Phase Diagram of spin 1/2 XXZ Model With Dzyaloshinskii-Moriya Interaction

Phase Diagram of spin 1/2 XXZ Model With Dzyaloshinskii-Moriya Interaction

R. Jafari Institute for Advanced Studies in Basic Sciences, Zanjan 45195-1159, Iran School of physics, IPM (Institute for Studies in Theoretical Physics and Mathematics),
P. O. Box: 19395-5531, Tehran, Iran
A. Langari Physics Department, Sharif University of Technology, Tehran 11155-9161, Iran [
July 7, 2019
Abstract

We have studied the phase diagram of the one dimensional XXZ model with Dzyaloshinskii-Moriya (DM) interaction. We have applied the quantum renormalization group (QRG) approach to get the stable fixed points, critical point and the scaling of coupling constants. This model has three phases, ferromagnetic, spin-fluid and Néel phases which are separated by a critical line which depends on the DM coupling constant. We have shown that the staggered magnetization is the order parameter of the system and investigated the influence of DM interaction on the chiral ordering as a helical magnetic order.

pacs:
75.10.Pq,73.43.Nq,03.67.Mn,64.60.ae

I Introduction

Quantum phase transition has been one of the most interesting topic in the area of strongly correlated systems in the last decade. It is a phase transition at zero temperature where the quantum fluctuations play the dominant role vojta (). Suppression of the thermal fluctuations at zero temperature introduces the ground state as the representative of the system. The properties of the ground state may be changed drastically shown as a non-analytic behavior of a physical quantity by reaching the quantum critical point (QCP). This can be done by tuning a parameter in the Hamiltonian, for instance the magnetic field or the amount of disorder. The ground state of a typical quantum many body systems consist of a superposition of a huge number of product states. Understanding this structure is equivalent to establishing how subsystems are interrelated, which in turn is what determines many of the relevant properties of the system. In Mott insulators the Heisenberg interaction is in most cases the dominant source of coupling between local moments, and most theoretical investigations are based on modeling in which only this type of interaction is included. Recently some novel magnetic properties in antiferromagnetic (AF) systems were discovered in the variety of quasi-one dimensional materials that are known to belong to an antisymmetric interaction of the form which is known as the Dzyaloshinskii-Moriya (DM)interaction. The relevance of antisymmetric superexchange interactions in spin Hamiltonian which leads to either a week ferromagnetic (F) or helical magnetic distortion in quantum AF systems, has been introduced phenomenologically by DzyaloshinskiiDzyaloshinskii (). A microscopic model of antisymmetric exchange interaction was first proposed by MoriyaMoriya () which showed that such interactions arise naturally in perturbation theory due to the spin-orbit coupling in magnetic systems with low symmetry and is essentially an extension of the Anderson superexchange mechanismAnderson () that shows for spin-flip hopping of electrons. Since it (DM interaction) breaks the fundamental SU(2) symmetry of the Heisenberg interactions, it is at the origin of many deviations from pure heisenberg behavior such as cantingCoffey () or small gapsDender1 (); Oshikawa1 (); Zhao (); Fouet (); Chernyshev (). A number of AF systems expected to be described by DM interaction, such as Dender1 (); Dender2 (), Kohgi (); Fulde (); Oshikawa2 (), Tsukada (), , Grande () and Greven (), which exhibit unusual and interesting magnetic properties in the presence of quantum fluctuations and/or applied magnetic fieldGrande (); Yildirim (); Katsumata (). Also belonging to the class of DM antiferromagnets is , which is a parent compound of high-temperature superconductorsKastner (). This has stimulated extensive investigation on the physical properties of the DM interaction. However, This interaction is rather difficult to Handel analytically, which has brought much uncertainty in the interpretation of experimental data and has limited our understanding of many interesting quantum phenomena of low-dimensional magnetic materials.

In the present paper, we have considered the one dimensional XXZ model with DM interaction by implementing the quantum renormalization group (QRG) method. In the next section the QRG approach will be explained and the renormalization of coupling constant are obtained. In section (III), we will obtain the phase diagram, fixed points, critical points and calculate the staggered magnetization as the order parameter of the underling quantum phase transition. We will also introduce the chiral order as an ordering which is produced by DM interaction. The exponent which shows the divergence of correlation function close to the critical point , the dynamical exponent and the exponent which shows the vanishing of staggered magnetization near the critical point will also be calculated. In Sec (IV), discussion concludes the paper.

