Antiferromagnetic Heisenberg Model [26P]

(1)

University of Regensburg Summer Term 2018

Quantum Theory of Condensed Matter I

Prof. John Schliemann

Dr. Paul Wenk, M.Sc. Martin Wackerl Mo. 08:00-10:00 c.t., PHY 5.0.21

Sheet 5

Antiferromagnetic Heisenberg Model [26P]

A spin S Heisenberg model on an arbitrary periodic D-dimensional lattice (coordination number z) is given by

H =J X

hmni

S_m·S_n, J >0, S²_α|m_αi=S(S+ 1)|m_αi, (1)

whereh.idenotes the summation over nearest neighbors only, andmαthe magnetic quantum number at siteα.

In a system whereS1 holds quantum fluctuations of the spin become less important (proof this statement by considering Heisenberg’s uncertainty relation for|h[Si, S_j]i|). The ground state under this condition (N´eel state,

|Nei) is a staggered spin configuration, i.e., all neighboring spins are antiparallel. However, the true ground state exhibits zero-point fluctuations so that the antiparallel configuration will only serve as a reference state from which we start our analysis.

(a)(6P) First, we try to find the upper and lower bound for the ground state of a system withLsites.

– Calculate the upper bound by evaluatinghNe|H|Nei. Why can’t|Neibe the ground state?

– For the lower bound, examine the interaction ofSiwith itszneighborsSm,Hi=Si·P

mSm. Rewrite Hi such that one can maximize (P

mSm)² and minimize the remaining term. Why can’t this minimum be reached by the true ground state?

(b)(10P) Show that the antiferromagnon spectrum, by neglecting magnon-magnon interaction, is given by ω(k) =J zS

q

1−γ_k², withγk= 1 z

X

δ

e^ik·δ. (2)

To accomplish this you can apply the following steps: Define two kinds of Holstein-Primakoff bosons (explain choice!):

On sublattice A

S_Aj⁺ =√

2S 1−a^†_jaj

2S

!^1/2

a_j, S_Aj⁻ =√

2Sa^†_j 1−a^†_jaj

2S

!^1/2

, (3)

on sublattice B

S⁻_Bl=√

2S 1−b^†_lb_l 2S

!^1/2

bl, S⁺_Bl=√

2Sb^†_l 1−b^†_lb_l 2S

!^1/2

, (4)

where [a_i, a_j] = [a^†_i, a^†_j] = 0 and [a_i, a^†_j] =δ_ij, accordingly for the operators of sublattice B. Thus, our ansatz for each sublattice is equivalent to the case of ferromagnetic coupling.

Express the model Hamiltonian using this transformation and expand it assuming

ha^†_ja_ji/S1. (5)

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Neglect terms which are not bilinear in the magnon operators, i.e., neglect magnon-magnon interac- tions. Apply a Fourier transformation,a_j^FT→c_k, b_j^FT→d_k, (a^†_j ^FT→c^†_k, b^†_j ^FT→d^†_k). (Assume the reduced Brillouin zone to containL/2k-vectors.) Show that this leads to

H0=J zSX

k

(γk(ckd_−k+d^†_kc^†_−k) + (c^†_kck+d^†_kdk)). (6)

RewriteH0in a symmetrized form consisting of two parts (2/(zJ S))H0=P

kH_k⁽¹⁾+P

kH_k⁽²⁾ with H_k⁽¹⁾ containing operators of sublattice A only with (k) index and operators of sublattice B only with (−k) index and accordingly forH⁽²⁾. Show that theBogoliubov transformation,

αk=ukck−vkd^†_−k, α^†_k=ukc^†_k−vkd−k (7) bringsH_k⁽¹⁾ to diagonal form. Why is the conditionu²_k−v²_k= 1 necessary? Apply a corresponding procedure toH_k⁽²⁾ with new operatorsβk, β_k^† and express, finally, H0 in this new basis.

(c)(4P) What is the long-wavelength limit of the spectrum? Connect this finding with the Goldstone theo- rem¹.

(d)(6P) Calculate the staggered magnetization density

m= 1 L

* X

i∈A

S_i^z−X

j∈B

S_j^z +

. (8)

Show that for the one-dimensional Heisenberg model quantum fluctuations destroy the antiferromagnetic long-range order for any finite spinS. Hint: Express the sum overkas an integral.

What is the effect inD= 3 for a simple cubic lattice?

1When a continuous symmetry is spontaneously broken, a massless boson appears (gapless in relation to the vacuum).