Holographic encoding of universality in corner spectra National - - PowerPoint PPT Presentation
Holographic encoding of universality in corner spectra National Center for Theoretical Sciences (NCTS),Taiwan Ching-Yu Huang ( ) work with Romn Ors (Donostia International Physics Center) Tzu-Chieh Wei (Stony Brook university)
National Center for Theoretical Sciences (NCTS),Taiwan Ching-Yu Huang (黃靜瑜)
(TNSAA) 2018-2019 2018/12/03
work with Román Orús (Donostia International Physics Center) Tzu-Chieh Wei (Stony Brook university)
Reference: PRB 95, 195170 (2017)
CTMs and CTs.
[Landau ]
Symmetry, Order parameters, Critical points, Critical exponents and Universality classes,… h=0
disordered
H = − X
i
(σz
i σz i+1 + hσx i )
2D classical Ising model
[Gu, Levin,& Wen’ 08]
disordered
2D quantum Ising model with Transverse field h
H = − X
i
(σz
i σz i+1 + hσx i )
[Sachdev,’11]
physical properties and observables
H ∼ dN
Find the ground state (approximation) Measure physical observables (approximation) exact diagonalization (ED), Density matrix renormalization group (DMRG) Tensor network state (simple update, full update)+ Tensor network algorithm (PEPS,TRG,SRG,HOTRG,TNR,…..) …………
based on the matrix/tensor product states (MPS/TNS).
…
MPS TNS
|Ψi = X
s1,s2,...,sN
tTr(As1As2...AsN )|s1, s2, ..., sNi
finite TNS
[J. Jordan, R. Orus, G. Vidal, F. Verstraete, I. Cirac, 08’]
infinite TNS
[F. Verstraete, I. Cirac 06’]
Unit cell of tensors is repeated periodically over the whole PEPS: translational invariance
… … … …
Find the best ground stats
iterative optimization
(energy minimization) imaginary time evolution
Compute observable
Contraction of the tensor network exact / approximate
Structure (Anstaz)
1d: MPS 2d: TNS
|Ψi = X
s1,s2,....,sn
tTr(As1As2...Asn)|s1, s2, ...., sni
hΨ|Ψi
|Ψi
… … … …
hΨ|
… … … …
hΨ| ˆ O|Ψi
|Ψi
… … … …
ˆ O hΨ|
… … … …
… … … … … … … … … … … …
(double indices) Contraction of this infinite lattice To determine observables
Norm Observable
[R. Orús, G. Vidal,09’, R. Orús,12’]
Renormalized Corner Transfer Matrices CTM method
…
T1
T2 T3 T4
C4
C1 C2
C3
… … … … … … … … … … … half-row transfer matrix corner transfer matrix half-column transfer matrix
from infinite TNS
The half-row/half-column tensors T → half an infinite row/column of TNS
absorbing rows and columns towards the left, up, right, and down directions until convergence is reached
renormalize with isometries
C represent one quadrant of the 2D TNS, and the half-row/ half-column tensors T to the renormalization of half an infinite row/column of TNS
[R. Orús, G. Vidal,09’]
holographically in numerical CTMs and CTs.
method can be used to study physical system.