Ii Quantum renormalization group

The main idea of the RG method is the elimination or the thinning of the degrees of freedom caried out step by step in an iteration procedure. Here, we used the well known Kadanoff block method as it is both well suited to perform analytical calculations in the lattice models and is conceptually straight-forward to be extended to the higher dimensionsmiguel1 (); miguel-book (); Langari (); Jafari (). In the Kadanoff’s method, the lattice is divided into blocks in which the Hamiltonian can be exactly diagonalized. Selecting a number of low-lying eigenstates of the blocks the full Hamiltonian is projected onto these eigenstates and an the effective (renormalized) Hamiltonian is obtained.

The Hamiltonian of XXZ model with DM interaction in the direction on a periodic chain of sites is

 H(J,Δ)=J4N∑i[σxiσxi+1+σyiσyi+1+Δσziσzi+1 (1) +D(σxiσyi+1−σyiσxi+1)],

where the is exchange constant, is the strength of component of DM interaction and the easy-axis anisotropy defined by which can be positive and negative. The positive and negative corresponds to the antiferromagnetic and ferromagnetic (F) cases, respectively. refers to the -component of the Pauli matrix at site . By implement rotation around axis on odd or even sites, the AF case of Hamiltonian () is mapped on the F case () with opposite sign of anisotropy,

 H(J,Δ)=J4N∑i[σxiσxi+1+σyiσyi+1−Δσziσzi+1 (2) +D(σxiσyi+1−σyiσxi+1)],  J>0.

So we can restrict ourselves to AF case () with and arbitrary anisotropy ( and ) without loss of generality.

The effective Hamiltonian to the first order RG approximation is

 Heff=Heff0+Heff1, Heff0=P0HBP0   ,   Heff1=P0HBBP0.

We consider a three-site block procedure defined in Fig.(1). The block Hamiltonian , its eigenstates and eigenvalues are given in Appendix A. The three-site block Hamiltonian has four doubly degenerate eigenvalues (see Appendix A). is the projection operator of the ground state subspace defined by , Where and are the doubly degenerate ground states, and are the renamed base kets in the effective Hilbert space. For each block we keep two states ( and ) to define the effective (new) site. Thus, the effective site can be considered as having a spin 1/2. Due to the level crossing which occurs for the eigenstates of the block Hamiltonian, the projection operator () can be different depending on the coupling constants. Therefore, we must specify different regions with the corresponding ground states. As Fig.(5) shows there are two regions with different eigenstates which are separated by where a level crossing occurs. In region (A) the ground state is the doubly-degenerate ferromagnetic state and while in region (B) and are the degenerate ground states. At the level crossing () the ground state is 4-fold degenerate (, , , ). A summary of this information is given in Fig.(5) of appendix A.

In the following, we will classify the RG equation of the regions where each of this states represent the ground state.

ii.1 Region (A): e0 is the ground state.

In this region the effective Hamiltonian in the first order correction is similar to the initial one, i.e,

 Heff=J′4N∑i[σxiσxi+1+σyiσyi+1+Δ′σziσzi+1 +D′(σxiσyi+1−σyiσxi+1)] (3)

where , and are the renormalized coupling constants. The new renormalized coupling constants are found to be functions of the original ones given by the following equations,

 J′=J(2q)2(1+D2),  D′=D (4) Δ′=Δ1+D2(Δ+q4)2.

ii.2 Region (B): e3 is the ground state.

In this region the effective Hamiltonian to the first order corrections leads to the ferromagnetic Ising model

 Heff=14⎡⎣Δ′N/3∑iσziσzi+1⎤⎦,

where

 Δ′ = JΔ,  J>0,  Δ<0.

Iii Phase Diagram

iii.1 Region (A)

For simplicity we have separated this region into positive anisotropy and negative anisotropy sectors.

• In the positive anisotropy sector the RG equations show that the coupling, representing the energy scale, approaching zero by iterating RG procedure. Thus, at the zero temperature,the quantum phase transition is the result of competition between the anisotropy () and the DM coupling constant (). In the region of planar anisotropy , the symmetric interactions () is known not to support any kind of long range order and the ground state is the so called spin-fluid (SF) state. Increasing the amount of anisotropy is necessary to stabilize the spin alignment. For the ground state is the Néel ordered state. In the case of , the anisotropy constant () and antisymmetric (DM) coupling are in competition with each other. The latter thus destroys the ordering tendency of the former and defers creating of Néel order. Our RG equations show that the phase boundary between the SF and Néel phases which depends on the DM coupling is (see Fig.(2)) which agrees with the phase boundary reported in Ref.Alcaraz, . This critical line coincide with boundary line which obtained by classical approximation (see appendix C). The RG equations (Eq.(4)) express the DM coupling dose not flow under RG transformations, and the anisotropy coupling goes to zero () in SF phase while it scales to infinity () in the Néel phase.

