Dynamic Properties of 1-D Ising Chain in a Transverse...

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Dynamic Properties of 1-D Ising Chain

in a Transverse Magnetic Field

Xun JiaDepartment of Physics, UCLA

December 2, 2005


Outline


Outline

z Motivation of this work


Outline


z Computational algorithm


Outline



z Test of programs— results for pure system


Outline




z Computation results for random system


Outline




z Computation results for random system

z Conclusion


Neutron Scattering Experiment

z Quantum phase tran-

sition in LiHoF4

H. M. Rønnow etc. al, Science, 308,389(2005)




sition in LiHoF4

z Coupling to nuclear

spin bath affects this

transition.

H. M. Rønnow etc. al, Science, 308,389(2005)




sition in LiHoF4

z Coupling to nuclear

spin bath affects this

transition.

z Full Hamiltonian

H =∑i

[HCF (Ji) + AJi · Ii

− gµBJi ·H]

− 1

2

∑ij

∑αβ

JDDαβ(ij)JiαJjβ

− 1

2

∑<ij>

J12Ji · Jj H. M. Rønnow etc. al, Science, 308,389(2005)


Motivation

z Dispersion relation are observed in neutron

scattering experiment.

• two dispersion branches

• finite excitation gap


Motivation

z Dispersion relation are observed in neutron

scattering experiment.

• two dispersion branches

• finite excitation gap

z Physically, coupling to the spin bath may

be mimicked by Ising model in a transverse

field, where magnetic fields are quenched

disorder.

• Effective magnetic fields

• Snapshot of the system


Motivation

z Model Hamiltonian

H = −∑i

J σzi σzi+1 −∑i

hiσxi

where hi are random variables following a certain

identical independent distribution.


Motivation

z Model Hamiltonian

H = −∑i


hiσxi



z Dynamic structure factor S(k, ω) directly related

to the neutron scattering experiment results.


Motivation

z Model Hamiltonian

H = −∑i


hiσxi



z Dynamic structure factor S(k, ω) directly related

to the neutron scattering experiment results.

z Transverse Ising model is the simplest model

with Quantum phase transition, it is meaningful

to study its dynamic structure factor S(k, ω).


Algorithm

z Jordon-Wigner transformation

σzi = c†i exp[iπ

∑j<i

c†jcj

]+ exp

[− iπ

∑j<i

c†jcj

]ci

σxi = 1− 2c†ici


Algorithm



∑j<i

c†jcj

]+ exp

[− iπ

∑j<i

c†jcj

]ci


maps this Hamiltonian to a free fermion Hamiltonian

H =∑i

hi(c†ici − cic

†i)−

∑i

J (c†i − ci)(c†i+1 + ci+1)


Algorithm



∑j<i

c†jcj

]+ exp

[− iπ

∑j<i

c†jcj

]ci


maps this Hamiltonian to a free fermion Hamiltonian

H =∑i

hi(c†ici − cic

†i)−

∑i

J (c†i − ci)(c†i+1 + ci+1)

z We take free, rather than the more usual periodic, bound-

ary condition, to avoid the complexity coming from the

problem of the number of fermions.


Algorithm

z We are seeking a transformation in the following form

to diagonalize the Hamiltonian:

c†i =∑µ

(aiµγ†µ + biµγµ) ci =

∑µ

(aiµγµ + biµγ†µ)


Algorithm

z We are seeking a transformation in the following form

to diagonalize the Hamiltonian:

c†i =∑µ

(aiµγ†µ + biµγµ) ci =

∑µ

(aiµγµ + biµγ†µ)

z Diagonalization gives us all eigenvalues εµ, µ = 1 . . . L,

where L is the size of the system:

• For pure system, this can be done analytically.

• For random system, we are limited to do numerically.


