Entanglement in quantum spin chains, symmetry classes of random matrices, and conformal field theory

spins and the rest of the system in the ground states of a general class of quantum spin-chains. We show that under certain conditions the entropy can be expressed in terms of averages over ensembles of random matrices. These averages can be evaluated, allowing us to prove that at critical points the entropy grows like

\kappa\log_2 N + {\tilde \kappa}

as

N\to\infty

, where

\kappa

and

{\tilde \kappa}

are determined explicitly. In an important class of systems,

\kappa

is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for

\kappa

therefore provides an explicit formula for the central charge.

Note:

4 pages

03.67.-a
75.10.Pq
02.10.Yn
11.25.Hf
model: spin
matrix model: random
field theory: conformal
Hamiltonian formalism
energy
entanglement

References(21)

Figures(0)

[1]

Concentrating partial entanglement by local operations

Charles H. Bennett
(
- IBM Watson Res. Ctr.
)
,
Herbert J. Bernstein
(
- Hampshire Coll.
)
,
Sandu Popescu
(
- Tel Aviv U.
)
,
Benjamin Schumacher
(
- Kenyon Coll.
)

- Phys.Rev.A 53 (1996) 2046-2052
•
e-Print:
- quant-ph/9511030
•
DOI:
- 10.1103/PhysRevA.53.2046

[2]

L. Amico
,
A. Osterloh
,
F. Plastina
,
R. Fazio
,
G.M. Palma

- Phys.Rev.A 69 (2004) 022304

[2]

Quantum to classical phase transition in noisy quantum computers

Dorit Aharonov
(
- LBL, Berkeley
)

- Phys.Rev.A 62 (2000) 6, 062311
•
DOI:
- 10.1103/PhysRevA.62.062311

[2]

Entanglement in a simple quantum phase transition

Tobias J. Osborne
(
- Queensland U. and
- Unlisted
)
,
Michael A. Nielsen
(
- Unlisted and
- Queensland U.
)

- Phys.Rev.A 66 (2002) 032110
•
e-Print:
- quant-ph/0202162
•
DOI:
- 10.1103/PhysRevA.66.032110

[2]

Scaling of entanglement close to a quantum phase transition

- Nature 416 (2002) 6881, 608-610
•
DOI:
- 10.1038/416608a

[3]

Entanglement in quantum critical phenomena

- Phys.Rev.Lett. 90 (2003) 227902
•
e-Print:
- quant-ph/0211074
•
DOI:
- 10.1103/PhysRevLett.90.227902

[4]

Quantum Spin Chain, Toeplitz Determinants and the Fisher—Hartwig Conjecture

- J.Statist.Phys. 116 (2004) 1, 79-95
•
DOI:
- 10.1023/B:JOSS.0000037230.37166.42

[5]

Universality of Entropy Scaling in One Dimensional Gapless Models

Vladimir Korepin
(
- YITP, Stony Brook
)

- Phys.Rev.Lett. 92 (2004) 096402
•
e-Print:
- cond-mat/0311056
•
DOI:
- 10.1103/PhysRevLett.92.096402

[6]

J. Stat

P. Calabrese
,
J. Cardy

- J.Stat.Mech. 2004 (2004) P06002

[7]

Entanglement and the Phase Transition in Single-Mode Superradiance

- Phys.Rev.Lett. 92 (2004) 073602
•
e-Print:
- quant-ph/0309027
•
DOI:
- 10.1103/PhysRevLett.92.073602

[8]

F. Verstraete
,
M.A. Martin-Delgado
,
J.I. Cirac

- Phys.Rev.Lett. 92 (2004) 087201

[9]

Geometric and renormalized entropy in conformal field theory

Christoph Holzhey
(
- Princeton U.
)
,
Finn Larsen
(
- Princeton U.
)
,
Frank Wilczek
(
- Princeton, Inst. Advanced Study
)

- Nucl.Phys.B 424 (1994) 443-467
•
e-Print:
- hep-th/9403108
•
DOI:
- 10.1016/0550-3213(94)90402-2

[10]

The central charge of a conformal field theory may be defined (see, e.g., [5]) to be the limit as z tends to zero of z4 < T (z)T (0) >, where < T (z)T (0) > is the autocorrelation function of the zz-component of the energymomentum tensor Tμ,ν (z), z is the complex space-time variable z = x + ivt, and v is the Fermi velocity. Roughly speaking, the central charge counts the number of gapless degrees of freedom in the field theory

[11]

J.P. Keating
,
F. Mezzadri

- Commun.Math.Phys. 252 (2004) 543

[12]

Two soluble models of an antiferromagnetic chain

- Annals Phys. 16 (1961) 407-466
•
DOI:
- 10.1016/0003-4916(61)90115-4

[13]

Relativistic Quantum Fields (MacGraw-Hill, New York,), p. 181

J.D. Bjorken
,
S.D. Drell

[14]

Handbuch der Kugelfunktionen G. Reimer

E. Heine

[15]

̋o. Orthogonal Polynomials (AMS, Providence,)

G. Szeg

[16]

M.E. Fisher
,
R.E. Hartwig

- Adv.Chem.Phys. 15 (1968) 333

[17]

in Toeplitz Matrices and Singular Integral Equations, A. Bottcher, I. Gohberg and P. Junghanns (eds.)

E.L. Basor
,
T. Ehrhardt

[18]

P.J. Forrester
,
N.E. Frankel

- J.Math.Phys. 45 (2004) 2003