We have linearized the RG flow at the critical line and found one relevant and one marginal directions. The eigenvalues of the matrix of linearized flow are , . The corresponding eigenvectors in the coordinates are , . The marginal direction corresponds to the tangent line of the critical line and the relevant direction shows the direction of anisotropy’s flow (Fig.(2)).

However we have found the boundary of the SF-Néel transition by calculating the staggered magnetization (see appendix B) in the -direction as an order parameter (Fig.(3) and Fig.(4)),

 SM=1NN∑i=1((−1)i2)⟨σzi⟩. (5)

is zero in the SF phase and has a nonzero value in the Néel phase. Thus the staggered magnetization is the proper order parameter to represent the SF-Néel transition. We have plotted versus and versus in Fig.(3) and Fig.(4), respectively. In Fig.(3) it is obvious that the staggered magnetization goes to zero continuously at the critical value of which shows the destruction of Néel order. The critical value where the staggered magnetization vanishes above it, increases by increasing of anisotropy, this means SF-Néel transition point () depends on the DM interaction. It is seen in Fig.(4) that the staggered magnetization is zero for (SF phase) and has a nonzero value for (Néel phase), while enhancing of DM coupling defer the creation of Néel order.

Moreover, to study the influence of DM coupling we have calculated the chiral order Kawamura (); Kaburagi ()() in the direction as the helical magnetization in one dimensionJafari2 () which is created by DM interaction.

 Ch=1NN∑i=114⟨(σxiσyi+1−σyiσxi+1)⟩.

Unfortunately the chiral order is not a self similar operator under RG transformations. In fact an term of Hamiltonian () shows up in the renormalized chiral order under RG. Thus, we have to calculate the sum of chiral and term which we have represented it by in the following equation,

 DC=1NN∑i=114⟨(σxiσxi+1+σyiσyi+1) +D(σxiσyi+1−σyiσxi+1)⟩. (6)

In Fig.(3) and Fig.(4) the DC order has plotted versus and respectively, for different values of and . The figures manifest that the order enhances by increasing of DM coupling and reduces with increasing of anisotropy parameter.

We have also calculated the critical exponents at the critical line . In this respect, we have obtained the dynamical exponent, the exponent of order parameter and the diverging exponent of the correlation length. This corresponds to reach the critical point from the Néel phase by approaching . The dynamical exponent is , the staggered magnetization goes to zero like with . The correlation length diverges with exponent . The remarkable result of these exponents is the independence of their values on the D value and equality of them with the correspondening ones in the model. The detail of this calculation is similar to what is presented in Ref.Langari ().

• In this sector, the effective Hamiltonian is similar to the positive anisotropy case with the same coupling constants. For the ground state is the spin-fluid phase and decreasing the anisotropy causes the ground state of the three site Hamiltonian changes by level crossing at where the RG equations should be reconstructed. However, the remarkable result is that the level crossing line which got by three site Hamiltonian coincides the critical line of this model in the thermodynamic systemAlcaraz (). The RG equations express the DM coupling dose not flow under RG transformations, and the anisotropy coupling goes to zero (). Thus the line is the position of stable fixed points where the RG flow is freezed.

iii.2 Region (B)

As we pointed out in sec.II-B the original Hamiltonian is mapped to the ferromagnetic Ising model. Ising model remains unchanged under RG as the stable fixed point and its properties are well known. We call this region as the ferromagnetic Ising phase.

Iv Summery and conclusions

We have applied the RG transformation to obtain the phase diagram, staggered magnetization and helical magnetization of model with DM interaction. In the positive anisotropy region, tuning the anisotropy coupling makes the system to fall into different phases, i.e Néel phase with nonzero order parameter and spin-fluid one with vanishing order parameter as characterized by the staggered magnetization. The RG equations state that the system has fixed points at , and . The fixed points and are attractive and correspond to the Spin-Fluid and Néel phases, respectively. The fixed points at are repulsive and correspond to the critical points of this model, in the other word it is the critical line of this Hamiltonian. However, in the negative anisotropy region, the level crossing line which is obtained by three site block Hamiltonian eigenvalues, is the critical line of infinite size system and separates the ferromagnetic and spin-fluid phases. Unfortunately we can not calculate the chiral order explicitly by RG method. To survey the influence of DM interaction and helical magnetization, the numerical Lanczos computation is in progress.

