Bulk Entanglement Spectrum in Gapped Spin Ladders
We study the bulk entanglement of a series of gapped ground states of spin ladders, representative of the Haldane phase. These ground states of spin ladders generalize the valence bond solid ground state. In the case of spin 1/2 ladders, we study a generalization of the Affleck-Kennedy-Lieb-Tasaki and Nersesyan-Tsvelik states and fully characterize the bulk entanglement Hamiltonian. In the case of general spin we argue that in the Haldane phase the bulk entanglement spectrum of a half integer ladder is either gapless or possess a degenerate ground state. For ladders with integer valued spin particles, the generic bulk entanglement spectrum should have an entanglement gap. Finally, we give an example of a series of trivial states of higher spin in which the bulk entanglement Hamiltonian is critical, signaling that the relation between topological states and a critical bulk entanglement Hamiltonian is not unique to topological systems.
Quantum entanglement has become a novel tool in the study of condensed matter systems due to its ability to reveal information about topological phases of matter, where there is no obvious order parameter kitaev2006 (); levin2006 (); jiang2012 (). Entanglement between two subsystems, and , can be characterized by the eigenvalues of the reduced density matrix of region (or equivalently ) which is given by , where is the density matrix of the system in the ground state, i.e. the projector on the ground state. The eigenvalues of are called the entanglement spectrum (ES) PhysRevLett.101.010504 () and have been used to characterize topological order fidkowski2010 (); Chandran2011 (); qi2012 ().
A recent trend in the study of the entanglement spectrum is to partition the system in ways other than the standard real-space bipartition. Examples include momentum-space partitions PhysRevLett.105.116805 (); PhysRevLett.113.256404 (); PhysRevLett.110.046806 (); 2014arXiv1412.8612L (); 1742-5468-2014-7-P07022 (), a partition between spin and orbital degrees of freedom Lundgren (), and an extensive real-space partition which is referred to as the bulk entanglement spectrum (BES) PhysRevLett.113.106801 (). It has been argued that the BES of a topological state encodes information about the transition between the topological and trivial phase within that system PhysRevLett.113.106801 (). This connection has since been explored in quantum Hall states zhu2014 (); fukui2014 (); lu2015 (), holographic and topological superconductors Belin2015 (); borchmann2014 (), the metal-insulator transition Vijay2015 () and symmetry protected topological states PhysRevB.90.085137 (); PhysRevB.90.075151 (); Santos2015 (); lu2015b ().
In this work, we investigate the BES of spin ladders of arbitrary spin. The states that we discuss are generalizations of the valence bond solid (VBS) ground state. In one dimension, the VBS state is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin chain model and its higher spin generalizations Xu2008 (). These states realize a symmetry protected topological phase when the spin of the particles, , in the chain is an odd integer PhysRevB.85.075125 (). For even, the system is topologically trivial. The AKLT spin chain does not have a long range order but instead posses nonlocal ordering characterized by a string order parameter Nijs1986 (), which is representative of the Haldane phase PhysRevLett.50.1153 (); Kennedy1992 (). Recently, it has been argued that the BES in a topologically non trivial state is critical, meaning that the low lying entanglement spectrum levels are either degenerate or gapless (in the thermodynamical limit) PhysRevLett.113.106801 (). This has been shown explicitly for the AKLT ground state and some generalizations PhysRevLett.113.106801 (); PhysRevB.90.085137 (); PhysRevB.90.075151 (); Santos2015 (). In these works, it has been shown that the BES of the VBS ground state is described by a conformal theory of central charge .