network
… … … …
C4 C1 C2 C3
(a)
… … … …
Z
Z = tr(C1C2C3C4),
|ψ(0)i hψ(0)| U(δτ) U(δτ) U(δτ)
U(δτ)∗ U(δτ)∗ U(δτ)∗
[R. Orús 2012] ✓ The Partition function of
classical lattice model
✓ The time-evolution of a 1d
quantum system
✓ The norm of 2d PEPS
hΨ|Ψi
[R. J. Baxte 1968; T. Nishino and K. Okunishi 1996; R. Orús 2012]
holographic information about the bulk properties of the system
“entanglement spectrum” and “entanglement Hamiltonians
2d classical Ising model Hc with an isotropic coupling K 1d quantum Ising model
Hq = −
L−1
σ [i]
x − δσ [L] x
− λ
L−1
σ [i]
z σ [i+1] z
,
[Hq, Hc] = 0
where δ = cosh 2K and λ = sinh2 K, matrix technique. The transverse field labeled
[I. Peschel, M. Kaulke, and Ö. Legeza 1999; I. Peschel 2012] [H. Li and F.D.M. Haldane 2008; J. I. Cirac, D. Poilblanc, N. Schuch, and F. Verstraete 2011]
quantum spin chain is identical to the spectrum of some Corner transfer matrices in 2d
study the properties of ground state
tensor (e.g. corner entropy) to pinpoint quantum phase transitions
[R. Orús, G. Vidal,09’, R. Orús,12’]
C4
C1 C2
C3
ρ
2d TN 3d TN
A reduced density matrix ρ of a system can be given, and we have corner spectra and corner entropy
|ψ(0)i hψ(0)| U(δτ) U(δτ) U(δτ)
U(δτ)∗ U(δτ)∗ U(δτ)∗
[R. Orús 2012]
T1
T2 T3 T4
C4 C1 C2 C3 C4 C1 C2 C3
ρ
corner spectrum
|ψfi hψf|
entanglement spectrum = CTM method
(-Ht) exp (-Ht) exp | Ψ Ψ = 〉 Ψ
τ
partition function ➜ 2d TN
from Hamiltonian ➜ (2+1)d TN from wave function (a) use the 2d quantum state renormalization ➜ 3d TN (b) the norm of PEPS ➜ 2d TN
partition function ➜ 3d TN
corner spectrum entanglement spectrum =
… … … …
C4 C1 C2 C3
(a)
… … … …
Z
C4 C1 C2 C3
ρ
A reduced density matrix ρ of a system can be given, and we have corner spectra and corner entropy
Hq = −
σ [i]
x σ [i+1] x
− h
σ [i]
z ,
…
MPS Get the ground state via imaginary time evaluation
(-Ht) exp (-Ht) exp | Ψ Ψ = 〉 Ψ
τ
Divide into small time-steps
δt ⌧ 1
|Ψ0i ! |Ψ1i ! ... ! |Ψmi ! |Ψm+1i ! ...
E0 ≥ E1 ≥ ... ≥ Em ≥ |Em+1 ≥ ...
iTEBD
… …
t(h) = Tc arcsin p 1/h
Entanglement spectra and the entanglement entropy obtained from iTEBD
Hq = −
σ [i]
x σ [i+1] x
− h
σ [i]
z ,
(-Ht) exp (-Ht) exp | Ψ Ψ = 〉 Ψ
τ
|ψ(0)i hψ(0)| U(δτ) U(δτ) U(δτ)
U(δτ)∗ U(δτ)∗ U(δτ)∗
T1 T2 T3 T4
C4 C1 C2 C3 C4
C1 C2 C3
ρ
t(h) = Tc arcsin p 1/h
Get the ground state via imaginary time evaluation
The corner spectrum obtained from the time-evolution of a 1d quantum system
… … … …
C4 C1 C2 C3
(a)
… … … …
Z
Z = tr(C1C2C3C4)
(b)
C4 C1 C2 C3
ρ
t = T/Tc
The partition function with classical Hamiltonian The critical point is
Zc = X
{s}
e−βHc({s}) Hc{s} = − X
hi,ji
s[i]s[j]
βc = 1/Tc = 1 2 log ⇣ 1 + √ 2 ⌘
hΨ|Ψi
…
C4 C1 C2 C3
…
[F. Verstraete, M. M.Wolf, D. Pérez-García, and J. I. Cirac 2006]
t(g) = − Tc log(sin(2g))
|ψ(β)i = 1 Zc e( β
2
P
hi,ji σ[i] z σ[j] z )|+, +, · · · , +i
The expectation values are the 2d classical Ising The transition occur at g ≈ 0.349596
g = 1 2 arcsin(e−β)
To set
hψ(β)|σ[i]
z σ[j] z |ψ(β)i = 1
Zc X
{s}
s[i]s[j]e−βHc({s}) = hs[i]s[j]iβ
A+
0000 = (cosh(β/2))4
A−
0010 = (cosh(β/2))3 (sinh(β/2))
A+
0110 = (cosh(β/2))2 (sinh(β/2))2
A−
1110 = (cosh(β/2)) (sinh(β/2))3
A+
1111 = (sinh(β/2))4
same type of branches on both the symmetric and the symmetry-broken phases
between the different calculations, since the different models can be mapped into each other exactly.
