Locality of temperature 1 / 17
Locality of temperaturestructural properties of thermal states
Christian Gogolin
ICFO - The Institute of Photonic Sciences
MPQ - Max Planck Institute of Quantum Optics
New trends in complex quantum systems dynamics2015-05-27
http://www.icfo.eu/http://www.mpq.mpg.de/
Locality of temperature 1 / 17
Locality of temperaturestructural properties of thermal states
Christian Gogolin
ICFO - The Institute of Photonic Sciences
MPQ - Max Planck Institute of Quantum Optics
New trends in complex quantum systems dynamics2015-05-27
Not as technical as it sounds!
http://www.icfo.eu/http://www.mpq.mpg.de/
Locality of temperature | Motivation 2 / 17
Motivation I: Measuring temperature
[1] Y. Gao and Y. Bando, Nature, 415.6872 (2002), 599
Locality of temperature | Motivation 2 / 17
Motivation I: Measuring temperature
[1] Y. Gao and Y. Bando, Nature, 415.6872 (2002), 599
Locality of temperature | Motivation 3 / 17
Motivation II: Coherent dynamics of large systems
DOI: 10.1126/science.1224953, 1318 (2012);337 Science
et al.M. GringRelaxation and Prethermalization in
an Isolated Quantum System
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t the last week in December, by the
Science
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337, 1318 (2012):
Relaxation and Prethermalization in
an Isolated Quantum SystemM. Gring,1 M. Kuhnert,
1 T. Langen,1 T. Kitagawa,2 B. Rauer,1 M. Schreitl,
1 I. Mazets,1,3
D. Adu Smith,1 E. Demler,2 J. Schmiedmayer
1,4*
Understanding relaxation processes is animportant unsolved problem in many are
as of
physics. A key challenge is the scarcity ofexperimental tools for the characterization
of complex
transient states. We used measurements of full quantum mechanical probability dis
tributions of
matter-wave interference to study the relaxation dynamics of a coherently split one
-dimensional
Bose gas and obtained comprehensive information about the dynamical states of
the system.
After an initial rapid evolution, the full distributions reveal the approach toward a
thermal-like
steady state characterized by an effectivetemperature that is independent from th
e initial
equilibrium temperature of the system before the splitting process. We conjecture t
hat this state
can be described through a generalized Gibbs ensemble and associate it with preth
ermalization.
Despite its fundamental importance, a gen-
eral understanding of how isolated quan-
tummany-body systems approach thermal
equilibrium is still elusive. Theoretical concepts
such as the quantum ergodic theory orthe ei-
genstate thermalization hypothesis (1–3) infer re-
quirements for a system to be able to undergo
relaxation, but it is still unclear onwhat time scale
this occurs. In situations in which conservation
laws inhibit efficient relaxation, many-body sys-
tems are expected to display a complex behavior.
An intriguing phenomenon that has been sug-
gested in this context is prethermalization (4), a
general concept that is predicted to be applicable
to a large variety of physical systems (5–9). In
the present understanding, prethermalization is
characterized by the rapid establishment of a
quasi-stationary state that already exhibits some
equilibrium-like properties. Full relaxation to the
1ViennaCenter forQuantumScienceandTechnology,Atominstitut,
Technische Universität (TU) Wien, Stadionallee 2, 1020
Vienna, Austria.2Harvard–Massachussets Institute of Tech-
nology Center for Ultracold Atoms (CUA),Department of
Physics, Harvard University, Cambridge, MA 02138, USA.
3Ioffe Physico-Technical Institute of the Russian Academy of
Science, 194021 St. Petersburg, Russia.4Zentrum für Mikro-
und Nanostrukturen (ZMNS), TU Wien, Floragasse 7, 1040
Vienna, Austria.