We discuss the connection between topologically trivial/nontrivial states and the critical/non-critical BES in spin ladders. As pointed out in Refs. Dagotto1992 () and gogolin2004bosonization (), the Haldane phase can be realized by coupling two identical spin chains. It is then natural to ask whether the BES of the Haldane phase realized in spin ladders is still critical. We find that for spin-ladder ground states representative of the Haldane phase, the BES is indeed gapless. More specifically, we analyze the most general spin 1/2 ladder ground state defined by a matrix product state (MPS) of bond dimension two, invariant under time reversal symmetry. This state contains the AKLT groundstate as a limit. We carry out a complete analysis of this generalized state, computing the two point correlation functions along the legs and the rungs. We show that degeneracies in the transfer matrix make the BES gapless, and compute the bulk entanglement Hamiltonian (BEH) for three inequivalent extensive partitions. In most cases, the BEH for all these partitions corresponds to an XYZ effective spin 1/2 model, where the coupling constants depend on the specific partition. For all partitions considered, the BEH becomes critical whenever the transfer matrix has degenerate eigenvalues.
We also consider a spin symmetric ladder, which is introduced by means of its MPS representation and is adiabatically connected to the VBS ground state. We show that it is possible to extract the BEH (which is an operator acting on virtual spins of size ) perturbatively from the MPS representation. Tracing every other rung of the ladder, we find that the BEH corresponds to a ferromagnetic effective Hamiltonian of a spin chain. Tracing every two other rungs, the BEH corresponds to a antiferromagnetic Hamiltonian. Thus, by the Haldane conjecture PhysRevLett.50.1153 (), we argue that generically the BEH in these systems is gapped for even and gapless for odd. A special deformation is then introduced that shows the non-universality of the bulk entanglement Hamiltonian Chandran2014 (). This deformation links the spin VBS ground state with an symmetric state (with ) in the same phase, keeping finite the correlation length along the deformation. It allows us to obtain exactly the BES by mapping the Rényi entropy to the partition function of a two dimensional Potts model at criticality, where the critical model represents a first order phase transition. We argue that this is a signal of a dimerization transition in the physical ground state.
Our paper is organized as follows. In Sec. II, the BES of spin 1/2 ladders is calculated. In Sec. III, we discuss spin ladders and their bulk entanglement properties. In Sec. IV, we investigate the non-universal properties of the BES. Finally, in Sec. V, we present our conclusions. Some technical information and details are presented in the Appendix.
Ii Spin Ladders
In this section, we consider the BES of gapped spin-half ladders. The ground state wave function of such states can be described in terms of MPS Perez-Garcia2007 (). We focus on MPS (of bond dimension two) that generalize the AKLT PhysRevLett.59.799 (); ALKT_LONG () ground state. More specifically, we consider the following symmetric ground state MPS wave function of the ladder of the form
where is the identity matrix. The Pauli matrices are given by
Here labels the singlet state composed of two spin 1/2 sitting in a rung and label in the triplet state of one rung.
This wave function can be thought as a generalization of the AKLT spin-1 chain where now each site hosts two real spin 1/2 particles and we allow for the existence of singlet states on each site (See Fig. 1 with ). The VBS wave function corresponds to PhysRevLett.59.799 (); ALKT_LONG (). At this point the MPS agrees with the AKLT ground state which belongs to the Haldane phase, with a non-zero string order parameter and fractionalized spin-half edge states Kennedy1992 (). For , the wave function (1) represents a spin-liquid phase PhysRevLett.80.2709 (), which was originally proposed by Nersesyan and Tsvelik PhysRevLett.78.3939 (). We refer to this phase as the NT phase. Both the AKLT and NT ground states are depicted in the bottom of Fig. 1.
The Haldane and NT phases are thermodynamically indistinguishable from each other, however their correlation functions differ drastically PhysRevLett.78.3939 (). In the Haldane phase the spinon excitations of the legs of the ladders confine and form triplet (magnon) and singlet excitations, with a triplet and singlet mass gap, and respectively. There is a coherent peak in the dynamical spin susceptibility, , at and due to the triplet excitations. In contrast, the NT phase has no coherent excitations PhysRevLett.78.3939 (). The entanglement properties of these two phases also have similarities and differences Lundgren (). For a bipartite partition, there is a protected two-fold degeneracy in the entanglement spectrum for both phases. For a partition between the legs of the ladder, the entanglement spectrum of the Haldane phase is described by a conformal field theory (CFT) with a central charge , while the entanglement spectrum for the NT phase consists only of two entanglement levels (when one takes a certain linear combination of the two possible dimer coverings). Combining the results of both of these partitions allows one to tell the difference between these two phases.
