Exact phase diagram and topological phase transitions of the XYZ spin chain
Within the block spin renormalization group we are able to construct the exact phase diagram of the XYZ spin chain. First we identify the Ising order along or as attractive renormalization group fixed points of the Kitaev chain. Then in a global phase space composed of the anisotropy of the XY interaction and the coupling of the interaction we find that the above fixed points remain attractive in the two dimesional parameter space. We therefore classify the gapped phases of the XYZ spin chain as: (1) either attracted to the Ising limit of the Kitaev-chain which in turn is characterized by winding number depending whether the Ising order parameter is along or directions; or (2) attracted to the Mott phases of the underlying Jordan-Wigner fermions which is characterized by zero winding number. We therefore establish that the exact phase boundaries of the XYZ model in Baxter’s solution indeed correspond to topological phase transitions. The topological nature of the phase transitions of the XYZ model justifies why our analytical solution of the three-site problem which is at the core of the renormalization group treatment is able to produce the exact phase diagram of Baxter’s solution. We argue that the distribution of the winding numbers between the three Ising phases is a matter of choice of the coordinate system, and therefore the Mott-Ising phase is entitled to host apprpriate form of zero modes. We further observe that the renormalization group flow can be cast into a geometric progression of a properly identified parameter. We show that this new parameter is actually the size of the (Majorana) zero modes.
pacs:75.10.Pq, 03.65.Vf, 64.40.ae
The XYZ spin chain is the most anisotropic from of the Heisenberg spin chain where the coupling between , and components of adjacent spins are generically different,
Baxter was able to solve the eight-vertex model BaxterPRL (); Baxter8V () from which also the exact solution of the XYZ model follows BaxterXYZ (); McCoy (). Recent progress in the off-diagonal Bethe ansatz has also enabled exact solutions for arbitrary boundary field ODBA (); odba-npb (). The limit is exactly solvable in terms of Jordan-Wigner (JW) fermions LSM () where the coupling translates into the hopping amplitude of the JW fermions, and the coupling , the deviation from isotropic limit induces p-wave superconducting pairing between the resulting spin-less JW fermions Mattis2006 (). In this limit this model and even generalizations of this model NXY (); Raghu2011 (); Sen () can be solved exactly where the non-trivial topology is encoded in a non-zero winding number (of the ensuing Anderson pseudo-vector) manifests itself as Majorana zero modes localized at the chain ends which is best pictured in terms of the Kitaev chain Kitaev2001 ().
When written in terms of the JW fermions which are particle excitations of the XY limit LSM (), the coupling will correspond to density interaction between the fermions and hence introduces further many-body effects into the problem Fradkin (). Luther noticed that the above JW mapping translates the XYZ model into the lattice version of the massive Thirring model Luther (). This enabled a Bethe ansatz solution for the massive Thirring model Bergknoff (). The equivalence between massive Thirring model and the sine-Gordon (bosonic) theory has its own rich literature Fradkin (); Tsvelik (); Gogolin ().
In the limit we are dealing with a liquid of JW fermions interacting through term which corresponds to massless Thirring model. In this limit if the coupling is below a certain critical value the system remains gapless, but if it is stronger than the system enters the Mott insulating phase and becomes gapped. The picture in the is therefore that of a critical line that ends at a a Berezinskii-Kosterlitz-Thouless (BKT) point and the algebraic correlations of the gapless phase are rendered exponential with a correlation length determined by the spectral gap. The field theory value of is while the exact solution gives Fradkin (). The XXZ limit that would correspond to massless Thirring model has been analyzed from spin systems and entanglement points of view Langari2009 (); Song2013 () where the critical value is obtained to be . In the situation a Dzyaloshinskii-Moriya interaction of strength can be added to the above form which results in a gapless line separating the spin-fluid phase from ferromagnetic and/or anti-ferromagnetic Ising phases depending on the sign of Langari2009 (). The XXZ model in external magnetic field was also found to posses a critical line separating the saturated magnetized phase from the Ising phase at large Langari98 ().