Algorithm

z Define new operators:

Ai ≡ c†i + ci =∑µ

φiµ(γ†µ + γµ)

Bi ≡ c†i − ci =∑µ

ψiµ(γ†µ − γµ)


Algorithm






z Real time spin-spin correlation function:

Si,j(t) =⟨σzi (t)σ

zj

⟩=

⟨eiHtσzi e

−iHtσzj⟩


Algorithm








zj

⟩=

⟨eiHtσzi e

−iHtσzj⟩

=⟨

exp[−iπ∑m<i

c†m(t)cm(t)][c†i(t) + ci(t)]

exp[−iπ∑n<j

c†ncn][c†j + cj]

⟩


Algorithm








zj

⟩=

⟨eiHtσzi e

−iHtσzj⟩

=⟨

exp[−iπ∑m<i

c†m(t)cm(t)][c†i(t) + ci(t)]

exp[−iπ∑n<j

c†ncn][c†j + cj]

⟩=

⟨[∏m<i

Am(t)Bm(t)]Ai(t)[∏n<j

AnBn]Aj

⟩


Algorithm

z Thus the real time spin-spin correlation function can be

evaluated via Wick’s theorem, in the form of a Pfaffian.

Si,j(t) =

⟨[∏m<i


AnBn]Aj

⟩

= Pf

0 〈A1(t)B1(t)〉〈A1(t)A2(t)〉 . . . 〈A1(t)Aj〉... 0 〈B1(t)A2(t)〉 . . . 〈B1(t)Aj〉... ... ... . . . ...... ... ... . . . 0


Algorithm

z Thus the real time spin-spin correlation function can be

evaluated via Wick’s theorem, in the form of a Pfaffian.

Si,j(t) =

⟨[∏m<i


AnBn]Aj

⟩

= Pf

0 〈A1(t)B1(t)〉〈A1(t)A2(t)〉 . . . 〈A1(t)Aj〉... 0 〈B1(t)A2(t)〉 . . . 〈B1(t)Aj〉... ... ... . . . ...... ... ... . . . 0

z Basic contractions are worked out, for example:

〈Ai(t)Bj〉 =∑µ

e−iεµtφiµψjµ 〈Ai(t)Aj〉 = . . .


Algorithm

z Evaluating the Pfaffian is a hard problem ∼ O((2N −1)!!).


Algorithm


z Pfaffian is square root of the determinant of the corre-

sponding antisymmetric matrix.

Si,j(t)2 =

∣∣∣∣∣∣∣∣∣0 〈A1(t)B1(t)〉〈A1(t)A2(t)〉 . . . 〈A1(t)Aj〉... 0 〈B1(t)A2(t)〉 . . . 〈B1(t)Aj〉... ... ... . . . ...

A.S. ... ... . . . 0

∣∣∣∣∣∣∣∣∣and determinant is easy to compute ∼ O(N 3).


Algorithm


z Pfaffian is square root of the determinant of the corre-

sponding antisymmetric matrix.

Si,j(t)2 =

∣∣∣∣∣∣∣∣∣0 〈A1(t)B1(t)〉〈A1(t)A2(t)〉 . . . 〈A1(t)Aj〉... 0 〈B1(t)A2(t)〉 . . . 〈B1(t)Aj〉... ... ... . . . ...

A.S. ... ... . . . 0

∣∣∣∣∣∣∣∣∣and determinant is easy to compute ∼ O(N 3).

z Sign problem, overcome based on continuity of Si,j(t) on

variable t.


Algorithm

z Finally, the dynamic structure factor is computed as

S(k, ω) ≡∑n

eikn∫ ∞

−∞dteiωtSi,i+n(t)

=∑n

eikn2Re

∫ ∞

0

dtei(ω+iδ)tSi,i+n(t) δ → 0+


Algorithm


S(k, ω) ≡∑n

eikn∫ ∞


=∑n

eikn2Re

∫ ∞

0


z Algorithm is suitable for both pure and disorder system.


Algorithm


S(k, ω) ≡∑n

eikn∫ ∞


=∑n

eikn2Re

∫ ∞

0


z Algorithm is suitable for both pure and disorder system.

z As for the disorder case, Si,j(t) was obtained by averaging

over many different disorder configurations,then S(k, ω) is

finally computed.


