Quantum Formation of Topological Defects

Mainak Mukhopadhyay Department of , Arizona State University (ASU)

Based on arXiv : 2004.07249 (Phys. Rev. D 102, 056021) Emergence of classical structures from the quantum vacuum MM, Tanmay Vachaspati and George Zahariade And arXiv : 2009.11480 (Phys. Rev. D 102, 116002) Quantum Formation of Topological Defects MM, Tanmay Vachaspati and George Zahariade

2021 Meeting of the Division of Particles and Fields of the American Physical Society (DPF21) July 12, 2021

1 Topological Defects

QFT

Small quantum excitations around a Large “classical” structures - true vacuum - Defects Particles

Stable, non-dissipative solutions

Have a topological charge that is conserved.

E.g.: Kinks Strings Magnetic Monopoles

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 2 Motivations

- Liquid He-4

- Liquid Crystals

- Superconductors

- Superfuids

- May be in Early cosmology

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 3 Topological Defects - How are they produced?

Phase transition : internal global symmetry is spontaneously broken

1 2 1 2 λ 4 L = dx (∂μϕ) − m2(t)ϕ − ϕ t ∫ ( 2 2 4 ) m (t) = − m2 tanh 2 ( τ )

“Quench time scale” External parameter

Often topological defect formation is studied by the Kibble-Zurek proposal in the case thermal phase transitions

T. Kibble, of cosmic domains and strings, J. Phys. A 9, 1387 (1976) W. Zurek, Cosmological experiments in superfuid ? Nature (London) 317, 505 (1980) Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 4 Topological Defects - How are they produced?

Phase transition : internal global symmetry is spontaneously broken

1 2 1 2 λ 4 L = dx (∂μϕ) − m2(t)ϕ − ϕ t ∫ ( 2 2 4 ) m (t) = − m2 tanh 2 ( τ )

symmetry Z2 spontaneously broken

Kinks

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 5 Topological Defects - How are they produced?

1 2 1 2 λ 4 L = dx (∂μϕ) − m2(t)ϕ − ϕ ∫ ( 2 2 4 )

Phase transition at t = 0

ϕ = 0 ϕ = − η ϕ = η t < 0 t > 0 Unique vacuum Degenerate vacua symmetry Z2 spontaneously broken

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 6 Kinks

ϕ(x) Degenerate vacua: ±η +η

x

−η

m mx ϕ±(x) = ± tanh t → ∞ λ ( 2 )

T. Vachaspati, Kinks and Domain Walls: An Introduction to Classical and Quantum (Cambridge University Press, Cambridge, England, 2010)

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 7 Kinks - Our Goal

ϕ(x) Degenerate vacua: ±η +η

x

Count the number of zeros −η ?

Number density of kinks

T. Vachaspati, Kinks and Domain Walls: An Introduction to Classical and Quantum Solitons (Cambridge University Press, Cambridge, England, 2010)

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 8 Kinks - Our Model

1 2 1 2 λ 4 L = dx (∂μϕ) − m2(t)ϕ − ϕ ∫ ( 2 2 4 )

V(ϕ) V(ϕ)

Phase transition at t = 0 m2 < 0

ϕ ϕ

m2 > 0

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 9 How to count the zeros?

Discretized feld theory

ϕ0 ϕN ϕ ϕ 1 N−1 ϕ2 N 1 · 1 1 L = a ϕ2 + ϕ ϕ − 2ϕ + ϕ + m (t)ϕ2 a disc. ∑ j 2 j( j+1 j j−1) 2 j j=1 [2 2a 2 ]

ϕj+1

ϕj 2 1 N 1 n̂ ≡ sgn ϕ̂ − sgn ϕ̂ Z ∑ [ ( j) ( j+1)] L j=1 4 Quantum operator Counts the number of sign changes between adjacent points on the lattice

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 10 How to count the zeros?

2 1 N 1 n̂ ≡ sgn ϕ̂ − sgn ϕ̂ Z ∑ [ ( j) ( j+1)] L j=1 4

Translational invariance

N n = ⟨n̂ ⟩ = 1 − sgn ϕ̂ ϕ̂ Z Z 2L [ ⟨ ( 1 2)⟩]

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 11 Mode Functions

N 1 · 1 1 L = a ϕ2 + ϕ ϕ − 2ϕ + ϕ + m (t)ϕ2 disc. ∑ j 2 j( j+1 j j−1) 2 j j=1 [2 2a 2 ]

4 πn c·· + sin2 + m (t) c = 0 n [ a2 ( N ) 2 ] n

Mode −1/4 −i 4 πn functions c (t ) = sin2 + m (t ) n 0 2 2 0 2L [ a ( N ) ] 1/4 1 4 πn c· (t ) = sin2 + m (t ) n 0 2 2 0 2L [ a ( N ) ]

After the phase transition the modes split into Stable and Unstable modes 4 πn depending on the sign of sin2 + m (t) , since m (t) can be both [ a2 ( N ) 2 ] 2 positive or negative

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 12 How to count the zeros?

N n = ⟨n̂ ⟩ = 1 − sgn ϕ̂ ϕ̂ Z Z 2L [ ⟨ ( 1 2)⟩]

Black box (Schrodinger equation, wave function, probability distribution…..)

