Dynamical topological transitions in the massive Schwinger model - - PowerPoint PPT Presentation

Dynamical topological transitions in the massive Schwinger model

Torsten V. Zache Cold quantum coffee May 21, 2019, Heidelberg

Based on TVZ et al, Phys. Rev. Lett., 122(5), 050403 (2019)

Motivation: quantum simulation/computing

Torsten V. Zache, ITP Heidelberg

Quantum systems are exponentially complex: Quantum speed up for computationally hard problems, e.g.

• Searching a large database („Grover‘s search“)

vs.

• Factorizing prime numbers („Shor‘s algorithm“)
• Quantum Fourier transform

vs. Quantum computation/simulation: simulate quantum physics with a quantum system!

Shor, SIAM review, 41(2), 303-332 (1999)

O( √ N)

O(N)

O(n2)

O(n2n)

dim

• H⊗p

= (dim H)p

Grover, arXiv:9605043 (1996) Manin, Sovetskoye Radio, Moscow, 128 (1980) Feynman, Int. J. Theor. Phys. 21(6), 467-488 (1982) 2

Outline

Torsten V. Zache, ITP Heidelberg

• Quantum simulation of lattice gauge theories
• The strong CP problem in QCD
• Dynamical quantum phase transitions
• Dynamical topological transitions in the massive

Schwinger model

3

Quantum simulation of high-energy physics

Torsten V. Zache, ITP Heidelberg

Many proposals with different approaches on various platforms: trapped ions, superconducting qubits,.. (digital) ultracold gases,.. (analog)

4 Ion chip trap, IONQ, picture from www.sciencemag.org (Jan. 10 2018) Google‘s superconducting chip, picture from physicsworlds.com (Jun. 10 2016) quantum gas mircoscope picture from Bakr et al. Nature, 462(7269), 74 (2009) Büchler et al., Physical review letters, 95(4), 040402 (2005) Wiese, Annalen der Physik, 525(10-11), 777-796 (2013) Tagliacozzo et al., Nature communications, 4, 2615 (2013) Zohar, Cirac & Reznik, Reports on Progress in Physics, 79(1), 014401 (2015) … Byrnes & Yamamoto, Physical Review A, 73(2), 022328 (2006) Jordan, Lee & Preskill, Science, 336(6085), 1130-1133 (2012) Marcos et al., Physical review letters, 111(11), 110504 (2013) Zohar et al., Physical review letters, 118(7), 070501 (2017) …

Implementation of lattice gauge theories

Torsten V. Zache, ITP Heidelberg

Limited realizations so far:

• ne-dimensional systems, few lattice sites, ground state or short time

dynamics, .. Real-world phenomena (higher dimensions, non- abelian gauge groups) lie in the far future – what can we do now?

TVZ et al., Quantum science and technology, 3(3), 034010 (2018) Martinez et al., Nature, 534(7608), 516 (2016) … Klco et al., Phys. Rev. A, 98(3), 032331 (2018) Kokail et al., arXiv:1810.03421 (2018) Schweizer et al., arXiv:1901.07103 (2019) … 5

The strong CP problem

Torsten V. Zache, ITP Heidelberg

LQCD = ¯ ψa (iγµDµ − m) ψa − 1 4F a

µνF a,µν − θg2Nf

32π2 F a

µν ˜

F a,µν

QCD with a ‘vacuum angle‘: experimental fact: QCD conserves CP … but why??? Peccei & Quinn: The angle could be a dynamical field (the ‘axion‘) …

|θ| . 10−10

θ = hˆ θi

Peccei & Quinn, Physical Review Letters, 38(25), 1440 (1977) Chupp et al Reviews of Modern Physics 91(1), 015001 (2019)

Violates combined charge (C) and parity (P) symmetry!

6

Setup: quench dynamics in QED2

Vacuum structure of QED2: Hamiltonian with a topological term: Use the axial anomaly to rewrite it: Consider a general quench and calculate real-time dynamics:

θ → θ0

|ψ(t)i = e−iHθ0t |Ω(θ)i

Torsten V. Zache, ITP Heidelberg

R ∼ S1 → U(1) ∼ S1

⌫ = e 4⇡ Z d2x ✏µνF µν

Hθ = Z

x

⇢ ¯ ψx ⇥ iγ1 (∂x − ieAx) + m ⇤ ψx + 1 2E2

x + eθ

2π Ex

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Hθ = Z

x

⇢ ¯ ψx ⇥ iγ1 (∂x − ieAx) + m eiθγ5⇤ ψx + 1 2E2

x

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Coleman, Annals Phys. 101, 239 (1976) 7

Partition function in thermal equilibrium: Thermal phase transition at

Dynamical quantum phase transitions

Torsten V. Zache, ITP Heidelberg

Z(β) = Tr ⇥ e−βH⇤ = e−V f(β)

L(t) = ⌦ ψ(0)|e−iHt|ψ(0) ↵ = e−V Γ(t)

Loschmidt amplitude out

• f equilibrium:

Dynamical quantum phase transition (DQPT) at

f(βc)

βc

Γ(tc)

tc

Non-analyticity

Heyl, Polkovnikov & Kehrein, Physical review letters, 110(13), 135704 (2013). 8

Topological insulators: the SSH model

Torsten V. Zache, ITP Heidelberg

Spinless fermions with staggered hopping as a model for polyacetylene: effective field theory: free Dirac fermions

ψx ← 1 √a ✓ cn,A cn,B ◆ (a → 0)

H → Z dx [ψx (σxm − iσy∂x) ψx]

Asboth et al. A short course on topological insulators arXiv:1509.02295

Two gapped (insulating) phases distinguished by their topology:

H = X

n

h v ⇣ c†

n,Acn,B + h.c.