Pure system

z For pure system, a

typical set of parame-

ters are

J = 1.0

hi = 1.4

so the system is in para-

magnetic phase.


Pure system

z For pure system, a

typical set of parame-

ters are

J = 1.0

hi = 1.4

so the system is in para-

magnetic phase.

z Density of states is

shown in the figure.


Pure system

z Dynamic structure

factor.

z The dispersion rela-

tion agrees with theo-

retical calculation(dash

line).

ω = 2√h2 + J 2 − 2hJ cos k


Distribution of hi in disorder system

z Two kinds of disorder distributions

are used in calculation.





• Rectangular distribution, magnetic field is

distributed around a certain average value

have with a width hw.

p(hi) =

{1/hw : |hi − have| ≤ 1

2hw

0 : otherwise





• Rectangular distribution, magnetic field is

distributed around a certain average value

have with a width hw.

p(hi) =

{1/hw : |hi − have| ≤ 1

2hw

0 : otherwise

• Binary distribution, magnetic field is al-

most constant at h0, it may take another

value h1 with a small probability q.

p(hi) = qδ(hi − h1) + (1− q)δ(hi − h0)


Density of states

z With the rectangular distri-

bution, density of states under

different hw are computed.

• Parameters are:

have = 1.4 J = 1.0

• The effect of the disorder in mag-

netic field is to broadening the

density of states.

• Only little density of states show

up at zero energy at strong disor-

der.

• However, excitation becomes gap-

less then.


Autocorrelation function

z Auto correlation function un-

der different hw are computed.





z Gapless excitation at strong

disorder.





z Gapless excitation at strong

disorder.

z Most weight concentrates at

zero energy.


Dynamic structure factor

z Dynamic structure

factor is computed at

strong disorder case.



z Dynamic structure



z Disorder closes the ex-

citation gap.



z Dynamic structure



z Disorder closes the ex-

citation gap.

z Stripe pattern implies

non-disperse modes—

localization


Effect of Rectangular Disorder

z Excitation gap closes with increasing width of disorder.




z Disorder shifts up the critical magnetic field, which captures

the characteristics of the coupling.




z Disorder shifts up the critical magnetic field, which captures

the characteristics of the coupling.

z Localization, tridiagonal matrix with random elements to be

diagonalized.

H̃ = 2

2(h2

1 + 1) −h1 0

−h1 2(h22 + 1) −h2

−h2. . . . . .

0 . . . . . .


Density of states

z Distribution of disorder is:

p(hi) = qδ(hi−h1)+(1−q)δ(hi−h0)

with parameters:

h1 = 0.1 h0 = 1.4 q = 0.05 ∼ 0.1


Density of states

z Distribution of disorder is:

p(hi) = qδ(hi−h1)+(1−q)δ(hi−h0)

with parameters:

h1 = 0.1 h0 = 1.4 q = 0.05 ∼ 0.1

z Density of states for pure sys-

tem is preserved, while a few

states are allowed around zero

energy. Thus, there might be

gapped excitation.



z Under such a disorder distri-

bution, autocorrelation func-

tion S(ω) is almost the same

as that in pure system except

that another peak appears at

zero frequency.








zero frequency.

z Finite gap is preserved.








zero frequency.

z Finite gap is preserved.

z Zero-energy peak increased

with disorder.



z Dynamic struc-

ture factor at:

q = 0.05



z Dynamic struc-

ture factor at:

q = 0.10


Conclusion


Conclusionz S(k, ω) could be computed successfully via this

algorithm in both pure and disorder system.




z Coupling between electronic spin and nuclear spin

bath is modelled by random fields.

• Random fields in rectangular distribution shift up the crit-

ical magnetic field.

• Binary distribution produces the non-dispersing peak.









z The non-dispersing peak may result from dilute

impurities.










impurities.

z What I am going to do:










impurities.


• quantitatively description










impurities.



• Model the long-range dipolar interaction.










impurities.



• Model the long-range dipolar interaction.

• Higher dimensional case


Thank you !

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