1d quantum Ising 2d Ising PEPS 2d classical Ising
t = T/Tc
t(g) = − Tc log(sin(2g))
t(h) = Tc arcsin p 1/h
→ the correspondence between d-dimensions quantum spin systems and classical systems in d+1 dimensions
[I. Peschel, ,1985] [M. Suzuki,1971]
1d quantum XY model 2d classical Ising model with coupling Kx Ky 1d quantum Ising model 2d classical Ising model with coupling Kx Ky Kx=Ky=k
the canonical quantum partition function can be written as a path integral in imaginary time
Zq = tr(e−βHq)
Zq = tr
= X
m
hm|e−βHq|mi
|mi : a given basis of the Hilbert space Zq = X
{m}
hm0|U|mL
−1ihmL −1|U|mL −2i · · · hm1|U|m0i
with U ⌘ e−δτHq, δτ ⌘ β/L ⌧ 1 (smaller than all time scales of Hq)
extra dimension (the imaginary-time evolution).
Zq = tr
= X
ηz
D {ηz}
E
The canonical quantum partition function of this model Splitting the imaginary time β into infinitesimal time step δτ
Hq = −Jz X
hi,ji
σ[i]
z σ[j] z − Jx
X
i
σ[i]
x = Hz + Hx
function of 2d anisotropic Ising model
Zc = X
{s}
e(
P
hi,ji Kxs[i,j]s[i,j+1]+Kys[i,j]s[i+1,j])
Comparing the result, we have Kx = Js = Jzδτ, Ky = Jτ = tanh−1(e−2δτJx) We can set Jz=1 and Jx=h and obtain the relation
tanh Ky = e−2Kxh
Zq ≈ X
{η}
C0eJs
P
α,hi,ji η[i] z (τα)η[j] z (τα) × eJτ
P
α,i η[i] z (τα+1)η[i] z (τα)
field quantum Ising model
Jτ = tanh−1 e−2δτJx Js = Jzδτ
⇒ δ = β/L → 0 (the number of sites L in
the imaginary time direction to be infinity)
⇒ Jτ →∞ and Js → 0
correspondence to a 1d quantum spin chain
be taken (in contrast to, e.g., the partition-function approach, where we had δτ → 0).
Zc = tr(D1D2...D1D2) = tr(D1D2)N/2 = tr(V )N/2
[Hq, V ] = 0 so we have δ = cosh 2K,
λ = sinh2 K
Hq = −
L−1
X
i=1
σ[i]
x − δσ[L] x
− λ
L−1
X
i=1
σ[i]
z σ[i+1] z
where δ = cosh 2K and λ = sinh2 K
Zc = X
{s}
e(K P
i,j(s[i,j]s[i,j+1]+s[i,j]s[i+1,j]))
(D1)φ,φ0 = eK(PM
j=1(sj+1s0 j+sjs0 j)),
(D2)φ0,φ00 = eK(PM
j=1(s0 js00 j +s0 js00 j+1))
[I. Peschel, ,1985]
The exact mapping is obtained in the limit kx → 0
(a) 1d quantum Ising model:
Hq = −
σ [i]
x σ [i+1] x
− h
σ [i]
z ,
(b) 2d classical isotropic Ising model from Peschel’s method
(c) (d) 2d classical anisotropic Ising model with the partition-function method tanh Ky = e−2Kxh
The exact mapping is obtained in the limit kx → 0 Via the partition-function method
Hq = −
σ [i]
x σ [i+1] x
− h
σ [i]
z ,
kz=0.1 kz=0.05 kz=0.01
for L sites
1d quantum system N=2 N=3 N=4 N=5 2d classical system
Kx = 0.01
Hq = −
L−1
X
i=1
N−1 X
n=1
⇣ Z[i]†Z[i+1]⌘n ! − h
L
X
i=1
N−1 X
n=1
⇣ X[i]⌘n ! = Hz + Hx,
where operators Z and X at every site satisfy
Z|qi = ωq|qi, X|qi = |q 1i
with ω = ei2π/N and q ∈ ZN
The partition-function method
square lattice
Zc = X
{s}
e
⇣P
hi,ji Kxδs[i,j],s[i,j+1]+Kyδs[i,j],s[i+1,j]
⌘
We have
Kx = Nδτ, tanh(δτh) = e−Ky
ground state of a 1d XY quantum spin model
classical correspondence
[M. Suzuki,1971]
There are three phases:
ferromagnetic (F), and paramagnetic (P).