*To whom correspondence should be addressed. E-mail:
Fig. 1. (A) An initial-phase fluctuating 1D Bose gas is split into two uncoupled
gases with almost identical phase distributions f1(z) and f2(z) (black solid
lines) and allowed to evolve for a time te. (B)At te = 0 ms, fluctuations in the
local phase difference ∆f(z) between the two gases are very small, and the
corresponding phase correlation length is very large. During the evolution,
these relative-phase fluctuations increase,and the correlation length de-
creases. The main question we address is whether or when this system will
reach the corresponding thermal equilibrium of uncorrelated phases as
characterized by the initial temperature T andthermal coherence length lT. In
experiment, this situation can be prepared on purpose by splitting a thermal
gas and cooling it into two independent gases (32). (C) Typical experimental
matter-wave interference patterns obtainedby overlapping the two gases in
time-of-flight after different evolution times.Differences in the local relative
phase lead to a locally displaced interferencepattern. Integrated over a length
L, the contrast C(L) is a direct measure of thestrength of the relative-phase
fluctuations. (D) Because of the stochastic nature of the phase distributions,
repeated experimental runs yield a characteristic distribution P(C
2) of con-
trasts, which allows one to distinguish between the initial state, an in-
termediate prethermalized state, and the true thermal equilibrium of the
system.
14 SEPTEMBER 2012 VOL 337 SCIENCE www.sciencemag.org
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Locality of temperature | Motivation 4 / 17
Motivation III: A new foundation for statistical mechanics
Thermodynamics
Statistical mechanics
?t
Second Law, ergodicityequal a priory probabilities
Classical mechanics ?
�
Locality of temperature | Motivation 4 / 17
Motivation III: A new foundation for statistical mechanics
Thermodynamics
Statistical mechanics
?t
Second Law, ergodicityequal a priory probabilities
Classical mechanics ?
�
“There is no line of argument proceding fromthe laws of microscopic mechanics to macroscopicphenomena that is generally regarded by physicistsas convincing in all respects.”
— E. T. Jaynes [2] (1957)
“Statistical physics [...] has not yet developeda set of generally accepted formal axioms [...]”
— Jos Uffink [3] (2006)
Locality of temperature | Motivation 4 / 17
Motivation III: A new foundation for statistical mechanics
Thermodynamics
Statistical mechanics
⇑
Quantum mechanics
Locality of temperature | Motivation 4 / 17
Motivation III: A new foundation for statistical mechanics
Thermodynamics
Statistical mechanics
⇑
Quantum mechanics
!!! Reviews !!! Reviews !!! Reviews !!! Reviews !!! Reviews !!!
Shallow but broad but overview:J Eisert, M Friesdorf, and C Gogolin, Nature Physics, 11(2014), 124–130
In depth review:C. Gogolin and J. Eisert (2015), arXiv: 1503.07538v1
http://arxiv.org/abs/1503.07538v1
Locality of temperature | Equilibration & thermalization 5 / 17
Equilibration & thermalization
Locality of temperature | Equilibration & thermalization 6 / 17
Equilibration
Theorem (Equilibration on average [12])
If H has non-degenerate energy gaps, then for every ρ(0) = |ψ0〉〈ψ0|there exists a ωS such that:
D(ρS(t), ωS
)≤ 1
2
√d2Sdeff
=⇒ If deff � d2S then ρS(t) equilibrates on average.
[10] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Physical Review Letters, 100.3 (2008), 30602[11] P. Reimann, Physical Review Letters, 101.19 (2008), 190403[12] N. Linden, S. Popescu, A. Short, and A. Winter, Physical Review E, 79.6 (2009), 61103[13] A. J. Short and T. C. Farrelly, New Journal of Physics, 14.1 (2012), 013063[14] P. Reimann and M. Kastner, New Journal of Physics, 14.4 (2012), 043020
Locality of temperature | Equilibration & thermalization 6 / 17
Equilibration
Theorem (Equilibration on average [12])
If H has non-degenerate energy gaps, then for every ρ(0) = |ψ0〉〈ψ0|there exists a ωS such that:
D(ρS(t), ωS
)≤ 1
2
√d2Sdeff
=⇒ If deff � d2S then ρS(t) equilibrates on average.