We now generalize Eq. (1) further. This will allow us to characterize the different phases of the entanglement Hamiltonian as a function of the MPS wave function. It is defined by
with In this section, we concentrate on the set of matrices , which are given by
where are real numbers and is the state on the rung, composed of two spin 1/2 particles. Eq. (5) is the simplest nontrivial set of matrices leading to a time reversal invariant ground state . Eq. (4) reproduces Eq. (1) for or .
It will prove useful to express Eq. (4) as function of the spin 1/2 particles explicitly. We label the rungs of the ladder by and the legs by (upper leg) and (lower leg). The MPS is explicitly
where is a compact notation for . is the eigenstate of of a spin 1/2 particle located on the th rung and on the th leg with eigenvalue (see Fig. 1). The matrix is an antisymmetric matrix such that . The matrices are in turn
ii.1 On-site symmetries of MPS
The symmetries of the MPS dictate the symmetries of the ground state. These symmetries can be used to classify the symmetry protected topological phases in one dimension Schuch2011 (). We now discuss the symmetries present in Eq. (4) which include time reversal, , and rotation symmetry.
ii.1.1 Time Reversal Symmetry
Under time reversal , changes as . If the ground state is invariant under time reversal, the matrices must transform as
with the complex conjugate of . Here is a matrix realizing a projective representation of . The projective representation acts on the matrix indices of and is independent of and . It is easy to see that for the matrices defined by Eq. (5), the matrix that realizes the projective representation of time reversal symmetry is simply . The simplest nontrivial solution of Eq. (8) is given by Eq. (5).
ii.1.2 and rotation symmetry
The symmetry that we consider is generated by the action of any pair of the operators , with the total spin operator in the rung. Eq. (5) is symmetric under the action of all operators, with , where are Pauli matrices. Eq. (5) also has rotation symmetry around the axis for vanishing or . Full rotational symmetry is achieved for (or and (or .
ii.2 Ground State correlation functions
The MPS representation of the ground state wave function in gapped one dimensional systems makes the computation of correlation functions straightforward (see Fig. (2)) and makes use of the one dimensional transfer matrix
We concentrate on two types of spin-spin correlations, the correlations of operators acting in the same rung, and the correlation of operators acting on different rungs. Note that the last case includes the correlation between different sites. As we will see next, the correlation lengths for the different correlators are connected with the eigenvalues of the transfer density matrix
ii.2.1 Correlations within a rung
The correlation function of two operators, ,, acting on the th rung is
where we have assumed that the ladder is infinite. The vector corresponds to the eigenvector of Eq. (9) with the largest eigenvalue. The matrix is given by
We denote the spin operators by . Two spins on the same rung have the following correlation functions,
Here, , and .
ii.2.2 Correlations at different rungs
The correlation function of two operators and (assuming ) in an infinite ladder is
Using the Eq. (5), we obtain the pair correlation function of spin operators,
with , and being
while , where . The correlation lengths , are explicitly
Finally, ferromagnetic-antiferromagnetic order of the correlation functions is dictated by , which corresponds respectively to , and , with the sign function.
ii.3 Bulk entanglement
In this section, we analyze the bulk entanglement of the spin ladder wave function defined in Eq. (4). In general, to obtain the entanglement of a given state, we need to define two complementary sets of sites which we denote by and . Together, these two sets contain all sites in the lattice. The density matrix that describes the ground state is (for normalized ). Tracing over one of the subsets, say , we obtain the reduced density matrix, . fully characterizes the entanglement between the regions and its complement . If the boundary between and covers the whole lattice, we then have access to the entanglement in the bulk of the system PhysRevLett.113.106801 (). There are of course an infinite number of ways of defining two complementary sets whose boundary traverses the whole lattice. We investigate the simplest of them, invariant under lattice translations.
ii.3.1 Tracing every other rung
We assign the sites in consecutive rungs to different sets, and (see Fig. (3a)). This partition allows us to investigate the entanglement between rungs in the ladder. The BES of this partition in the AKLT spin 1 point has been studied in several works PhysRevB.90.075151 (); PhysRevB.90.085137 (); Santos2015 (). The main result of these works is that the bulk entanglement Hamiltonian is described by a CFT.