Acknowledgements.
The authors would like to thank J. Abouie, M. Kargarian and M. Siahatgar for fruitful discussions. This work was supported in part by the Center of Excellence in Complex Systems and Condensed Matter (www.cscm.ir).

V Appendix

v.1 The block Hamiltonian of three sites, its eigenvectors and eigenvalues

We have considered the three-site block (Fig.(1)) with the following Hamiltonian

 hBI=J4[(σx1,Iσx2,I+σx2,Iσx3,I+σy1,Iσy2,I+σy2,Iσy3,I) +Δ(σz1,Iσz2,I+σz2,Iσz3,I) +D(σx1,Iσy2,I−σy1,Iσx2,I+σx2,Iσy3,I−σy2,Iσx3,I)]

The inter-block ( and intra-block () Hamiltonian for the three sites decomposition are

 HBB=J4N/3∑I[(σx3,Iσx1,I+1+σy3,Iσy1,I+1+Δσz3,Iσz1,I+1) +D(σx3,Iσy1,I+1−σy3,Iσx1,I+1)] ,HB=J4N/3∑I[(σx1,Iσx2,I+σx2,Iσx3,I+σy1,Iσy2,I+σy2,Iσy3,I) +Δ(σz1,Iσz2,I+σz2,Iσz3,I) +D(σx1,Iσy2,I−σy1,Iσx2,I+σx2,Iσy3,I−σy2,Iσx3,I)].

where refers to the -component of the Pauli matrix at site of the block labeled by . The exact treatment of this Hamiltonian leads to four distinct eigenvalues which are doubly degenerate. The ground, first, second and third excited state energies have the following expressions in terms of the coupling constants.

 |ψ0⟩=1√2q(q+Δ)(1+D2)[2(D2+1)|↓↓↑⟩−(1−iD)(Δ+q)|↓↑↓⟩−2[2iD+(D2−1)]|↑↓↓⟩], (7) |ψ′0⟩=1√2q(q+Δ)(1+D2)[2(D2+1)|↓↑↑⟩−(1−iD)(Δ+q)|↑↓↑⟩−2[2iD+(D2−1)]|↑↑↓⟩)], e0=−J4(Δ+q), (8) |ψ1⟩=1√2q(q−Δ)(1+D2)[2(D2+1)|↓↓↑⟩−(1−iD)(Δ−q)|↓↑↓⟩−2[2iD+(D2−1)]|↑↓↓⟩], |ψ′1⟩=1√2q(q−Δ)(1+D2)[2(D2+1)|↓↑↑⟩−(1−iD)(Δ−q)|↑↓↑⟩−2[2iD+(D2−1)]|↑↑↓⟩)], e0=−J4(Δ−q), |ψ2⟩=1√2(1+D2)[[2iD+(D2−1)]|↑↓↓⟩+(D2−1)|↓↓↑⟩], |ψ′2⟩=1√2(1+D2)[[2iD+(D2−1)]|↑↑↓⟩+(D2−1)|↓↑↑⟩], e2=0, (9) |ψ3⟩=|↑↑↑⟩   ,   |ψ′3⟩=|↓↓↓⟩, e3=J2(Δ),

where is .

and are the eigenstates of . In Fig.(5) we have presented the different regions where the specified state is the ground state of the block Hamiltonian. In the region (A) the projection operator is

 P0=|⇑⟩⟨ψ0|+|⇓⟩⟨ψ′0|.

The Pauli matrices in the effective Hilbert space have the following transformations

 PI0σx1,IPI0=−2q(σ′xI+Dσ′yI)   ,   PI0σx2,IPI0=4(D2+1)q(Δ+q)σ′xI   ,   PI0σx3,IPI0=−2q(σ′xI−Dσ′yI) PI0σy1,IPI0=−2q(Dσ′xI−σ′yI)   ,   PI0σy2,IPI0=−4(D2+1)q(Δ+q)σ′yI   ,   PI0σy3,IPI0=2q(Dσ′xI+σ′yI) PI0σz1,IPI0=PI0σz3,IPI0=−Δ+q2qσ′zI   ,   PI0σz2,IPI0=Δqσ′zI

In the region (B) () the projection operator is

 P0=|⇑⟩⟨ψ3|+|⇓⟩⟨ψ′3|.

and the Pauli matrices in the effective Hilbert space have the following transformations

 PI0σx1,IPI0=0   ,   PI0σx2,IPI0=0   ,   PI0σx3,IPI0=0 PI0σy1,IPI0=0   ,   PI0σy2,IPI0=0   ,   PI0σy3,IPI0=0 PI0σz1,IPI0=PI0σz3,IPI0=PI0σz2,IPI0=σ′zI.

