Spectral Form Factor as an OTOC Averaged over the Heisenberg Group - - PowerPoint PPT Presentation

Spectral Form Factor Application: OTOC Conclusion

Spectral Form Factor as an OTOC Averaged over the Heisenberg Group

Chen-Te Ma

Cape Town University and South China Normal University Robert de Mello Koch (SCNU and Witwatersrand), Jiahui Huang (SCNU), and Hendrik J. R. Van Zyl (Witwatersrand)

May 27, 2019

Spectral Form Factor Application: OTOC Conclusion

Spectral Form Factor

• The spectral form factor (SFF) is

g2(β, t) ≡ R2(β, t) R2(β, 0), (1) where R2(β, t) ≡ |Tr

• Z(β, t)
• |2

(2) is the unnormalized two-point SFF, β is the inverse temperature, H is the Hamiltonian of the system, and Z(β, t) ≡ exp(−βH − iHt). (3)

Spectral Form Factor Application: OTOC Conclusion

Motivation

• The motivation for the SFF has rooted in the random matrix

theory and information loss.

Spectral Form Factor Application: OTOC Conclusion

Motivation

• The motivation for the SFF has rooted in the random matrix

theory and information loss.

• It was conjectured that a generic quantized system with a

classical chaotic limit should exhibit the spectral statistics of a random matrix ensemble. This was confirmed from Sinai’s billiard.

Spectral Form Factor Application: OTOC Conclusion

Motivation

• The motivation for the SFF has rooted in the random matrix

theory and information loss.

• It was conjectured that a generic quantized system with a

classical chaotic limit should exhibit the spectral statistics of a random matrix ensemble. This was confirmed from Sinai’s billiard.

• The Sachdev-Ye-Kitaev (SYK) model provides the consistent

universal dynamical form with the random matrix theory.

Spectral Form Factor Application: OTOC Conclusion

Motivation

• The motivation for the SFF has rooted in the random matrix

theory and information loss.

• It was conjectured that a generic quantized system with a

classical chaotic limit should exhibit the spectral statistics of a random matrix ensemble. This was confirmed from Sinai’s billiard.

• The Sachdev-Ye-Kitaev (SYK) model provides the consistent

universal dynamical form with the random matrix theory.

• The issue of information loss can be probed by the late time

study in the SFF from the violation of bound.

Spectral Form Factor Application: OTOC Conclusion

Reference of the Spectral Form Factor

• E. Dyer and G. Gur-Ari, “2D CFT Partition Functions at Late

Times,” JHEP 1708, 075 (2017) [arXiv:1611.04592 [hep-th]].

• J. S. Cotler et al., “Black Holes and Random Matrices,”

JHEP 1705, 118 (2017) Erratum: [JHEP 1809, 002 (2018)] [arXiv:1611.04650 [hep-th]].

• O. Bohigas, M. J. Giannoni and C. Schmit, “Characterization
• f chaotic quantum spectra and universality of level

fluctuation laws,” Phys. Rev. Lett. 52, 1 (1984).

Spectral Form Factor Application: OTOC Conclusion

OTOC

• The out-of-time ordered correlation function (OTOC) is

defined by the square of commutator of two operators in a bosonic system C4(t) ≡ Tr

• ρW (t)V (0)W (t)V (0)
• Trρ

, (4) where ρ ≡ exp(−βH).

Spectral Form Factor Application: OTOC Conclusion

Reference of the OTOC

• A. I. Larkin and Yu. N. Ovchinnikov, “Quasiclassical Method

in the Theory of Superconductivity,” JETP 28, 1200 (1969).

Spectral Form Factor Application: OTOC Conclusion

Regularization

• It has been shown that the unregularized OTOC does not

share the universal Lyapunov exponent with the regularized OTOC due to the sensitivity of the infrared regulator. In the SYK model at the large-q limit, the universal Lyapunov exponent can be captured by the regularized OTOC. Hence the regularized OTOC should be better for the universal

• meaning. The regularized OTOC is

Cr4(t) ≡ Tr

• ρ1/4W (t)ρ1/4V (0)ρ1/4W (t)ρ1/4V (0)
• Trρ

. (5)

Spectral Form Factor Application: OTOC Conclusion

Reference of the Regularization

• J. Maldacena, S. H. Shenker and D. Stanford, “A bound on

chaos,” JHEP 1608, 106 (2016) [arXiv:1503.01409 [hep-th]].