In the following discussion, we will make use of the graphical representation of the MPS, already presented in the previous section for the computation of correlation functions. For this partition, has a graphical form given by Fig. (3c), which is an operator that acts in the Hilbert subspace generated by the sites in . The Rényi entropy, , can be obtained by stacking of these objects and contracting the corresponding spin indices, as seen in Fig. (4). The result is that the Rényi entropy is represented in terms of a classical partition function Santos2015 (). For the spin ladder ground state (Eq. (4)), with periodic boundary conditions, the Rényi entropy becomes the partition function of an eight vertex model on a torus PhysRevB.90.085137 (); Santos2015 (); baxter2013exactly ().
More concretely, the Rényi entropy for this rung partition becomes
with . Here, corresponds to the partition function of the eight vertex model generated by the one dimensional transfer matrix (Eq. (9)) (see also Fig. (5i)). The matrix, Eq. (9), defines the Boltzmann weights as
The relationship between the Boltzmann weights and the arrow configurations that define the eight vertex model are presented in Fig. (5ii).
For a matrix of Boltzmann weights of the form
we define two important quantities
The eight vertex model is critical for baxter2013exactly (). After tracing every other rung we have
For the AKLT point, and (or ), (or ), and the Rényi entropy indeed corresponds to the free energy of a critical model with central charge PhysRevB.90.075151 (); PhysRevB.90.085137 (); Santos2015 () as expected.
ii.3.2 Tracing one leg
Another partition that we consider is the partition between the legs of the ladder, which give us information about the entanglement between them. Critical entanglement spectrum has been shown to appear between the legs of spin-ladders PhysRevLett.105.077202 (); FurukawaKim:prb11 (); Lauchli:prb12 (); 2012JSMTE..11..021S (); Fradkin_Ladder (); PhysRevB.88.245137 (). Tracing out the spin states in (which is defined by the black dots in Fig. (6a)), we obtain as depicted in Fig. (6c). Stacking of them to form , we obtain again an eight vertex partition function, but with the Boltzmann weights
where the correspond to the different arrow configurations that define the eight vertex model, as in Fig. (5ii) rotated counterclockwise 45 degrees.
Defining the matrix , we have
in this case
For the AKLT point, signaling that the entanglement between the legs of the ladder in the AKLT-like ground state is also described by a CFT with central charge . This provides a new and additional analytical proof of the numerical results of Ref. PhysRevLett.105.077202 ().
ii.3.3 Zig-zag tracing
Finally, for the MPS defined by Eq. (6), we analyze the entanglement across an alternating partition, depicted in Fig. (7a). To our knowledge, this is the first time such a partition has been considered in spin ladder. The main difference from the previous partition is that the Boltzmann weights of the resulting classical partition function are flipped in consecutive columns, so obtaining the corresponding eight vertex model is not that straightforward as before.
The Boltzmann weights in this case become the same as in (II.3.2) in one row, while in the next row we have to interchange and . Let the column transfer matrix be . For the zig-zag partition, the Rényi entropy of a ladder of length, , becomes
Here the matrix and . In the computation of The matrix has dimension . From (31) it is clear that the role of and is interchanged from column to column. This modification does not change the value of the partition function, as the eight vertex model transfer matrix that construct the lattice column by column has eigenvectors that depend only on the combinations baxter2013exactly ()
This indicates that the column transfer matrix commutes with . The largest eigenvalue of does depend on symmetric combinations of and , so in the limit of we have
ii.4 Bulk Entanglement Hamiltonian
In the previous section, we have shown how the Rényi entropy for three different partitions of the ground state (described by Eq. (5)) is equivalent to an eight vertex model partition function. We now show this connection allows us to have access to the bulk entanglement Hamiltonian, . is defined as the logarithm of ,
where is the norm of the ground state, in our case , where is the length of the ladder. We set this normalization to one.
