2g2(l) : (a0 +a+

0)2 : +

1

4lg4 : (a0 +a+

0)4 :

h= hmini + h0+∫ L

0

mdx[g0(l) + g2(l) : φ

2 : +g4(: φ

4 : +4 : φ3 : φ0 +6 : φ

2 : φ20

)]where:

h0 = h0−hmini0 ; hmini

0 = ω0 a+0a0




Technical implementation

Has to �nd lowest eigenvalues for large, symmetric (real)sparse matrices

Great iterative, general purpose eigensolver: PRIMME

do not diagonalize whole matrix, only �rst O(10) eigenpairsUse a re�ned version of Jacobi-Davidson method(�JDQMR-ETOL�)

C++ program to calculate matrix elements

can handle up to 200000 dimensional problems on single 32-bitPCfurther scaling-up possible





periodic BC:

generate basis at �x mL0 = 1compute lowest eigenvalues of hmini (L)obtain matrix elements of H(L) as a direct product; use exacteigenstates in the minimal basiscompute lowest eigenpairs of H(L)perform this to various energy truncations and volumes

antiperiodic BC:

generate basis at �x mL0 = 1obtain matrix elements of H(L) → eigenpairs





basis generating volume: the spectrum is close to theconformal spectrum, energy approx. measurable in 2π

mLunits

mode (a+n )jn contributes to this �conformal energy� as jn|n|.

Energy cuto�s introduced as integer multiples of this unit,taken between 14 and 26 (oscillator content remains same)

Volume dependence measured from l = 1 to l = 30, usuallyinteger steps

minimal space contains 2000 basis vectors. We keep the lowest6 �exact� minimal eigenstates

symmetries help: we work in zero-momentum sector.Even/odd particle number sectors treated separately. Currentlynot using space parity.




Extrapolation in energy cuto�

Rychkov et al.: leading correction to H due to truncation:

∆H =− g2

e2max

{(c1 log

2 emax + c2 logemax + c3

)V0

+(c4 logemax + c5)V2 + c6V4}

c1 . . .c6 known

we extrapolate (�t) using the leading ecut dependence:

E (ecut) = a+blog2 ecute2cut

+ clogecute2cut

obtanining an error:

E (ecut) = a+blog2 ecute2cut




Raw spectrum at g = 1.2

5 10 15 20 25 30mL

-10

-5

5

10

E�m




GS energy density extrapolation

5 10 15 20 25 30mL

-6.268

-6.266

-6.264

-6.262

e0�l

The energy density obtained at l = 7 is indicated with a dashed line.




GS energy density extrapolation

100 120 140 160Ecut�m

-62.659

-62.658

-62.657

-62.656

-62.655

-62.654

E0�m

Data points as a function of the truncation energy together withthe �tted function for the ground-state energy density at g = 0.06and volume l = 10.




Low-lying spectrum at g = 0.06

5 10 15 20 25 30mL

2

4

6

8

E�m

Even particle number sector; vacuum energy density at l = 7substracted




BY description of B2

5 10 15 20 25 30mL

2.790

2.795

2.800

E2 - E0

m

Imaginary BY description of B2 at g = 0.06 with leading �nite sizecorrection (gray). Fitting sensitive to α , hereα = 0.06.




Low-lying spectra with BY lines

5 10 15 20 25 30mL

2

4

6

E�m

5 10 15 20 25 30mL

1

2

3

4

5

6

E�m

for g = 0.06 (left) and g = 0.6 (right)




coupling dependence

0.5 1.0 1.5g

1

2

3

Α; m1�m; m2�m; 2M�m

m1 (red), m2 (green), together with the scattering parameter α ,(blue) . 2M shown in gray