• A. M. Garc´

ıa-Garc´ ıa, B. Loureiro, A. Romero-Berm´ udez and

• M. Tezuka, “Chaotic-Integrable Transition in the

Sachdev-Ye-Kitaev Model,” Phys. Rev. Lett. 120, no. 24, 241603 (2018) [arXiv:1707.02197 [hep-th]].

• N. Tsuji, T. Shitara and M. Ueda, “Bound on the exponential

growth rate of out-of-time-ordered correlators,” Phys. Rev. E 98, 012216 (2018) [arXiv:1706.09160 [cond-mat.stat-mech]].

Spectral Form Factor Application: OTOC Conclusion

Reference of the Regularization

• Y. Liao and V. Galitski, “Nonlinear sigma model approach to

many-body quantum chaos: Regularized and unregularized

• ut-of-time-ordered correlators,” Phys. Rev. B 98, no. 20,

205124 (2018) [arXiv:1807.09799 [cond-mat.dis-nn]].

• A. Romero-Berm´

udez, K. Schalm and V. Scopelliti, “Regularization dependence of the OTOC. Which Lyapunov spectrum is the physical one?,” arXiv:1903.09595 [hep-th].

Spectral Form Factor Application: OTOC Conclusion

Reference of the Observation

• B. Swingle, G. Bentsen, M. Schleier-Smith and P. Hayden,

“Measuring the scrambling of quantum information,” Phys.

• Rev. A 94, no. 4, 040302 (2016) [arXiv:1602.06271

[quant-ph]].

• N. Y. Yao, F. Grusdt, B. Swingle, M. D. Lukin,
• D. M. Stamper-Kurn, J. E. Moore and E. A. Demler,

“Interferometric Approach to Probing Fast Scrambling,” arXiv:1607.01801 [quant-ph].

• M. G¨

arttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall,

• J. J. Bollinger and A. M. Rey, “Measuring out-of-time-order

correlations and multiple quantum spectra in a trapped ion quantum magnet,” Nature Phys. 13, 781 (2017) [arXiv:1608.08938 [quant-ph]].

Spectral Form Factor Application: OTOC Conclusion

SFF and OTOC in Qubit Models

• Consider a quantum system in an L-dimensional Hilbert space.

Recall the average over L × L unitary matrices with the Haar measure is

• dA Aj

kA† l m = 1

Lδj

mδl k .

(6) The integral over A is over all possible unitary operators on the Hilbert space.

• In terms of the regularized two-point OTOC

O(t) ≡ Tr

• A(0)√ρA†(t)√ρ
• /L, it is clear that
• dA O(t)

= 1 L

• dA Tr(A√ρe−iHtA†eiHt√ρ)

= R2(β/2, t) . (7)

Spectral Form Factor Application: OTOC Conclusion

Heisenberg Group Averaging

• A general element of the Heisenberg group is specified by the

variables, q1, q2, as follows U(q1, q2) ≡ exp(iq1X + iq2P). By direct computation, we find ∞

−∞

dq1 2π ∞

−∞

dq2 x1|U(q1, q2)|x2y1|U†(q1, q2)|y2 = δ(x2 − y1) δ(x1 − y2) . (8) What we obtained precisely follows the properties: exp(iqX)|x = exp(iqx)|x and exp(iqP)|x = |x − q.

Spectral Form Factor Application: OTOC Conclusion

Heisenberg Group Averaging

• A general element of the Heisenberg group is specified by the

variables, q1, q2, as follows U(q1, q2) ≡ exp(iq1X + iq2P). By direct computation, we find ∞

−∞

dq1 2π ∞

−∞

dq2 x1|U(q1, q2)|x2y1|U†(q1, q2)|y2 = δ(x2 − y1) δ(x1 − y2) . (8) What we obtained precisely follows the properties: exp(iqX)|x = exp(iqx)|x and exp(iqP)|x = |x − q.