When both the pairing gap and the interaction parameter compete with each other, in the limit where dominates we expect a gapped phase that corresponds to the Mott insulating phase of the corresponding Thirring model. When is negligible the parameter being a pairing strength of the JW fermions, gives rise to a (p-wave) pairing gap. The above gapped phases must be separated by a gapless line in the plane of denNijs (). Indeed Ercolessi and coworkers using the exact solution of Baxter BaxterPRL (); Baxter8V (); BaxterXYZ () calculated the Renyi entropy and identify lines of essential singularity Ercolessi2011 () ending at tricritical points where the lines join. They find that two of the tricritical points are conformally invariant Ercolessi2011 (); Ercolessi2013 ().
Earlier attempt using block-spin renormalization group (RG) was undertaken by Langari Langari2004 () who used a two site cluster to study the phase diagram of the model in presence of a transverse field. However clusters with even number of sites are not able to provide a true Kramers doublet degenerate ground states. Therefore the two low-energy states used in the above work consists in, one singlet, and one triplet (split by the Zeeman coupling) which are not connected by the time reversal operation. In this work, by introducing a conserved charge and making use of the mirror symmetry we are able to break the Hilbert space of the three site problem into blocks of maximum three dimension which can then be analytically diagonalized. Such an analytical solution of the three site cluster of the XYZ model enables us to construct a phase portrait of the XYZ model when both and are present. Before discussing the result let us note that in the Ising limit, ferromagnetic (FM) and antiferromagnetic (AFM) Ising chains are related by a simple unitary transformation at alternating sites. We therefore colloquially use the term ”Ising order” (IO) to refer to both magnetization of the FM Ising case and the staggered magnetization in the AFM Ising case. Now let us summarize the outcome of our block spin RG: (1) First within the XY model (i.e. case) we are able to identify Ising limits corresponding to IO along or as two attractive RG fixed points of the Kitaev chain. This therefore attaches topological significance to IO along or which are characterized with a winding number , respectively. (2) When we turn on the coupling for a region that dominates again we have an attractive Ising fixed point for very large which however is characterized with a zero winding number. To emphasize the non-zero winding number, we call states with IO along and the Kitaev-Ising (KI) fixed points, while the sate with IO along direction the Mott-Ising (MI) fixed point. (3) The gapless lines that separate regions with winding numbers of within our block spin RG treatment using three-site cluster coincides with the exact phase boundaries obtained from the Baxter’s exact solution.
The main message of this paper will be that the XYZ model has essentially three different gapped phases characterized with winding numbers . The phase portrait of the model can be described by BKT repellers and Ising attractors with different topological charge. In the JW representation the phase corresponds to a Mott insulating phases, while the other two correspond to p-wave superconducting states. While the Mott phase of JW fermions corresponds to IO order along , the topologically non-trivial superconducting phase will correspond to IO along or . The winding number of the gapped phases changes between the above three values upon crossing the critical (gapless) lines. The essential significance of the topology is that the topological charges are not sensitive to many details, including the size of the cluster as long as it does not miss the essential symmetries of the Hamiltonian. This explains why a three-site problem that correctly embeds the Kramers doublet structure of the ground states is capable of capturing the exact phase diagram of the model.
Let us start by stating simple, but very important property of the XYZ model. For the XYZ Hamiltonian the quantity is a constant of motion. This is straightforward to see: Assume any arbitrary state with some arrangements of and spins. Operating with the XYZ Hamiltonian on it since there are two consecutive or two consecutive operations on the spins of the system, the total number of spin flips is even and hence either two are turned into two (or vice versa) or the is turned into which does not change the value of . This observation indeed will allow us to analytically nail down the three-site problem and write down its ground state properties in the closed form. Two possible values correspond to number parity of JW fermions. Indeed the JW transformation Mattis2006 (),
where is the phase string defined as converts the above Hamiltonian to,
The XY part of the above Hamiltonian () when rewritten in terms of the following Majorana fermions and becomes,
In the Ising limit , the above Hamiltonian couples every Majorana fermion (MF) with a Majorana fermion to its right (left), leaving a MF at the left (right) of the chain, and one MF at the right (left) of the chain Kitaev2001 (); NXY (); Alicea (), as depicted in the inset of Fig. 1 (See also discussion following Eq. (17)).