v.2 Order Parameter and Chiral Order

v.2.1 Staggered magnetization

Generally, any correlation function can be calculated in the QRG scheme. In this approach, the correlation function at each iteration of RG is connected to its value after an RG iteration. This will be continued to reach a controllable fixed point where we can obtain the value of the correlation function. The staggered magnetization in direction can be written

 SM=1NN∑i⟨O|(−1)i2σαi|O⟩, (10)

where is the Pauli matrix in the th site and is the ground state of chain. The ground state of the renormalized chain is related to the ground state of the original one by the transformation, .

 SM=1NN∑i⟨O′|P0((−1)i2σαi)P0|O′⟩.

This leads to the staggered configuration in the renormalized chain. The staggered magnetization in direction is obtained

 S0M = 1NN∑i=1⟨0|(−1)i2σzi|0⟩ (11) = 161N3N/3∑I=1[⟨0′|PI0(−σz1,I+σz2,I−σz3,I)PI0|0′⟩ − ⟨0′|PI+10(−σz1,I+1+σz2,I+1−σz3,I+1)PI+10|0′⟩] = −2Δ+q3q1N3N/3∑I=1⟨0′|(−1)i2σzI|0′⟩=−γ03S1M,

where is the staggered magnetization at the th step of QRG and is defined by .

This process will be iterated many times by replacing with . The expression for is similar to where the coupling constants should be replaced by the renormalized ones at the corresponding RG iteration (). The result of this calculation has been presented in Fig.(3) and Fig.(4).

v.2.2 Chiral Order

The chiral order which is the proper function to detect the helical magnetization in the systems can be written

 Ch=1NN∑i=114⟨(σxiσyi+1−σyiσxi+1)⟩. (12)

As we mentioned in the section III, the term of the Hamiltonian shows up to the chiral order under RG. The term order is written

 DXX=1NN∑i=114⟨(σxiσxi+1+σyiσyi+1)⟩. (13)

In this case the calculating of the chiral order being elaborate, because of the unknown effect of the term on the ground state of system at fixed point (). To simplification of the calculation, we transform the with DM interaction Hamiltonian (Eq.(1)) to the Ising model with DM interactionJafari2 () (IDM) by implement a non-local transformation which shows a rotation about the axis at site where . We have calculated the chiral order (Eq.(12)) of IDM in Ref.[Jafari2 ()]. By implement the inverse of transformation, the chiral order in IDM model transforms to the DC order where introduced in Eq.(6). The DC order has been plotted in Fig.(3) and Fig.(4) versus and .

v.3 Classical Approximation

In the classical approximation the spins are considered as classical vectors which form the spiral structure with a pitch angle between neighboring spins and canted angle

 σxn=cos(nφ)sinθ  ,σyn=sin(nφ)sinθ  ,σzn=cosθ,

The classical energy per site for the with DM interaction Hamiltonian (Eq.(1)) is

 EclN=J4[(cosφ+Dsinφ)sin2θ+Δcos2θ].

The minimization of classical energy with respect to the angles and shows that there are two different regions. (I) , the minimum of energy is obtained by arbitrary and which show the spins projection on axis is nonzero and spins have the helical structure (see Fig.6) in the plain. In this region the minimum classical energy is

 EIclN=J4(√1+D2sin2θ+Δcos2θ). (14)

(II) , the energy is minimized by and arbitrary or arbitrary and , which correspond respectively to the configurations with nonzero value of spins projection on -axis with helical structure of spins projection in the -plain and disorder configuration. In this region the minimum classical energy is

 EIIclN=J4(cosφ+Dsinφ)=J4Δ. (15)

One can see from Eq.(14) and Eq.(15) that the transition between phase (I) and (II) takes place at .

Vi Canonical Transformation

The Hamiltonian of model with DM interaction (Eq.(1)) has the global symmetry. This Hamiltonian is mapped to the well known chain via a canonical transformationAristov (); Alcaraz (),

 U=N∑j=1αjσzj , αj=j−1∑m=1mtan−1(D), ~σ±j=e−iUσ±jeiU , ~σzj=σz, ~H = e−iUHeiU, (16)

which gives

 ~H=J√1+D24[∑i~σxi~σxi+1+~σyi~σyi+1+(Δ√1+D2)~σzi~σzi+1]. (17)

The symmetry of initial Hamiltonian survive in the transformed Hamiltonian too, but at the symmetry breaks to the local symmetry.

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