• This already implies

∞

−∞

dq1 ∞

−∞

dq2 2π ∞

−∞

dx O(x, t, q1, q2) = ∞

−∞

dx ∞

−∞

dx1 x1|e−iHt|x1x|eiHt|x . (9)

Spectral Form Factor Application: OTOC Conclusion

Non-Interacting Scalar Field Theory

• Rewrite this computation in terms of oscillators since this

generalizes easily to non-interacting scalar field theory, which is an assembly of non-interacting oscillators. Using a = (P − iωX) √ 2ω , a† = (P + iωX) √ 2ω , (10) the unitary operators that we have considered are given by U(q1, q2) = ea

• iq2√ ω

2 − q1 √ 2ω

• ea†

iq2√ ω

2 + q1 √ 2ω

• e

q2 1 4ω + q2 2ω 4 .

Spectral Form Factor Application: OTOC Conclusion

Non-Interacting Scalar Field Theory

Now consider a non-interacting scalar field theory, in a box (with the periodic boundary condition), so that momenta k are discrete, with an oscillator for every k.

Spectral Form Factor Application: OTOC Conclusion

Non-Interacting Scalar Field Theory

Now consider a non-interacting scalar field theory, in a box (with the periodic boundary condition), so that momenta k are discrete, with an oscillator for every k. The Hamiltonian is HNS = 1 V

• k

1 2 ˜ a†( k)˜ a( k), (11) where V is the volume of the box. The ˜ a† and ˜ a are the usual creation and annihilation operators in the box, and they satisfy the commutation relation [˜ a( k1), ˜ a†( k2)] = 2V ω

k1δ k1 k2, where

ω2

• k1 ≡ |

k1|2 + m2 with m the mass of the non-interacting scalar

• field. Hence we can perform the field redefinition

˜ a( k) ≡

• 2V ω(

k)a( k) and apply the result of the harmonic

• scillator to the non-interacting scalar field theory.

Spectral Form Factor Application: OTOC Conclusion

Coherent State

• We consider the exactly solvable model from the two-photon

non-degenerate Jaynes-Cummings (JC) model with the rotating wave approximation, which ignores the oscillating fast term.

Spectral Form Factor Application: OTOC Conclusion

Coherent State

• The effective Hamiltonian is

HJC ≡ N1 + N2 + M, (12) where Nj = ωj

• a†

j aj + (σz + 1)

2

• (13)

and M ≡ ∆(σz + 1) 2 + ga(a1a2σ+ + a†

1a† 2σ−),

(14) where σ+ ≡ σx + iσy 2 , σ− ≡ σx − iσy 2 . (15)

Spectral Form Factor Application: OTOC Conclusion

Coherent State

• The coherent states that we use are:

a1|α1α2 = α1|α1α2, a2|α1α2 = α2|α1α2, and |α1α2 = exp

• − (|α1|2 + |α2|2)/2
• exp
• α1a†

1 + α2a† 2

• |0, 0.

Completeness of the coherent states is d2α1 π d2α2 π |α1α2α1α2| = 1 . (16)

Spectral Form Factor Application: OTOC Conclusion

Coherent State

• In terms of the unitary operator

U(q1, q2, r1, r2) = exp(iq1X1 + iq2P1 + ir1X2 + ir2P2), (17) we compute the regularized two-point OTOC (repeated indices a, b are summed over 1,2) C(t) = α1α2|U(q1, q2, r1, r2)[e−βHJC/2−iHJCt]aaU(q1, q2, r1, r2)† ×[e−βHJC/2+iHJCt]bb|α1α2 , (18) where [· · · ]aa is the matrix element of the row-a and the column-a with the repeated summation.