ii.1 Three site problem
Consider three sites labeled by for which we would like to construct the matrix representation of the Hamiltonian (I) in the basis which is a dimensional space. Conservation of breaks the Hilbert space into two dimensional blocks. The spin configuration corresponds to and hence . In the sector, there are four states, , , and . The first two are even with respect to reflection with respect to the middle site. Therefore the last two better be combined into even and odd combinations to give the following symmetry adopted basis in sector:
Similarly for the sector all we need is to replace and spins. The Hamiltonian being invariant under reflection with respect to the middle site does not mix even and odd-parity states and hence is already an eigen-state. Straightforward application of the multiplication table 1 to all bonds on the above state reveals the energy of state to be zero.
Using this table we can operate with the Hamiltonian in the three dimensional space of even states which (for both sectors) gives,
The above form is suggestive as it corresponds to two levels at coupled by ”hybridization” of strengths and to a third level at energy . The observation at the three-site level is that the transformation accompanied by does not change the spectrum. However this simple observation helps us to identify a symmetry of the XYZ spin chain in general of any size: Indeed in Eq. (I) this comes from the unitary transformation , , that preserves the algebra. The above transformation when translated into the language of JW fermions is actually a particle-hole transformation only in one sub-lattice which induces the change in the role of and .
ii.2 Block spin renormalization group transformations
When we are dealing with odd number of sites the ground state will belong to a Kramers doublet. In the language of JW fermions this correspond to odd number-parity of JW fermions which has a chance to produce Majorana fermions. In the case of XYZ model where the quantity is conserved, the two degenerate ground states correspond to . The basic idea of block-spin RG is to construct the matrix elements of the operator in the space of the these doublet , as
which being matrix can again be rewritten in terms of Pauli matrices . This can be interpreted as new spin-half degrees of freedom on a coarse-grained lattice Langari98 (). The relation between the new couplings and the old couplings is the block-spin RG transformation. In the following two sections we proceed with implementation of this RG program first for the limiting cases from which we learn about topology of the attractors, and next for a general case.
Iii Limiting cases: XY and XXZ chains
Before considering the solution of the problem in the most general case , it is instructive to consider the special cases first to establish the Ising limit of the Kitaev-chain Hamiltonian as a renormalization group fixed point. We provide analytic solutions for the RG flow and will identify a length scale associated with the MFs. Then we will proceed to construct the whole picture for the phase diagram.
iii.1 The XY limit: and
This limit is exactly solvable by the JW transformation giving a half-filled chain of JW fermions hopping with amplitude and with p-wave superconducting pairing of strength between them Kitaev2001 (). This superconductor belongs to BDI NXY () class of topological superconductors characterized by a winding number. Changing the sign of amounts to changing the winding number. For any non-zero we have a topologically non-trivial gapped (superconducting) phase. The gapped phases corresponding to positive and negative values of are separated by a gap closing at . To set the stages for the general case of the XYZ spin chain, first of all, let us see how can the above picture be reproduced in the block spin RG language.
In the XY limit where , the eigenvalue equation (49) gives three eigenvalues,
with corresponding eigen-states in sectors,
Obviously the Kramers doublet of ground states is given by for , which explicitly reads,
Evaluating the matrix elements of the original spin variables linking the adjacent blocks gives,
where we only need the sites at the boundary of three site cluster. The denotes the coarse-grained Pauli matrices for a given block which describe the fluctuations in the ground state of Kramers doublets . These coarse-grained Pauli matrices are mutually connected to neighboring blocks only through sites . Note that it is not surprising that the spin variables at sites of the cluster transform identically under coarse-graining, as the three site cluster has a reflection symmetry with respect to the middle site which has been nicely manifested in the Kramers doublet ground states (7) and (8). Therefore the interaction terms connecting neighboring blocks are transformed as
Using Eq. (9) allows to identify the coarse-grained couplings and as,
These equations are invariant under which is actually due to symmetry operation discussed under Eq. (5). Since the physics of XY Hamiltonian is given only by the ratio of the above parameters, dividing the above flow equations gives,
Note that the above flow equation is invariant under . This is natural since the above transformation is the same as . Obviously is a fixed point, which then due to this symmetry implies that it is equivalent to fixed point. The invariance of fixed points under implies that there could be fixed points at as well. This can be explicitly obtained by constructing the difference equation Strogatz () for the variable that reads,
The zeros of the right hand side give the fixed points Strogatz () which are both attractive fixed points along with a repulsive fixed point at . The invariance of the flow equations under imply that the infinity is also repulsive fixed point. Under this symmetry operation, the attractive fixed points at map to themselves. This has been plotted in Fig. 1.