kk Phase Shifts

0.5 1.0 1.5 2.0 2.5 3.0Θ

2

4

6

8

∆

0 1 2 3 4Θ0

2

4

6

8

10∆

left: g = 1.8, right: g = 2.4. Obtained from BY equation fordi�erent lines




kk Phase Shifts

0.5 1.0 1.5 2.0 2.5 3.0

2

4

6

8

0 5 10 15 20 25 30mL0

1

2

3

4

5E�m

for g = 1.8. Fitted function: δ (θ) = aθ −bθ2 + cθ3




Exp correction to E0

2 4 6 8 10mL

-0.025

-0.020

-0.015

-0.010

-0.005

E0�m

leading exponential correction to GS energy at g = 0.06




Exp correction to B1

0 5 10 15 20 25 30mL

1.390

1.395

1.400

1.405HE1-E0L�m

at g = 0.06. Blue dots: B1 energies from the rescaled TCSAspectrum of the sine-Gordon theory at pπ = 0.06




Comparison with sG TCSA

5 10 15 20 25 30mL

3

4

5

6

7

8E�m

sG data: pπ = 0.06 and ecut = 20; φ4 data: g = 0.06. The φ4 dataare indicated by solid lines. All data are normalized wrt. thecorresponding ground state energies.




vacuum splitting

2 3 4 5 6mL

0.01

0.02

0.03

0.04

0.05

0.06

0.07

DE 0�m

3 4 5 6mL

-2.2

-2.0

-1.8

-1.6

DE 0�m

E even0 (L)−E odd

0 (L) =

√2M

πLe−ML + . . .




kink mass coupling dependence

0.0 0.5 1.0 1.5 2.0g

1

2

3

4

5M�m

solid line: SC kink mass. Red dots: mass extracted from vacuumsplitting




Lowest antiperiodic state

5 10 15 20mL

1.6

1.8

2.0

2.2

E�m

after substracting periodic vacuum. g = 0.6




Low-lying AP sector

0 5 10 15 20mL0

1

2

3

4

5

6

7E�m

at g = 0.6. Two-particle BY lines for n = 1

2, 32, 52. Resonance seems

3-particle involving a steady kink and two B1sMárton Lájer THSA for ϕ4 model



Kink mass

1.5 2.0g

0.5

1.0

1.5

M�m

Red dots: vacuum splitting result. Blue dots: antiperiodic result.Gray curve: SC expectation




Critical point

vanishing kink mass → critical point around g = 3

Critical point: Ising universality class, governing CFT:

HCFT =2π

L(L0 + L0−

c

12) ; PCFT =

2π

L(L0− L0)

L0, L0: Virasoro generators

Low-lying periodic spectrum:

state scaling dim. state scaling dim. state scaling dim.

|0〉 0 | 12, 12〉 1 | 1

16, 1

16〉 1

8

L−2L−2 |0〉 4 L−1L−1 | 12 ,1

2〉 3 L−1L−1 | 116 ,

1

16〉 1

8+2




Lowest odd state

0 5 10 15 20 25 30mL

0.05

0.10

0.15

0.20

0.25EL

This should become | 116, 1

16〉 at critical point. From g = 2.76

(purple) to g = 3.48 (gray). l2π

(e1− e0) is depicted. le2π

= 1

8is

shown by continous lineMárton Lájer THSA for ϕ4 model



Spectrum at g = 3.12

5 10 15 20 25 30mL

1

2

3

4

EL

energies multiplied by l2π. Solid lines indicate L→ ∞ CFT

expectationMárton Lájer THSA for ϕ4 model



Conclusions

Investigated 1+1D scalar ϕ4 in symmetry breaking case

numerical side: THSA, e�ective eigenpair calculation for largebases; used minimal space in periodic sector

analytical side: extrapolate in�nite volume quantities from�nite size e�ects

at most two neutral excitations: Mussardo's conjecturecon�rned

we see B1, B2 disappear from the spectrum at largercouplings, even observe phase transition

two methods to obtain kink mass: vacuum splitting,antiperiodic lowest state




Coupling dependence of masses

0.5 1.0 1.5g

1

2

3

Α; m1�m; m2�m; 2M�m




Outlook

In the near future, the renormalization technique of Rychkov etal. should be included

Numerical algorithm can be scaled up to larger matrices

Measure non-integrable e�ects

longer term: other theories, higher dimensions, etc.




Thank you for your attention


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