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. Jansen H. Saito et al. PoS Lattice, 302 (2014), arXiv:1412.0596 M. C. Bañuls et al, arXiv:1505.00279
34
Embed
Tensor network approach for chiral symmetry restoration of 1 ......June 22, 2015 @ DESY & HU Lattice seminar H. Saito (NIC, DESY Zeuthen) Tensor network approach for chiral symmetry
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
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
8
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
9
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)
11
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
12
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
13
=
=
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)
14
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
16
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
18
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:
19
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)