The interpretation of this flow diagram is as follows: first of all, corresponding to gapless phase of the filled Fermi sea of JW fermions is a repulsive (unstable) fixed point. In the language of the JW fermions this is nothing but the instability of a Fermi system with respect to the superconducting pairing interactions (i.e. ). Any non-zero flows ultimately to the depending on the sign of . In terms of it means that the smallest of , is renormalized to zero, and the fixed point is an Ising chain polarized along or direction. In such an Ising ground state, every MF of a given type is paired with a MF of opposite type in either right (red, fixed point ) or left (blue, fixed point ). To clarify this let us repeat the analysis of Kitaev Kitaev2001 (): For the fixed point at denoted by red in Fig. 1 the Majorana representation of the Hamiltonian is,
In terms of new fermions denoted by red links in the inset of Fig. 1 , i.e. the above Hamiltonian is simply which does not include the nor which are denoted as unpaired circles in the inset of Fig. 1. Similarly the other fixed point at corresponds to a Kitaev chain where every MF of type is paired with the MF to its left, leaving again two unpaired MFs at the chain end Akbari (). The two fixed points are related by the transformation which in the language of original spins amounts to a rotation around axis of the spins, and .
Therefore the KI fixed points at with their non-trivial topology that ultimately spawn two sharply localized Majorana zero modes at the two ends of the chain are the fixed points of the XY (Kitaev) chain. This interpretation can be understood intuitively: For e.g. the point the resulting MFs are sharply (Dirac delta) localized at the two ends of the chain. If one solves the zero mode eigenvalue problem with a simple Z-transform method NXY () one finds that away from the fixed point, one still has the MFs at the two ends, but this time they are exponentially decaying instead of being Dirac-delta localized (see Eq.21). Take any chain with away from with its exponentially localized pair of MFs at the chain ends, and look at the MFs at larger length scales: After every scale transformation the MF wave function will be more and more localized. This means that at very large length scales the MF will look like a Dirac-delta localized MF. This is the meaning of flowing towards Dirac-delta-localization – i.e. the – point.
Let us now show that the recursive relation (16) can be explicitly solved. Let us change of variable as,
which after a little algebra renders the recursive Eq. (16) to a simple geometric progression for the new variable,
whose solution implies,
where the system size at the ’th RG level, appears quite naturally in this solution.
The relation suggests that must be some sort of length scale. The question is, what kind of length scale is it? The answer to this question is surprisingly simple and physical: Assume that Eq. (4) has a zero mode solution of the form which has amplitude at every site of the lattice. Then it will satisfy the recursive relation,
This equation is solved by the exponentially decaying ansatz, , where,
This is nothing but the transformation (18) in disguise. This relation enables us to interpret the quantity as the length scale associated with the zero modes.
