# Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model

## Abstract

Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time {tau}{sub Q} across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation length {xi} proportional to the square root of the quench time {tau}{sub Q}: {xi}={radical}({tau}{sub Q}). The inverse of this correlation length is known to determine average density of defects (e.g., kinks) after the transition. In this paper, we show that this same {xi} controls the entropy of entanglement, e.g., entropy of a block of L spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large L, this entropy saturates at (1/6) log{sub 2} {xi}, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturatesmore »

- Authors:

- Institute of Physics and Centre for Complex Systems Research, Jagiellonian University, Reymonta 4, 30-059 Krakow, (Poland)
- (United States)
- Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)

- Publication Date:

- OSTI Identifier:
- 20982490

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevA.75.052321; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; CORRELATION FUNCTIONS; CORRELATIONS; DECAY; DEFECTS; DISTRIBUTION; ENTROPY; EXACT SOLUTIONS; GROUND STATES; ISING MODEL; OSCILLATIONS; PARAMAGNETISM; PHASE TRANSFORMATIONS; PROBABILITY; PULSES; QUANTUM ENTANGLEMENT; SIMULATION; SPIN; WAVELENGTHS

### Citation Formats

```
Cincio, Lukasz, Dziarmaga, Jacek, Rams, Marek M., Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, and Zurek, Wojciech H.
```*Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.75.052321.

```
Cincio, Lukasz, Dziarmaga, Jacek, Rams, Marek M., Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, & Zurek, Wojciech H.
```*Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model*. United States. doi:10.1103/PHYSREVA.75.052321.

```
Cincio, Lukasz, Dziarmaga, Jacek, Rams, Marek M., Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, and Zurek, Wojciech H. Tue .
"Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model". United States.
doi:10.1103/PHYSREVA.75.052321.
```

```
@article{osti_20982490,
```

title = {Entropy of entanglement and correlations induced by a quench: Dynamics of a quantum phase transition in the quantum Ising model},

author = {Cincio, Lukasz and Dziarmaga, Jacek and Rams, Marek M. and Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 and Zurek, Wojciech H.},

abstractNote = {Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time {tau}{sub Q} across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation length {xi} proportional to the square root of the quench time {tau}{sub Q}: {xi}={radical}({tau}{sub Q}). The inverse of this correlation length is known to determine average density of defects (e.g., kinks) after the transition. In this paper, we show that this same {xi} controls the entropy of entanglement, e.g., entropy of a block of L spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large L, this entropy saturates at (1/6) log{sub 2} {xi}, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size L{approx_equal}{xi}, but -- in the subsequent evolution in the ferromagnetic phase -- a somewhat larger length scale l={radical}({tau}{sub Q}) ln {tau}{sub Q} develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when L{approx_equal}l. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation length {xi}, but (as for entropy of entanglement) in the following evolution length scale l gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined by {xi}. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.},

doi = {10.1103/PHYSREVA.75.052321},

journal = {Physical Review. A},

number = 5,

volume = 75,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}