[10] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Physical Review Letters, 100.3 (2008), 30602[11] P. Reimann, Physical Review Letters, 101.19 (2008), 190403[12] N. Linden, S. Popescu, A. Short, and A. Winter, Physical Review E, 79.6 (2009), 61103[13] A. J. Short and T. C. Farrelly, New Journal of Physics, 14.1 (2012), 013063[14] P. Reimann and M. Kastner, New Journal of Physics, 14.4 (2012), 043020
Setting
dB � dSdS
ρS(t) = TrB[ρ(t)]
Locality of temperature | Equilibration & thermalization 6 / 17
Equilibration
Theorem (Equilibration on average [12])
If H has non-degenerate energy gaps, then for every ρ(0) = |ψ0〉〈ψ0|there exists a ωS such that:
D(ρS(t), ωS
)≤ 1
2
√d2Sdeff
=⇒ If deff � d2S then ρS(t) equilibrates on average.
[10] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Physical Review Letters, 100.3 (2008), 30602[11] P. Reimann, Physical Review Letters, 101.19 (2008), 190403[12] N. Linden, S. Popescu, A. Short, and A. Winter, Physical Review E, 79.6 (2009), 61103[13] A. J. Short and T. C. Farrelly, New Journal of Physics, 14.1 (2012), 013063[14] P. Reimann and M. Kastner, New Journal of Physics, 14.4 (2012), 043020
Locality of temperature | Equilibration & thermalization 6 / 17
Equilibration
Theorem (Equilibration on average [12])
If H has non-degenerate energy gaps, then for every ρ(0) = |ψ0〉〈ψ0|there exists a ωS such that:
D(ρS(t), ωS
)≤ 1
2
√d2Sdeff
=⇒ If deff � d2S then ρS(t) equilibrates on average.
[10] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Physical Review Letters, 100.3 (2008), 30602[11] P. Reimann, Physical Review Letters, 101.19 (2008), 190403[12] N. Linden, S. Popescu, A. Short, and A. Winter, Physical Review E, 79.6 (2009), 61103[13] A. J. Short and T. C. Farrelly, New Journal of Physics, 14.1 (2012), 013063[14] P. Reimann and M. Kastner, New Journal of Physics, 14.4 (2012), 043020
Effective dimension
deff =1∑
k |〈Ek|ψ0〉|4
Intuition: Dimension of supporting energy subspace
Locality of temperature | Equilibration & thermalization 6 / 17
Equilibration
Theorem (Equilibration on average [12])
If H has non-degenerate energy gaps, then for every ρ(0) = |ψ0〉〈ψ0|there exists a ωS such that:
D(ρS(t), ωS
)≤ 1
2
√d2Sdeff
=⇒ If deff � d2S then ρS(t) equilibrates on average.
[10] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Physical Review Letters, 100.3 (2008), 30602[11] P. Reimann, Physical Review Letters, 101.19 (2008), 190403[12] N. Linden, S. Popescu, A. Short, and A. Winter, Physical Review E, 79.6 (2009), 61103[13] A. J. Short and T. C. Farrelly, New Journal of Physics, 14.1 (2012), 013063[14] P. Reimann and M. Kastner, New Journal of Physics, 14.4 (2012), 043020
Locality of temperature | Equilibration & thermalization 7 / 17
Thermalization
t
Yes
t
No
Equilibrate?
Locality of temperature | Equilibration & thermalization 7 / 17
Thermalization
t
T
Yes
T
No
Yes
t
No
Equilibrate?
Thermalize?