This solution enables us to figure out the flow of the energy gap per site (),
The symmetry of the problem under is reflected at this stage in a symmetric functional dependence on . For the above equation gives . On the other hand the flow equation (15) for will become which gives vanishing for and hence giving a gapless system, in agreement with the exact solution of the XY model NXY (). For any non-zero value of , with the behavior of gap for in mind, the ratio of terms in the above expression for large enough length scales is always close to and therefore the essential factor that determines the behavior of gap is the behavior of at large length scales. Therefore we need to solve the recursive relation (15) for which is,
where is a geometric progression corresponding to the size of MF. The solution to the above equation is,
where we have used the fact that due to geometric progression nature of , the rapidly converges to zero in steps. This causes the terms in parenthesis to come close to which prevents vanishing of the in large length scales. A lower bound for the in large limit is obtained for very small by setting in all the terms in the parenthesis which gives the lower bound for and hence at large length scales as,
Numerical evaluation of the iterative equation quickly converges and results in plots represented in Fig. 2. The trends of the gap curves as function of the initial gap parameter are indicated by arrows in both panels. In the right panel we have magnified the range of to which the limiting (blue) curve given by gives a perfect fit. Indeed zooming in by a further order of magnitude does not change the exponent up to third decimal point, which indicates the quality of fit. It is remarkable that the above exponent is so close to the lower bond estimated in Eq. (26). The exact solution from the JW transformation gives a superconducting state with pairing potential proportional to and hence the exact gap exponent is actually . However, the value of obtained above gives the finite size value of the gap exponent.
A final note is that the behavior of flow Eq. (16) near the critical point (where the gap is zero) is given by . Assuming a divergent correlations length and demanding gives the correlation length exponent for the XY model by .
iii.2 XXZ limit: and
This limit indeed has been studied much in its spin, fermionic (massless Thirring model), and its bosonic disguise (Sine-Gordon model). To slightly generalize it, let us add a Dzyaloshinskii-Moriya (DM) interaction Langari2009 () and on every link we consider the operators,
The Hamiltonian (27) not only conserves the the fermion number parity , but also is invariant with respect to rotation around axis, and hence the total is also conserved. To search for our Kramers doublets we are interested in a sector with total equal to . In the sector we define the following states,
The space is similarly obtained by flipping every spin. With respect to the DM interaction as can be seen from table 1, an imaginary factor as is involved. In this case, changing the sign of amounts to , i.e. the complex conjugation. A formal way to see this in general is that is the eigenvalue of the operator . The label of a state with odd number of sites changes if the operator acts on it. It is readily seen that commutes with and terms while it anti-commutes with DM terms of type. Therefore for any state , there exists a state whose values of are negative of each other, and hence in the sector instead of one has . More compactly for any , the DM interaction upgrades to .
The effect of each individual term of the above Hamiltonian on a two-spin state is summarized in table 1. For sector, the effect of the above Hamiltonian on various states can be easily seen to be,
which gives the matrix representation,
where as emphasized combines into a single complex parameter. The eigenvalues of the above matrix for the and sector are,
For both negative and positive values of the ground state corresponds to , and asymptotically approaches the first excited state at for , but never touches it. Therefore the ground state doublet is given by,
We have used the fact that switching between is equivalent to complex conjugation which replaces and . Let us emphasize again that as far as the multiplication table 1 is concerned the effect of introducing the DM interaction is to replace . This looks like a global gauge transformation by an angle when expressed in terms of the JW fermions that modulates the hopping. Let us see how does it show up in the RG language.
The matrix elements of operator in the Kramers doublet space is summarized as,
Note that the spin at site of a given cluster is connected to spin at site of the neighboring cluster which leads to cancellation of the DM phases , thereby making and scale identically. The above equation in terms of the combined complex variable becomes,
where and are real and incorporate no phase to through the RG scaling. The above equation is remarkable in that it implies that the magnitude scales the the same way as , but the phase does not change:
The fact that in the one-dimensional version of the DM interaction, the does not change as the length scale is changed is actually a manifestation of the fact that the non-zero is basically a global gauge transformation of the limit. As in the rest of the paper we wish to compare the results of the XXZ model against the XYZ, let us fix (corresponding to ).
In terms of , the flow equations (37) becomes,
This naturally suggests to define,
in terms of which the flow equations (38) is simplified to,
In terms of the energy gap per site can be written as or equivalently,
The difference equation describing the flow of is,
which has been plotted in Fig. 3.
The above equation has a repulsive fixed point at which corresponds to the BKT point or .