Hq = − X
i
⇣ Jxσ[i]
x σ[i+1] x
+ Jyσ[i]
y σ[i+1] y
⌘ + h X
i
σ[i]
z
βHc = − X
i,j
⇣ Kxs[i,j]s[i,j+1] + Kys[i,j]s[i+1,j]⌘
is the 1d transfer matrix of the system, which contains all the Boltzmann weight factors of the spin in the adjacent rows Tφi,φi+1
Tφ,φ0 = e(Kx
P
i sisi+1+Ky
P
i sis0 i)
= eKx
P
i sisi+1 × eKy
P
i sis0 i
≡ V1V2
Zc = tr(V 1/2
2
V1V 1/2
2
)N = tr(V )N
we have The partition function can thus be thought of as a function of , and can be rewritten as
φ1,.....,φN
Zc = X
φ1
... X
φN
Tφ1,φ2...TφN−1,φN TφN.φ1
φ1
φ2 φ3
Tφ1,φ2 Tφ2,φ3
V2 = e(Ky
PM
i=1 σi zσi+1 z
)
V1 = (2 sinh 2Kx)M/2e(K∗
x
PM
i=1 σi z)
The commutator of V and Hq can be zero [V, Hq] = 0
γ = 0.5 γ = 0.9
γ = 0.99
χ = 40
1d quantum state MPS 2d classical model
h ≈ 0.85 h ≈ 0.4 h ≈ 0.1
Hq = − X
i
⇣ Jxσ[i]
x σ[i+1] x
+ Jyσ[i]
y σ[i+1] y
⌘ + h X
i
σ[i]
z
where γ = (Jx − Jy) is the anisotropy, and h the magnetic field
Jy Jx = e−4KX, h Jx = 2e−2Kx coth(2Ky)
spontaneous symmetry-breaking => Local order parameters: distinguish different phases
No local order parameters No symmetry breaking
[Landau] [Tsui, Stormer, & Gossard ’82]
intrinsic Topological Order
Symmetry protected topological order 2D Z2 Toric code 1D Haldane phase Ground state degeneracy NO Fractional statistics of quasiparticles NO Topological entanglement entropy NO Long range entanglement Short range entanglement
Topological order → Tensor category Symmetry protected topological order → Group cohomology
[Chen, Gu,Liu & Wen 2013]
phase transition?
SPT1 SPT0
tune parameter
λ
continuous first-order √ √
SPT1 SPT1 SPT1 SPT0 SPT0 SPT0
Modular matrices
need to first es {|ψai}N
a=1
be given by
hψa| ˆ S|ψbi = eαSV +o(1/V )Sab hψa| ˆ T|ψbi = eαT V +o(1/V )Tab,
:degenerate ground state
(SPT)
Quasiparticle excitations with different braiding statistics To use the symmetry twist to simulate degenerate ground state hx hy [Hung & Wen 2014]
hx,hy∈G
No degenerate ground state Modular matrices is trivial
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
t |t| |t| t
where is a tensor
|Ψi = X
si
tTr(A ⌦ A... ⌦ A)|s1, s2, ...i.
