I JUST WANT YOU TO FIND HOW WE GET FROM this: II. QUANTUM PHASE TRANSITION IN THE HEISENBERG CHAIN A. Approach description We begin by explaining the improvements we made in this work to the...

I JUST WANT YOU TO FIND HOW WE GET FROM this: II. QUANTUM PHASE TRANSITION IN THE HEISENBERG CHAIN A. Approach description We begin by explaining the improvements we made in this work to the zero-field bond-mean-field theory9
BMFT for the Heisenberg chain. Then, we extend the BMFT to the case when an external magnetic field is applied to the chain. 1. Zero-field BMFT The 1D Heisenberg model is H1D = J i,j Si · Sj,
2 where Si is the spin-1/ 2 operator at site i and i, j designates the sum over adjacent sites only. Using the well-known 1D Jordan-Wigner
JW transformation Si - = ci eii , i = j=0 i-1 ni, Si z = ni - 1/2,
3 with ni=ci † ci , the Heisenberg chain maps onto a Hamiltonian of 1D interacting spinless fermions; i.e., H1D = J 2 i
ci † ci+1 + H.c. + J i ci † ci - 1 2 ci+1 † ci XXXXXXXXXX .
4 In the framework of the BMFT, we consider the bond parameter ci ci+1 †  to decouple the Ising quartic term in Eq.
4. The procedure is as follows. By neglecting the fluctuations in ci ci+1 † , we write ci † ci ci+1 † ci+1  ci ci+1 † ci † ci+1 + ci+1ci † ci+1 † ci + ci ci+1 † ci+1ci †  = Qi ci † ci+1 + Qi * ci+1 † ci + Qi 2,
5 with Qi=ci ci+1 †  the mean-field bond parameter. Note that Si - Si+1 +  = ci ci+1 †  , which means that the parameter Qi accounts for the quantum fluctuations in the Heisenberg Hamiltonian. To recover a result comparable to the des Cloiseaux–Pearson10 spin-excitation spectrum for the single Heisenberg chain, namely, E
k = J 2 sin k ,
6 one has to choose Qi=
-1 i Q Qeixi , where Q is site independent. Here, xi is an integer that labels the site coordinate along the Heisenberg chain. This yields a phase configuration along the chain of the form xi ...,,0,,0,..., which will in turn give9 E
k = J1 sin k ,
7 when the single-fermion term
the XY term is also written within the alteranted phase scheme. The hopping amplitude for the JW fermions resulting from this treatment is
-1 i J1, with J1=J
1+2Q indicating that a renormalization of the single-particle term in Eq.
4 by the Ising interaction took place. Choosing a uniform Qi and neglecting the alternation of the sign of XY-hopping amplitude gives a cos k spectrum that does not agree with the des Cloiseaux–Pearson result. Physically, we can justify the choice of the staggered Qi in the following way. Consider two adjacent bonds
2i, 2i + 1
2i+1,2i+ 2. Because of the AF correlations, the bonds
2i, 2i+ 1
2i+1,2i+ 2 may be in the spin configuration
?2i ,?2i+1
?2i+1 ,?2i+2. Using the argument of only shortrange and short-lived AF order, one has to view the object
?2i ,?2i+1 as a renormalized spin singlet, rather than a static spin arrangement, because of the important spin fluctuations. 74, 174422
XXXXXXXXXXIn the mean-field treatment based on the JW fermions, any two adjacent bonds can be quantitatively represented by the product c2i † c2i+1c2i+1c2i+2 † , which turns out to be -Q2 , since Q=c2i+1c2i+2 †  and c2i † c2i+1=-c2i+1c2i † =-Q. It is thus justified to use opposite signs for Q on adjacent bonds. We assumed that creating a JW fermion is equivalent to having a spin-up state and destroying a JW fermion corresponds to a spin-down state. With all this and Eq.
5, the mean-field Hamiltonian takes on the form H1D = NJQ2 + k k † H1Dk,
8 where the spinor  is given by k † =
ck A† ck B†
9 and the Hamiltonian density by H = 0 e
k e*
k 0 ,
10 with e
k=iJ1 sin k. ck is the Fourier transform of ci . =A or B is the label of the two AF sublattices. The eigenenergies are E±
k= ±J
1+2Q sin k . The ground-state energy per site is EGS = JQ XXXXXXXXXXdk 2 p=± ln 1 + e- Ep
k ,
11 which gives EGS
T= 0=JQ2- 1 J
1+2Q at zero temperature because only the band with negative energies is filled at this temperature. Minimizing EGS with respect to Q gives Q =1/ at T= 0. At nonzero temperature Q is given by Q = - 1 2 dk 2 sin k  p=± pnF Ep
k .
12 Here, nF Ep
k =1/ 1+e Ep
k is the Fermi-Dirac factor. In the high-T limit with kBT J, we get Q J 8kBT 1 1-4kBT/J , a result that rules out any finite-T phase transition, in agreement with the Mermin-Wagner theorem.11 Next, we will extend the analysis we just finished to the case of a Heisenberg chain coupled to an external magnetic field.