∑ |c |2 cos(2πn/N) N 2 −1 all modes n nZ = 1 − sin 2 2L π ∑all modes |cn | Number density of zeros

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 13 How to count the number density of kinks? Number density of zeros

∑ |c |2 cos(2πn/N) N 2 −1 all modes n nZ = 1 − sin 2 2L π ∑all modes |cn |

Stable, oscillating modes correspond to small fuctuations, hence not Kinks

∑ |c |2 cos(2πn/N) N 2 −1 unstable modes n nK = 1 − sin 2 2L π ∑unstable modes |cn | Number density of kinks

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 14 Log-Log plot of vs. t for diferent ⟨nk⟩ τ nK τ = 0.1 τ = 0.5 0.100 τ = 1.0

τ = 5.0 0.050 τ = 10.0

∼t-1/2

0.010

0 10 50 100 250 550 1050 t

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 15 Log-Log plot of vs. t for diferent ⟨nk⟩ τ nK

0.100

0.050

∼t-1/2 Phase transition triggers creation of kinks 0.010

0 10 50 100 250 550 1050 t

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 16 Log-Log plot of vs. t for diferent ⟨nk⟩ τ nK Faster Phase transition “more” kinks 0.100

0.050

∼t-1/2 Phase transition triggers creation of kinks 0.010

0 10 50 100 250 550 1050 t

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 17 Log-Log plot of vs. t for diferent ⟨nk⟩ τ nK

0.100 Kinks and anti kinks annihilate

0.050

∼t-1/2

0.010

0 10 50 100 250 550 1050 t

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 18 Number Density of Kinks

nK

ΔnK 0.100

0.050 0.100 ∼t-1/2

0.010

t 0.010 0 10 50 100 250 550 1050 At late times….,

-3/2 0.001 ∼t

-4 10 m −3/2 nK(t) = C + �(t ) 10-5 τ1 = 0.1 t 10-6 t 0 10 50 100 250 550 1050 1 Log-Log plot of vs. t ΔnK = ⟨nk(t, τ1)⟩ − ⟨nk(t, τ2)⟩ C = ≈ 0.22 2 π Kibble-Zurek proposal : Thermal phase transitions Independent of At late times…, τ ∼ τ−1/4

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 19 Number Density of Kinks

nK

(n ) 0.100 K max t 0.050 max 0.175 -1/2 0.150 ∼t 5 0.010 0.125

0 10 50 100 250 550 1050 t 0.34 0.100 ∼τ 2 0.075 ∼τ-0.33 1 0.050

0.5

τ 0.1 0.5 1 5 10 50 100 τ 0.1 0.5 1 5 10 50 100 Log-Log plot of vs. Log-Log plot of vs. (nK)max τ tmax τ

Faster the phase transition (smaller quench timescale), the more kinks are produced and quicker they start annihilating

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 20 Topological Defects in Higher

nV nM 0.050 0.010

0.010 0.001 0.005

-1 10-4 -3/2 0.001 ∼t ∼t 5.×10-4 10-5

1.×10-4 t 0 10 50 100 250 550 1050 t 0 10 50 100 250 550 1050

2 Dimensions: 3 Dimensions: Vortices Monopoles

In d dimensions: d/2 d! m n = + � (t−(d+2)/2) D 2d/2πd ( t )

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 21 Efect of self-interactions

Hartree Approximation : λϕ4 ∼ 3λ⟨ϕ2⟩ϕ2

3 3 Efective mass : mef(t) = m (t) + λ⟨ϕ2⟩ − λ⟨ϕ2⟩ 2 2 2 in 2 Mass renormalization ef m2 (t) ≈ m2(t) ‘in’ denotes Initial time, t → − ∞

2 2 3λ [⟨ϕ ⟩ − ⟨ϕ ⟩in] ≪ 2|m2 |

Violated at t ∼ 0, the time for the violation needs to be small λτ/m ≪ 1

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 22 Summary

- A detailed numerical and analytical analysis of the dynamics of topological defect formation in a QFT where the only interactions are with external parameters that induce the phase transition.

- The number density of topological defects in d spatial dimensions scales as t−d/2 and does not depend on the quench timescale, in the late time limit.

- The analytical results for sudden phase transition is a universal attractor.

- For, d=1 dimensions the limit where our theory is a good approximation even for λ ≠ 0 is, λτ/m ≪ 1.

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 23 Summary

Initial State: Translationally invariant “quantum” vacuum.

Final State: Translationally invariant “quantum” state. But each “classical” realization/observation breaks translational invariance and we have objects with defnite positions and velocities.

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 24 Thank You!

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 25 Backup

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 26 Efect of self-interactions

λτ/m ≪ 1

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 27 Classical Quantum Correspondence (CQC)

p2̂ 1 Ĥ = + mω(t)2q2̂ 2m 2

Time dependent frequency

T. Vachaspati and G. Zahariade, Phys. Rev. D 98, 065002 (2018) T. Vachaspati and G. Zahariade, J. Cosmol. Astropart. Phys. 09 (2019) 015

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 28 Classical Quantum Correspondence (CQC)

p2̂ 1 Ĥ = + mω(t)2q2̂ 2m 2

† · · † Operators: q̂ = z*a0̂ + za0̂ p̂ = z*a0̂ + za0̂

Equation of Motion: ··z + ω(t)2z = 0

Initial Conditions:

1 mω z(0) = z·(0) = 0 i 2mω0 2

T. Vachaspati and G. Zahariade, Phys. Rev. D 98, 065002 (2018) T. Vachaspati and G. Zahariade, J. Cosmol. Astropart. Phys. 09 (2019) 015

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 29 Classical Quantum Correspondence (CQC)

Quantum 1D Harmonic Oscillator in its ground state (q)̂

Classical 2D Harmonic Oscillator with specifed angular momentum (z(t))

Works for arbitrary backgrounds ω(t)

T. Vachaspati and G. Zahariade, Phys. Rev. D 98, 065002 (2018) T. Vachaspati and G. Zahariade, J. Cosmol. Astropart. Phys. 09 (2019) 015

Quantum Formation of Topological Defects July 12, 2021 Mainak Mukhopadhyay 30