SPT1 SPT0 tune parameter t
t
1
∞
≈ ≈
symmetry breaking symmetry breaking
Z2 SPT0 Z2 SPT1
1 1 1 1 1 1 1 1 1 1 1 1
Where is the transition point?
the norm via the directional CTM approach
C4 C1 C2 C3
ρ
2 1 1 2 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0
SB SPT0 SPT1 SB
and modular T matrix
[C.-Y. Huang and Tzu-Chieh Wei 2016]
hΨ|Ψi
C4 C1 C2 C3
z-axis magnetic field
phase transition from Neel to spin- flipping phase
hs = 2(1 + Δ), the fully polarized state is reached.
Hq = −
x σ [j] x
+ σ [i]
y σ [j] y
− σ [i]
z σ [j] z
σ [i]
z ,
Neel phase spin-flipping phase fully polarized phase 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0
<M> h
<Mx> <Mz> <Mz
s>
hΨ|Ψi
C4 C1 C2 C3
nonuniversal short-range entanglement related to the microscopic details of the system
we make use of corner tensors
CTM approach but this time acting directly on the PEPS, which is single layer, and not
double layer.
≈
T1
T2 T3 T4
C4 C1 C2 C3
… … … … … … … … … … … … … … … … … … … … … …
|Ψ =
…
(e)
˜ T4
˜ C4
(a) C1
T1
C4
T3 T4 (d) (c) (f) (g)
˜ C1
(b)
=
W
U ˜ C1 C1 C4
W
U † ˜ C4
T4
W
U † U
˜ T4
C1
T4
C4
= = =
U † U
U
DU
+
˜ C1 ˜ C1
†
( ˜ C4
T )†
˜ C4
T
D
W
W †
+ =
˜ C1 ˜ C1
†
˜ C1
T
( ˜ C1
T )†
known to be critical and has a chiral gapless edge described by a SU(2) Wess-Zumino- Witten (WZW) CFT.
PEPS with D = 3 on a square lattice exactly.
been characterized by studying the entanglement spectrum
but finite circumference cylinder
A−1/2
2,0,1,1 = −λ1 − iλ2,
A−1/2
2,1,1,0 = −λ1 + iλ2,
A−1/2
2,1,0,1 = −λ0;
A−1/2
1,1,2,0 = −λ1 − iλ2,
A−1/2
1,0,2,1 = −λ1 + iλ2,
A−1/2
0,1,2,1 = −λ0;
A−1/2
1,2,0,1 = λ1 + iλ2,
A−1/2
0,2,1,1 =
λ1 − iλ2, A−1/2
1,2,1,0 = λ0;
A−1/2
0,1,1,2 = λ1 + iλ2,
A−1/2
1,1,0,2 = λ1 − iλ2,
A−1/2
1,0,1,2 = λ0;
A1/2
2,1,0,0 = λ1 + iλ2,
A1/2
2,0,0,1 = λ1 − iλ2,
A1/2
2,0,1,0 = λ0;
A1/2
0,0,2,1 = λ1 + iλ2,
A1/2
0,1,2,0 = λ1 − iλ2,
A1/2
1,0,2,0 = λ0;
A1/2
0,2,1,0 = −λ1 − iλ2,
A1/2
1,2,0,0 = −λ1 + iλ2,
A1/2
0,2,0,1 = −λ0;
A1/2
1,0,0,2 = −λ1 − iλ2,
A1/2
0,0,1,2 = −λ1 + iλ2,
A1/2
0,1,0,2 = −λ0,
(66) where λ0 = −2, λ1 = 1,and λ2 = 1. Here we have computed the entanglement spectrum of this
SU(2)1 SU(2)2
Using 2d QSRG
[D. Poilblanc, N. Schuch, and I. Affleck 2016
quantum phase transitions
C4 C1 C2 C3
ρ
γ = 0.5 γ = 0.9 γ = 0.99
C4
C1 C2
C3