We have numerically evaluated the flow equation (40) and have used it to generate a flow for the gap of the XXZ model, Eq. (41) in Fig. 4. The flow of the energy gap for the left (right) of BKT point has been plotted in blue (red). For clarity of presentation we have normalized the blue plots to , while the red plots are normalized to which is the dominant and natural energy scale on the right side of the BKT point. The direction of the flow has been indicated by gray arrows. To the left of the BKT point , every thing flows to (corresponding to ) and hence the relevant energy scale is which sets the scale of the energy gap. But on the other hand since for every one always has , the second of Eq. (40) indicates that flows faster than the geometric progression which implies that the energy scale in the large (long length) limit approaches to zero and therefore the left of the BKT point (blue) lines is a gapless phase. Indeed one can do a formal expansion around the non-interacting attractive fixed point at which gives,
where for large , the quantity is exponentially small at large length scales. With this the behavior of gap function for the left of BKT point can be understood and it can be seen why all the blue curves settle on the line.
Now let us move to the right of BKT point in Fig. 4 that corresponds to the Mott phase. Expanding around the BKT repulsive fixed point , we can write,
where and we have used along with the fact that for large , . The first of the above equation shows that in the large limit where diverges, the relevant energy scale is . The second equation indicates how does the scale fade away at large length scales, and the third equation shows why in the where , the ratio of energy gap per site and approaches the in agreement with Fig. 4. However since the energy scale in the right of BKT point is not renormalized to zero, the right of BKT point is actually a gapped phase corresponding to the Mott insulating phase of the underlying JW fermions.
Fig. 4 nicely shows that for every corresponding to the gap settles on which eventually approaches to zero as does in the long length limit. The closer is to zero, earlier in RG steps the gap reaches the zero. For values of that are closer to , a larger RG iteration, i.e. a longer length is required to attain the zero gap. But eventually for long enough length scales, all Hamiltonians with end up in a gapless state. This gapless state terminates at the BKT point .
A further hallmark of the BKT transition is the behavior of the gap to the right of BKT point. In the MI phase the gap is essentially given by the at every length scale. The closer the is to the right of BKT, it takes a longer length to settle the gap to asymptote. Combining the asymptotic behavior of Eqs.(44), (45), (46) we obtain
Equipped with block-spin RG interpretation of the phase transitions of XY and XXZ models, we are now prepared to construct a global phase diagram of the XYZ model.
Iv Phase diagram of the XYZ spin chain
The role of when is to open up a topologically non-trivial bulk-gap. The role of when on the other hands is to open up a an Ising gap, or in the language of the massive Thirring model to open up a Mott gap. To distinguish this Ising limit from the Ising limit of the XY model, we call the former Mott-Ising gap, while for the gapped state due to p-wave pairing of JW fermions we use the term KI gap. Now it is desirable to have both these gap opening mechanisms together, and to study the critical (gapless) line separating these regions. For this purpose we need to analytically diagonalize the three-site Hamiltonian (5) which gives the following equation for the eigenvalues :
This is already in its canonical form , which admits three trigonometric solutions,
where . In the MI limit where hybridizations and vanish, the roots of the cubic equation given by trigonometric formula (50) reduce to,
In the MI limit the ground state is the root that starts at and always remains the ground state, except for the special point where the first excited state touches the ground state. This is shown in Fig. 5. This indicates that always remains the ground state.
Therefore the ground state wave function corresponding to this energy is,
where the naming is chosen such that will have the dimension of energy. Note that the above ground state is non-degenerate as long as . On the boundary of each cluster only the spins and are living which might be connected to neighboring blocks. The symmetry under exchange of site indices in the cluster is manifest in the above Kramers doublet ground states. So we only need to compute the transformation of one of them, e.g. . The computation is straightforward noting that every operation changes the conserved quantity and hence only has off-diagonal components between the two ground states with . For the same reason, operator does not change the charge (fermion parity in the language of JW fermions) and hence only has diagonal components. This gives,
which result in the flow equations,
To proceed further, let us define the dimensionless version of and in units of with . Then the dimensionless version of the flow equations become,