Locality of temperature | Equilibration & thermalization 8 / 17
Thermalization is a complicated process
Thermalization implies:
1 Equilibration [10–15]
2 Subsystem initial state independence [16, 17]
3 Weak bath state dependence [18]
4 Diagonal form of the subsystem equilibrium state [19]
5 Thermal state ωS = TrB[ω] ≈ gSHS (β) ∝ e−β HS [18, 20, 21]
[10] M. Cramer, C. M. Dawson, J. Eisert, and T. J. Osborne, Physical Review Letters, 100.3 (2008), 30602[11] P. Reimann, Physical Review Letters, 101.19 (2008), 190403[12] N. Linden, S. Popescu, A. Short, and A. Winter, Physical Review E, 79.6 (2009), 61103[13] A. J. Short and T. C. Farrelly, New Journal of Physics, 14.1 (2012), 013063[14] P. Reimann and M. Kastner, New Journal of Physics, 14.4 (2012), 043020[16] C. Gogolin, M. P. Müller, and J. Eisert, Physical Review Letters, 106.4 (2011), 40401[15] J. Gemmer, M. Michel, and G. Mahler, vol. 784, Lecture Notes in Physics, Berlin, Heidelberg: Springer, 2009[17] A. Hutter and S. Wehner, Physical Review A, 87.1 (2013), 012121[18] A. Riera, C. Gogolin, and J. Eisert, Physical Review Letters, 108.8 (2012), 080402[19] C. Gogolin, Physical Review E, 81.5 (2010), 051127[20] M. P. Mueller, E. Adlam, L. Masanes, and N. Wiebe (2013), arXiv: 1312.7420[21] F. G. S. L. Brandão and M. Cramer (2015), arXiv: 1502.03263v1. . .
http://arxiv.org/abs/1312.7420http://arxiv.org/abs/1502.03263v1
Locality of temperature | Locality of temperature 9 / 17
Locality of temperature
Locality of temperature | Locality of temperature 10 / 17
The setting
Local Hamiltonian (spins or fermions)
H
S
:=∑λ∈E
(S)
hλ
Thermal state
g
B
(β) :=e−β H
B
Tr [e−β H
B
]
Introduce buffer region
TrSc [gB(β)] ≈ TrSc [g(β)] ?
Locality of temperature | Locality of temperature 10 / 17
The setting
Local Hamiltonian (spins or fermions)
H
S
:=∑λ∈E
(S)
hλ
Thermal state
g
B
(β) :=e−β H
B
Tr [e−β H
B
]
Introduce buffer region
TrSc [gB(β)] ≈ TrSc [g(β)] ?
1/β
Locality of temperature | Locality of temperature 10 / 17
The setting
Local Hamiltonian truncated to S ⊂ V
HS :=∑λ∈E(S)
hλ
Thermal state
g
B
(β) :=e−β H
B
Tr [e−β H
B
]
Introduce buffer region
TrSc [gB(β)] ≈ TrSc [g(β)] ?
S
E(S)
Locality of temperature | Locality of temperature 10 / 17
The setting
Local Hamiltonian truncated to S ⊂ V
HS :=∑λ∈E(S)
hλ
Thermal state
g
B
(β) :=e−β H
B
Tr [e−β H
B
]
Introduce buffer region
TrSc [gB(β)] ≈ TrSc [g(β)] ?
S
????
?
1/β
Locality of temperature | Locality of temperature 10 / 17
The setting
Local Hamiltonian truncated to S ⊂ V
HS :=∑λ∈E(S)
hλ
Thermal state
gB(β) :=e−β HB
Tr [e−β HB ]
Introduce buffer region
TrSc [gB(β)] ≈ TrSc [g(β)] ?
S
B
d(S, ∂B)
????
?
Locality of temperature | Locality of temperature 11 / 17
This can be made rigorous:
Generalized covariance
covτρ(A,A′) := Tr[ρτAρ1−τA′]− Tr[ρA] Tr[ρA′]
exactly captures the response of local expectation values.
Theorem (Truncation formula [22])
For any observable A = AS ⊗ 1
Tr[AgB(β)]− Tr[Ag(β)] = β∫ 1
0
∫ 10
covτg(s,β)(H∂B, A) dτ ds ,
where g(s, β) is thermal state of H(s) := H − (1− s) H∂B .
[22] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert (2013), arXiv: 1309.0816v2
http://arxiv.org/abs/1309.0816v2
Locality of temperature | Locality of temperature 11 / 17
This can be made rigorous:
Generalized covariance
covτρ(A,A′) := Tr[ρτAρ1−τA′]− Tr[ρA] Tr[ρA′]
exactly captures the response of local expectation values.
