June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Tensor network approach for chiral symmetry restoration of
1-flavor Schwinger model at finite temperature
Hana Saito (NIC, DESY Zeuthen)
with M. C. Bañuls, K. Cichy, J. I. Cirac and K. JansenH. Saito et al. PoS Lattice, 302 (2014), arXiv:1412.0596
M. C. Bañuls et al, arXiv:1505.00279
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
QCD Phase diagram• Quark: one of elementary particles • Confinement at low T (QCD Phase diagram)⇒ non-perturbative aspect
• Lattice QCD simulation ✴ At finite temperature and zero chemical potential : Successful!!
✴ At finite chemical potential μ : failing
• But, a lot of interests in dense QCD • critical point at finite μ, • unknown phases at large μ
2
T
Confinement
Ex. Early universe after Big Bang
Fukushima, Hatsuda, Rept. Prog. Phys. 74, 014001 (2011), arXiv:1005.4814[hep-ph]
sQGP
uSCdSCCFL
2SC
Critical Point
Quarkyonic Matter
Quark-Gluon Plasma
Hadronic Phase
Color Superconductors?Te
mpe
ratu
re T
Baryon Chemical Potential mB
Inhomogeneous ScB
Liquid-Gas
Nuclear SuperfluidCFL-K , Crystalline CSCMeson supercurrentGluonic phase, Mixed phase
0
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Sign problem• Complex action at finite μ spoils Markov Chain Monte Carlo built on positive probability measure
• To avoid the problem in convention lattice QCD • Reweighting: • Taylor expansion of μ:
• For real time dynamics of high energy physics, alternative approaches required
3
to change the distribution of Monte Carlo ensemble → overlap problem
coefficients calculated with μ = 0 ensemble → a problem of convergence
Complex quark determinant
To establish a promising numerical tool ⇒ our strategy: to employ Hamiltonian approach
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Outline • Motivation • 1-flavor Schwinger model • Hamiltonian of Schwinger model
• Tensor Network • Chiral condensate at T = 0 • Chiral symmetry restoration at finite T • Thermal calculation in TN and outcomes • Summary
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June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
1-flavor Schwinger model• 1+1 dimensional QED model
✴ not QCD, but similar to QCD : confinement, chiral symmetry breaking (via anomaly for Nf =1)
✴ exactly solvable in massless case
• Hamiltonian of 1-flavor Schwinger model
5
hopping term
gauge part
mass term
J. Schwinger Phys.Rev. 128 (1962)
T. Banks, L. Susskind and J. Kogut, PRD13, 4 (1973)
N. L. Pak and P. Senjanovic, Phys.Let.B71, 2 (1977), K. Johnson Phys.Let. 5, 4(1963)
Gauss law
⇒ a good test case
where inverse coupling x=1/a2g2, dimensionless mass μ=2m/ag2 and
in spin language(staggered discrieization)
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Hamiltonian of Schwinger model in continuum• Lagrangian of Schwinger model:
• Hamiltonian via Legendre transformation:
• Gauss’s law: 6
, : 2-component fermion field
∵ i) , ii) temporal gauge
: covariant derivative, : gamma matrix: Field strength
For details, e.g. T. Byrnes, et al Phys. Rev. D66 (2002), 013002
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Hamiltonian of Schwinger model on lattice• For fermion, Kogut-Susskind type discretization:
• For gauge field, discretization: • Jordan-Wigner transformation:
• Gauss law on Lattice: • Hamiltonian:
7
1-component fermion field:2-component fermion field:
n n+1
for keeping anti-commutative relation
in continuum
hopping term gauge partmass termzero back ground field
For details, T. Byrnes, et al Phys. Rev. D66 (2002), 013002
s0 s1 s2 s3
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Tensor network (TN)• Hamiltonian approach with exact diagonalization available for only small size: Ex. In 1D, with chain length N ~ O(10), not enough to take thermodynamic limit
• Tensor Network: An efficient approximation of quantum many-body state from quantum information
• Matrix product state (MPS): tensor network for 1d
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ik: physical indices at site k, m, n (=1,…, D) : indices from this approximation, D : bond dimension
physical indexvirtual indices
site
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
An example of MPS• 1/2-spin 2 particle system:
• Supposing D = 2 for tensors
• By computing trace of products
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ik = ↑ , ↓ for k = 1, 2
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
• An efficient way to express sub-space of Hilbert space
• From quantum information aspect, much smaller value D than dN/2 is enough to derive ground state