⌘ + w ⇣ c†

n,Bcn+1,A + h.c.

⌘i

m = v − w , a = 1 w

Su, Schrieffer, Heeger.

• Phys. Rev. Lett. 42(25), 1698 (1979)

9

Torsten V. Zache, ITP Heidelberg

Results

10

Real-time topology without interactions

Time-ordered correlation function: count (anti-)vortices in plane: g(x, t) = hψ†

x(t) e−ie R x

0 dx0Ax0(t) ψ0(0)i

n±(t) = I

C±(t)

dz 2π ⇥ ˜ g†(z)rz˜ g(z) ⇤

z = (k, t)

˜ g(z) = g(k, t) |g(k, t)|

Torsten V. Zache, ITP Heidelberg 11

Dynamical topological quantum phase transitions

(a) (b)

Rate function of the Loschmidt amplitude: Topological invariant of the time-ordered correlator: In the absence of interactions: ν(t) = n+(t) − n−(t)

Lθ→θ0(t) = Y

k

gθ→θ0(k, t)

Related work: Budich and Heyl, Phys. Rev. B 93, 085416 (2016)

|Lθ→θ0(t)| = |hψ(0)|ψ(t)i| = e−V Γθ!θ0(t)

Torsten V. Zache, ITP Heidelberg 12

Interpretation with equal-time correlations

• Loschmidt amplitude & two-time correlators are difficult to measure!
• gaussian state:
• spin vector :
• Loschmidt echo:

F = (Fs, F1, F5)

Fxy(t) = h ⇥ ψx(t), ¯ ψy(t) ⇤ i

|Lθ→θ0(t)|2 = Kθ→θ0(t) = Y

k

[F(k, t) + F(k, 0)]2

Torsten V. Zache, ITP Heidelberg 13

Results of a numerical simulation (exact diagonalization) on a lattice with sites and lattice spacing ü order parameter robustly predicts a shift of the topological transition ü good agreement of the transition times from different rate functions

am = 0.8

Numerics: Towards strong coupling

(a) (b) (c) N = 8

Torsten V. Zache, ITP Heidelberg 14

Numerics: Lattice size dependence

4 6 8 10 12 14 16 18 20 Lattice size N 0.8 1.0 1.2 Time tm

e/m = 0.5 Loschmidt amplitude L equal time correlator K unequal time correlator g

• rder parameter ν

4 6 8 10 12 14 16 18 20 Lattice size N 0.8 1.0 1.2 1.4 Time tm

e/m = 1 Loschmidt amplitude L equal time correlator K unequal time correlator g

• rder parameter ν
• all rate functions indicate

similar transition times

• rate functions exhibit

fluctuating finite size errors

• order parameter robust due

to topological origin

Torsten V. Zache, ITP Heidelberg 15

• Noninteracting case: correlation functions and DQPT

ü Rigorous connection between correlators and DQPT ü Physical interpretation of dynamical transitions ü Definition of a dynamical topological invariant

• Interacting case: dynamical topological order parameter

ü gauge invariant ü integer-valued ü accessible on small lattices

• To Do:

Ø Experimental realization (cold atoms and/or trapped ions) Ø Application to other models (condensed matter) Ø Extension to more realistic scenarios (e.g. QCD with dynamical axions)

Summary & outlook

(a) (b)

(a) (b

Torsten V. Zache, ITP Heidelberg 16

Presented work done in collaboration with

• N. Mueller, J. T. Schneider, F. Jendrzejewski, J. Berges, and P. Hauke
• Phys. Rev. Lett., 122(5), 050403 (2019)

Torsten V. Zache, ITP Heidelberg

Supplementary material

18

Fermions in a gauge field background: Chiral rotation and axial vector current: (Non-)conservation law: Fermion path integral:

The axial anomaly in QED

Torsten V. Zache, ITP Heidelberg

jµ

5 = ¯

ψγµγ5ψ

γ5 = iγ0γ1γ2γ3

ψ → eiαγ5ψ

∂µjµ

5 = 2im ¯

ψγ5ψ + e2 8π2 Fµν ˜ F µν

eiS[A] = Z DψD ¯ ψ eiS[A,ψ, ¯

ψ]

S[A, ψ, ¯ ψ] = Z d4x ¯ ψ [iγµ (∂µ − ieAµ) − m] ψ

Adler, Physical Review, 177(5), 2426-2438 (1969) Bell & Jackiw, Il Nuovo Cimento A, 60(1), 47-61 (1969)

˜ F µν = 1 2✏µνρσFρσ

19

Yang-Mills Hamiltonian (in temporal-axial gauge): (classical) zero-energy solutions are ‘pure gauge‘: Topological classification: ‘Theta‘ vacua:

Topology of gauge fields

Torsten V. Zache, ITP Heidelberg

LYM = −1 4F a

µνF a,µν Aa

0=0

= ⇒ HYM = 1 2

• E2 + B2

AU (x) = i g U (x) rU † (x) ) E = 0 , B = 0 , HYM = 0

U (x) : R3 ∼ S3 → SU(2) ∼ S3

|θi = X

ν∈Z

eiνθ|νi

ν = g2 32π2 Z d4x F a

µν ˜

F a,µν ∈ Z

Violates combined charge (C) and parity (P) symmetry!

Jackiw & Rebbi. in Instantons In Gauge Theories (25-28) (1994) 20