Field Induced Reentrant Phase Transitions in Quasi-One-Diemnseional
Random Quantum Heisenberg Chains
Saitama University Kazuo HIDA
1 Introduction
1.1 Quantum Magnets in the Magnetic Field
1. Magnetization Plateau
Field induced spin gap= Disordered Phase
2. Field Induced Antiferromagnetic Phase（a）Quasi-One-Dimensional SystemMagnetic Field > Spin Gap⇒ Tomonaga-Luttinger Liquid+ Weak interchain interaction⇓Antiferromagnetic Long Range OrderDisorder eﬀects on the quantum spin systems in the magnetic field• Revived spins⇒magnetic long range order• Localized singlet pairs = plateau formation⇒Suppresss the magnetic
orderCompetition⇓Field induced reentrant transition

2 ModelQuasi-1-dimensional Random S = 1/2 Heisenberg model with bond alternation

N/2
X N/2X
X
H=JS 2i−1,j S 2i,j +Jij S 2i,j S 2i+1,j
ji=1

i=1intrachain
N XX+JintS i,j S i,j 0 interchain
i=1 <j,j 0>• Intrachain Interaction:
J = 1 Ji,j =(JS probability p
JW probability 1 − p

JS > J > JW > 0Method DMRG: Keeping 60 ∼ 160 states.

[Bond Configuration]

3 Magnetization Curve of an Isolated Chain at T = 0.
JS = 2 JW = 0.1 J = 1
1. Uniform Chain - DMRG

2. Magnetization Curve of Mixed Chain - Cluster picture

Averaged over 64 samples with N = 120 .

4 Eﬀect of Interchain Interaction
Mean Field Approximation for the Interchain Interaction(ED(−1)im Jint < 0 Interchain ferromagnetic interaction
xSi,j =(−1)iPj m Jint > 0 Interchain antiferromagnetic interaction
forJint > 0

Pj = +1 j ∈ A-sublattice

Pj = −1 j ∈ B-sublatticeInterchain mean field Hamiltonian
H IMF =NX

JS 2i−1S 2i +i=1NX
i=1NX− Hst(−1)iSix
i=1JiS 2iS 2i+1

Hst = λm(Hst) Self-consistent equation λ ≡ z|Jint|λc = lim Hst/m(Hst)Hst→0λc: critical interchain interaction λp = 0.2
Hst = 0.0005
p = 0.8
Multiple reentrant transition
N = 120. Averaged over 512 samples.

• Fine peak structure• Hc-dependence of λc

N = 240, 480 Averaged over 256 samples (middle 240 sites).
λc =0 only for discrete points?

5 Intrachain spin-spin correlation function

Averaged over 512 samples for N = 240Exponential decay even for λc = 0
Non plateau state : Dense excited states near the ground state
Staggered mean field mixes the excited state into the ground state⇓
Divergence of χst⇓
Long range order with weak interchain interaction

6 Summary
1. Quasi-one-dimensional random alternating bond S = 1/2 Heisenberg
model exhibits multiple reentrant transitions in the magnetic field.
Method：DMRG+Interchain mean field approximation
2. In the absence of interchain coupling, the spin-spin correlation function
decays exponentially even in the non-plateau regime.
Mix up the low energy excited states by interchian interaction.
⇒Long range order
3. Bose glass phase(Nohadani et al) is not found
Limitation of interchain mean field approximation
4. • Reentrant transition in 3-D random dimer system
Random destruction of singlet dimer⇒Local moment• Quasi-One-dimensional random alternating bond system
Competition of 2 types of dimer patterns⇒local moment
More complicated features ＝ Multiple reentrant transition

5. Speculated finite temperature phase daigram

6. Possibility of experimental observation：random substitution of anions
on the superexchange path
Ref. K. Hida : J. Phys. Soc. Jpn. 75, 074709 (2006). cond-mat/0602016;