• Hilbert space growing exponentially as increasing system size ⇔ With TN, one can investigate sub-space growing polynomially
• systematically improved
Bond dimension
10
F. Verstraete et al. PRL 93, 227204
dN ⇔ NdD2 d : d.o.f of physical index at each site, N : chain length
By using MPS with D,
⇒ If D ~ dN/2, no advantage of TN
size of the whole Hilbert space
Ex.) 1d spin system
polynomial
Hilbert space
exponential Ex.) In our studies, D ~ 100 is enough (<< dN/2~1015 )
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Graphical representation of TN• Ex.1)
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i k
j
ij
klm
j
i kl
m
l
m
j
i
(i) A general tensor Tijklm
(ii) A tensor with three indices Mijk (iii)Product of two tensors Mijk Aklm = Bijlm
• Ex.2) MPS state:
i1 i2 i3 i4
MPS state
:physical indices
bond indices from TN approximation
ex. N = 4open boundary condition
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
• For ground (and some excited) state search • Ground state derived by searching minimum of trial energy, computed by trial MPS state:
• the minimum searched with variational approach
Variational search
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with fixing the other elements
: a trial MPS state
Hamiltoniantrial MPS
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
• More concretely,updating through the whole chain, until convergence
• Techniques to solve it efficiently, i) canonical form derived by SVD: ii)MPO for Hamiltonian
Variational search
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=
=
for given in, kn, kn+1
=
MPS state:: a function in terms of with fixing the others
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
• Hamiltonian of Schwinger model
• Basis of Hamiltonian dN ⇒ Hamiltonian dN x dN matrix
• Another way to express H mapping into operator space
• Ex.) MPO of one hopping term with N = 4, open boundary
Matrix Product Operator (MPO)
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hopping term gauge partmass term
:hopping
where ,,
, , , ,
for left boundary
for right boundaryfor bulk n = 2,3
Useful to compute the trial energy
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Our previous study• 1-flavor Schwinger model at zero T • With variational method, computing:
✴ mass spectrum ✴ (subtracted) chiral condensate:
• Continuum extrapolation:
15
0 0.05 0.1 0.15 0.2 0.250.16
0.165
0.17
0.175
0.18
0.185
0.19
0.195
1/√
x
m/g = 0
exact 0.1599290.159930(8)
0 0.05 0.1 0.15 0.2 0.250.09
0.095
0.1
0.105
0.11
0.115
0.12
1/√
x
m/g = 0.125
0.092023(4)(H.) 0.0929
in spin language
M. C. Bañuls et al JHEP 1311, 158, LAT2013, 332
(H.) Y. Hosotani arXiv:9703153
Logarithmic correction from analytic form of free theory
Fit function:
x = 1/g2a2 ⇒ continuum limit
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Lattice gauge theory (LGT) with TN approach
• Earlier Study: critical behavior of Schwinger model with Density Matrix Renormalization Group
• Nowadays: various branches ✴ Our previous studies ✴ On higher dimension TN ✴ Real time evolution ✴ With quantum link model ✴ Tensor Renormalization Group
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T. Byrnes, et al. PRD.66.013002 (2002)
B. Buyens, et al. PRL, 113, 091601
Y. Shimizu, Y. Kuramashi, PRD90, 014508 (2014), PRD90, 074503 (2014)
E. Rico, et al. PRL112, 201601 (2014)
M. C. Bañuls et al JHEP 1311, 158, LAT2013, 332 (2013)
T. Pichler, et al. arXiv:1505.04440
S. Kühn, et al. arXiv:1505.04441
This study
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Chiral symmetry restoration of Schwinger model for Nf = 1• Chiral symmetry breaking at T = 0 (via anomaly) ⇔ At high T, the symmetry restoration
• Analytic formula
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0
0.05
0.1
0.15
0.2
0 0.2 0.4 0.6 0.8 1 1.2
Y
`
zero T limit
smooth curve
I. Sachs and A. Wipf, Helv. Phys. Acta, 65, 652 (1992)
where
Euler constant
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Thermal state calculation• Expectation value at finite T :
• How to calculate the ρ(β) ✴ to approximate ρ(β) ✴ ρ (β/2) to ensure positivity: ✴ imaginary time evolution:
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thermal density operator where
high T → low T
Ex. ) For fixed δ, larger Nstep corresponds to lower T
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Thermal state calculation✴ Each factor ,
• Approximation (global optimization) (1st step)(2nd step)
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dN x dN elements tensors M[k] in size of d2 x D2
obtained by variational search
and so on
In β = 0, ρ (β) is identity.
