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
De�nition of modelFinite size physics
Numerics and data evaluationResults
Outline
1 De�nition of model
2 Finite size physics
3 Numerics and data evaluation
4 Results
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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)
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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)
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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)
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
2-particle S-matrices
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)
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
2-particle S-matrices
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
Technical implementation
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
Technical implementation
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.
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
Raw spectrum at g = 1.2
5 10 15 20 25 30mL
-10
-5
5
10
E�m
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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.
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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.
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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.
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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)
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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.
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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 + . . .
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
Lowest antiperiodic state
5 10 15 20mL
1.6
1.8
2.0
2.2
E�m
after substracting periodic vacuum. g = 0.6
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
De�nition of modelFinite size physics
Numerics and data evaluationResults
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
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
Coupling dependence of masses
0.5 1.0 1.5g
1
2
3
Α; m1�m; m2�m; 2M�m
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
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.
Márton Lájer THSA for ϕ4 model
De�nition of modelFinite size physics
Numerics and data evaluationResults
Thank you for your attention
Márton Lájer THSA for ϕ4 model
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