Theorem (Truncation formula [22])
For any observable A = AS ⊗ 1
Tr[AgB(β)]− Tr[Ag(β)] = β∫ 1
0
∫ 10
covτg(s,β)(H∂B, A) dτ ds ,
where g(s, β) is thermal state of H(s) := H − (1− s) H∂B .
[22] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert (2013), arXiv: 1309.0816v2
http://arxiv.org/abs/1309.0816v2
Locality of temperature | Locality of temperature 11 / 17
This can be made rigorous:
Generalized covariance
covτρ(A,A′) := Tr[ρτAρ1−τA′]− Tr[ρA] Tr[ρA′]
exactly captures the response of local expectation values.
Theorem (Truncation formula [22])
For any observable A = AS ⊗ 1
Tr[AgB(β)]− Tr[Ag(β)] = β∫ 1
0
∫ 10
covτg(s,β)(H∂B, A) dτ ds ,
where g(s, β) is thermal state of H(s) := H − (1− s) H∂B .
[22] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert (2013), arXiv: 1309.0816v2
http://arxiv.org/abs/1309.0816v2
Locality of temperature | Locality of temperature 12 / 17
Clustering of correlations
Theorem (Clustering of correlations at high temperature [22])
Let J := maxλ ‖hλ‖∞, then for every τ ∈ [0, 1] and β < β∗(J, α)∣∣covτg(β)(A,A′)∣∣ ≤ C e− d(A,A′) / ξ(β J,α)with α = α(E) the lattice animal constant.
=⇒ D(gS(β), gB
S(β))≤ C ′ e−d(S,∂B)/ξ(β J,α)
[22] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert (2013), arXiv: 1309.0816v2
http://arxiv.org/abs/1309.0816v2
Locality of temperature | Locality of temperature 12 / 17
Clustering of correlations
Theorem (Clustering of correlations at high temperature [22])
Let J := maxλ ‖hλ‖∞, then for every τ ∈ [0, 1] and β < β∗(J, α)∣∣covτg(β)(A,A′)∣∣ ≤ C e− d(A,A′) / ξ(β J,α)with α = α(E) the lattice animal constant.
=⇒ D(gS(β), gB
S(β))≤ C ′ e−d(S,∂B)/ξ(β J,α)
[22] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert (2013), arXiv: 1309.0816v2
Lattice animal constant
#animals(m) ≤ αm
http://arxiv.org/abs/1309.0816v2
Locality of temperature | Locality of temperature 12 / 17
Clustering of correlations
Theorem (Clustering of correlations at high temperature [22])
Let J := maxλ ‖hλ‖∞, then for every τ ∈ [0, 1] and β < β∗(J, α)∣∣covτg(β)(A,A′)∣∣ ≤ C e− d(A,A′) / ξ(β J,α)with α = α(E) the lattice animal constant.
=⇒ D(gS(β), gB
S(β))≤ C ′ e−d(S,∂B)/ξ(β J,α)
[22] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert (2013), arXiv: 1309.0816v2
Lattice animal constant
#animals(m) ≤ αm
http://arxiv.org/abs/1309.0816v2
Locality of temperature | Locality of temperature 12 / 17
Clustering of correlations
Theorem (Clustering of correlations at high temperature [22])
Let J := maxλ ‖hλ‖∞, then for every τ ∈ [0, 1] and β < β∗(J, α)∣∣covτg(β)(A,A′)∣∣ ≤ C e− d(A,A′) / ξ(β J,α)with α = α(E) the lattice animal constant.
=⇒ D(gS(β), gB
S(β))≤ C ′ e−d(S,∂B)/ξ(β J,α)
[22] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert (2013), arXiv: 1309.0816v2
Lattice animal constant
#animals(m) ≤ αm
http://arxiv.org/abs/1309.0816v2
Locality of temperature | Locality of temperature 12 / 17
Clustering of correlations
Theorem (Clustering of correlations at high temperature [22])
Let J := maxλ ‖hλ‖∞, then for every τ ∈ [0, 1] and β < β∗(J, α)∣∣covτg(β)(A,A′)∣∣ ≤ C e− d(A,A′) / ξ(β J,α)with α = α(E) the lattice animal constant.