Spectral Form Factor Application: OTOC Conclusion

Coherent State

• Direct computation gives

α1α2|U(q1, q2, r1, r2)|γ1

1γ1 2

= e

¯ α1

• iq2√ ω1

2 + q1

√

2ω1

• e

¯ α2

• ir2√ ω2

2 + r1

√

2ω2

• ×e

γ1

1

• iq2√ ω1

2 − q1

√

2ω1

• +γ1

2

• ir2√ ω2

2 − r1

√

2ω2

• ×e−

|α1|2+|α2|2+|γ1 1|2+|γ1 2|2 2

+¯ α1γ1

1+¯

α2γ1

2e− q2 1 4ω1 − q2 2ω1 4

−

r2 1 4ω2 − r2 2 ω2 4

. This matrix element is common for any two-particle problem - it is the coherent state expectation value of an element of the two-particle Heisenberg group. The integrations that we need to perform over coherent state parameters are Gaussian integrals, which is a nice simplification that will always be present.

Spectral Form Factor Application: OTOC Conclusion

Coherent State

In general, we will not be able to carry things out exactly. Nevertheless, given that t is a large parameter, the final integration naturally lends themselves to saddle point evaluations.

Spectral Form Factor Application: OTOC Conclusion

Large-N Matrix QM

Concretely, consider the model HQMN = PjPj 2 + µ2 X jX j 2 + g (X jX j)2 4 , (19) where j = 1, 2, · · · , N, and g is the coupling constant.

Spectral Form Factor Application: OTOC Conclusion

Large-N Matrix QM

Concretely, consider the model HQMN = PjPj 2 + µ2 X jX j 2 + g (X jX j)2 4 , (19) where j = 1, 2, · · · , N, and g is the coupling constant. Using the simplifications of the large-N, we replace this Hamiltonian with the approximate form (σ is a constant.) HQMNM = PjPj 2 + µ2 X jX j 2 + λσX jX j 2 . (20) The ’t Hooft coupling constant λ ≡ gN is fixed as we scale N → ∞, and we determine σ = N

j=1X jX j/N from the

two-point function. The large-N theory is harmonic oscillators but now with a modified frequency.

Spectral Form Factor Application: OTOC Conclusion

Large-N Matrix QM

The SFF is g2(β, t) =

• 1 + e−2√

µ2+λσβ − 2e−√ µ2+λσβ

1 + e−2√

µ2+λσβ − 2 cos(

• µ2 + λσt)e−√

µ2+λσβ

N .

Spectral Form Factor Application: OTOC Conclusion

Large-N Matrix QM

5 10 15 t 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 g2(t)

β=1, g=2 Perturbation

5 10 15 t 0.2 0.4 0.6 0.8 1.0 g2(t)

β=1, g=1 Perturbation

5 10 15 t 0.2 0.4 0.6 0.8 1.0 g2(t)

β=1, g=2/3 Perturbation

Figure: We fix the inverse temperature β = 1 while choosing the ’t Hooft coupling constant λ=gN=2. The lattice sizes are 8 in N=1 and 4 in N=2, 3. The numbers of lattice points are 128 in N=1 and 32 in N=2,

• 3. We compute the two-point spectral form factor g2(t) from 16

low-lying eigenenergy modes for N=1, 2, and 3 in the left, middle, and right figures respectively. The numerical solution in N=3 matches the large-N perturbation quantitatively.

Spectral Form Factor Application: OTOC Conclusion

Conclusion

• We link the spectral statistics to the OTOC through the

Heisenberg group averaging in bosonic QM and QFT.

Spectral Form Factor Application: OTOC Conclusion

Conclusion

• We link the spectral statistics to the OTOC through the

Heisenberg group averaging in bosonic QM and QFT.

• The late time limit is also the classical limit. Therefore, we

apply our study to coherent state, which is a quantum state closest to a classical regime, and large-N matrix QM. It is useful for understanding the late time behavior of the SFF.

Spectral Form Factor Application: OTOC Conclusion

Conclusion

• We link the spectral statistics to the OTOC through the

Heisenberg group averaging in bosonic QM and QFT.

• The late time limit is also the classical limit. Therefore, we

apply our study to coherent state, which is a quantum state closest to a classical regime, and large-N matrix QM. It is useful for understanding the late time behavior of the SFF.

• Because the uncertainty principle forbids the infinitesimal

perturbation, the OTOC cannot have the late time chaos. The link between the spectral statistics and OTOC gives the late time chaos to the OTOC.