d :physical indices
tensors M’[k] in size of d2 x D2
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Bond dimension and Step size dependence
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0
0.05
0.1
0.15
0.2
0 1 2 3 4 5
Y
g`
Sachs and WipfD=20, b=0.001 D=20, b=0.0001D=40, b=0.001 D=40, b=0.0001
difference
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
The other errors
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0
0.05
0.1
0.15
0.2
0.25
0 1 2 3 4 5
Y
g`
N= 20, x=1N= 50, x=6.25N= 80, x=16N=100, x=25N=120, x=36N=160, x=64Sachs and Wipf
continuum extrapolation with fixed
approaching
(fixed physical length)
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
• Four simulation parameters
• Four extrapolations • To avoid too large finite size effect: • Open boundary condition
Simulation setup1.From MPS approx., bond dimension D 2. From T evol., step size δ 3. chain length N 4. inverse coupling x
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D → ∞
δ → 0
N → ∞
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Extrapolation ① & ②
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① D → ∞ with fixed δ, N, x ② δ → 0 with fixed N, xfrom a mathematical proof, convergence in D
linear convergence in δ theoretically predicted
at gβ = 0.4
0 5 10 15 200
1
2
3
4x 10ï7
(1/D) × 103
0 10 200246x 10ï5
δ = 5 · 10−4
δ = 7 · 10−4
δ = 10−3
0 0.5 1 1.5x 10ï3
0.014
0.0145
0.015
δ
N =40N =50N =60
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Extrapolation ③
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③ N → ∞ with fixed x
0 0.005 0.01 0.015 0.02 0.0250.01
0.012
0.014
0.016
1/N
x =5x =11x =20x =32x =45
at gβ = 0.4
0 0.005 0.01 0.015 0.02 0.0250.1
0.12
0.14
0.16
0.18
0.2
1/N
x =5x =11x =20x =32x =45
at gβ = 4.0
In high/low T, the slope changing between them.
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
0 0.2 0.4 0.6
0.14
0.16
0.18
0.2
1/√
x
Extrapolation ④
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gβ = 4.0
④ continuum extrapolation
(solid blue)
(dashed red)
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Chiral condensatein continuum limit
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After eliminating those systematic errors …
0 2 4 60
0.05
0.1
0.15
gβ
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Cut-off effect• continuum extrapolation in high T
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0 0.2 0.4 0.60
0.005
0.01
1/√
x
gβ = 0.4 (solid)
(dashed)
large x required→ large N for finite size effect → large norm of H → extremely small δ for approx. exp (-δHz) ~ 1 - δHz
→ the huge number of steps in Suzuki-Trotter exp.
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Another approach• Massless Schwinger Hamiltonian:
• basis states in strong coupling limit
• Exponential of Hamiltonian:
• Gauss’s law for the :
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s0 s1 s2 s3
l0 l1 l2
where
with l-1 = 0,
exponentially suppressed
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Another approach• chiral condensate at high T
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0 0.5 1 1.5 20
0.02
0.04
0.06
gβ
• large electric flux is exponentially suppressed in thermal state:
• additional extrapolation needed
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Summary• Computing chiral condensate at finite T in Hamiltonian formalism with tensor network methods
• Dependence of each parameters: bond dimension, step size, system size, inverse of coupling
• By taking continuum limit, we obtained results consistent with an analytic formula
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I. Sachs and A. Wipf, arXiv:1005.1822
Backup slides
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
• Hamiltonian of Schwinger model
• Basis of Hamiltonian dN ⇒ Hamiltonian dN x dN matrix
• Matrix Product Operator (MPO) form of Hamiltonian: a mapping into operator space (locally expressed form of H )
Matrix Product Operator 1
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hopping term gauge partmass term
possible to decompose indices into physical (i, j) and virtual one (l, r)l r
i
j wherep
June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen)
Matrix Product Operator 2• Ex. For only hopping term with N = 4, open boundary
• Automata • useful to compute the efficiently
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:hopping
where ,,
, , , ,
for left boundary
for right boundaryfor bulk n = 2,3
For the detail, G. M. Crosswhite and D. Bacon, PRA78, 012356