=⇒ D(gS(β), gB
S(β))≤ C ′ e−d(S,∂B)/ξ(β J,α)
[22] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert (2013), arXiv: 1309.0816v2
Lattice animal constant
#animals(m) ≤ αm
http://arxiv.org/abs/1309.0816v2
Locality of temperature | Locality of temperature 12 / 17
Clustering of correlations
Theorem (Clustering of correlations at high temperature [22])
Let J := maxλ ‖hλ‖∞, then for every τ ∈ [0, 1] and β < β∗(J, α)∣∣covτg(β)(A,A′)∣∣ ≤ C e− d(A,A′) / ξ(β J,α)with α = α(E) the lattice animal constant.
=⇒ D(gS(β), gB
S(β))≤ C ′ e−d(S,∂B)/ξ(β J,α)
[22] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert (2013), arXiv: 1309.0816v2
http://arxiv.org/abs/1309.0816v2
Locality of temperature | Locality of temperature 13 / 17
Implications
β < β∗(J, α) =⇒ D(gS(β), gB
S(β))≤ C ′ e−d(S,∂B)/ξ(β J,α)
Local stability of thermal states
gS(β) only depends exponentially weakly on far away terms of theHamiltonian.
Classical simulability with cost independent of total system size
Local expectation values can be calculates with cost independent of thetotal system size.
Locality of temperature | Locality of temperature 13 / 17
Implications
β < β∗(J, α) =⇒ D(gS(β), gB
S(β))≤ C ′ e−d(S,∂B)/ξ(β J,α)
Local stability of thermal states
gS(β) only depends exponentially weakly on far away terms of theHamiltonian.
Classical simulability with cost independent of total system size
Local expectation values can be calculates with cost independent of thetotal system size.
Locality of temperature | Locality of temperature 13 / 17
Implications
β < β∗(J, α) =⇒ D(gS(β), gB
S(β))≤ C ′ e−d(S,∂B)/ξ(β J,α)
Local stability of thermal states
gS(β) only depends exponentially weakly on far away terms of theHamiltonian.
Classical simulability with cost independent of total system size
Local expectation values can be calculates with cost independent of thetotal system size.
Locality of temperature | Locality of temperature 14 / 17
A universal bound on phase transitions
Universal critical temperature
The critical temperature
1
β∗J=
2
ln((1 +
√1 + 4/α)/2
)upper bounds the physical critical temperatures of all possible models.
Example: 2D square lattice (α ≤ 4 e)The bound:1/(β∗ J) = 2/ ln((1 +
√1 + 1/ e)/2) ≈ 24.58
Ising model (ferromagnetic, isotropic) phase transition at:1/(βc J) = 2/ ln(1 +
√2) ≈ 2.27
Locality of temperature | Locality of temperature 14 / 17
A universal bound on phase transitions
Universal critical temperature
The critical temperature
1
β∗J=
2
ln((1 +
√1 + 4/α)/2
)upper bounds the physical critical temperatures of all possible models.
Example: 2D square lattice (α ≤ 4 e)The bound:1/(β∗ J) = 2/ ln((1 +
√1 + 1/ e)/2) ≈ 24.58
Ising model (ferromagnetic, isotropic) phase transition at:1/(βc J) = 2/ ln(1 +
√2) ≈ 2.27
Locality of temperature | Conclusions 15 / 17
Conclusions
Locality of temperature | Conclusions 16 / 17
Conclusions
Ongoing program reconsidering the foundations of statisticalmechanics
Well connected to exciting experiments
Rigorous results on equilibration, thermalization, locality oftemperature, . . .
Locality of temperature | References 17 / 17
References
Thank you for your attention![1] Y. Gao and Y. Bando, Nature, 415.6872 (2002), 599.
[2] E. T. Jaynes, Physical Review, 106.4 (1957), 620–630.
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