As discussed previously, the Rényi entropy is build up by stacking -times vertically. The operator is in turn built from the one dimensional transfer matrix , where is defined accordingly for each partition. As builds up the lattice, it is equivalent to the transfer matrix of the 8-vertex model. The logarithmic derivative of the 8-vertex model transfer matrix corresponds to the XYZ Hamiltonian (except at certain parameter points, which we discuss later) baxter2013exactly (). But the logarithmic derivative of the transfer matrix, and consequently , is nothing more than . This establish the correspondence , where represents the different partitions. This is in agreement with earlier work on the. More explicitly in terms of the Pauli matrices and we have
As pointed out in PhysRevB.90.085137 (), inherits the same symmetries as the original MPS. At the isotropic point, and , Eq. (35) has full rotational symmetry and corresponds to the spin 1/2 Heisenberg model. In general, depending on the parameters, can be ferro or antiferromagnetic. In both cases the spectrum of has a degenerate ground state (in the thermodynamic limit) or a unique ground state with a gapless excitations. We see explicitly that for this model, time reversal symmetry or imply a BES either gapless or with degenerate ground state levels.
ii.5 Critical phases of bulk entanglement
In this section, we analyze the relation between the critical phases of for the three types of partitions discussed so far. For the MPS we consider, the critical phases of for rung, leg and zig-zag partitions coincide. After some straightforward algebra, it is possible to show that implies and that implies . So the bulk entanglement becomes critical in all partitions considered above simultaneously. We write this relation as
Similarly, the relation also holds.
We can understand why becomes critical when by considering the transfer matrix (Eq. (9)). The eigenvalues of the transfer matrix are
The condition (and consequently a critical for all three partitions) is simply the condition that . The correlation functions and in particular the correlation lengths in the ladder are precisely controlled by the ratios (Eqs. (19-21)), so we conclude that whenever two correlation lengths coincide, the entanglement in the bulk becomes critical for this MPS.
We now discuss the BES of the NT point in detail. If , which is the case at the NT point, the connection between and the XYZ model breaks down. As mentioned earlier, the NT wave function is simply one possible dimer covering of a spin ladder (corresponding to the red bonds in Fig. (1), without the symmetrization in each rung). Thus, the BES is flat and the bulk entanglement entropy for the alternating rung partition (Sec. (II.3.1)) is simply a measure of how many dimers are present across the partition. This also implies that the zig-zag partition and the leg partition are not longer equivalent. For the leg partition (Sec. (II.3.2), all the singlets belong to the two complementary sets and making the entanglement entropy maximal between these sets. For the zig-zag cut (Sec. (II.3.3), the opposite is true. The singlets belong to either or so the ground state wave function is separable and the entanglement entropy is zero. is proportional to the identity for the leg partition, and a projector onto the ground state in the zig-zag partition. This indicates that for , while . This result is consistent with the results in Ref. Lundgren () at the NT point, up to an unitary transformation in every other rung of the groundstate that permutes the spin states. This unitary transformation translates the MPS representation of our work into the ground state studied in Ref. Lundgren () at the NT point.
Iii Spin- ladder state and connection to spin Aklt
In this section, we will argue that, given that the AKLT state is in the Haldane phase, the structure of the BES is a consequence of the Lieb-Schultz-Mattis theorem Lieb1961 (); Affleck1986 (). We note, unlike the spin-1 AKLT state, the spin- AKLT state can be connected to a topologically trivial state for even spin PhysRevB.85.075125 (); PhysRevB.87.235106 (). As the spin- AKLT state can be realized in a spin ladder, we would like to see how this difference between even and odd integer spins is reflected in the BES of spin ladders with . To this end, we extend the previous discussion to spin particles on each leg of the ladder. We consider partitions where groups of rungs are traced out. When specialized to the AKLT state, these partitions reduces to tracing every other or every two other sites.
We consider the following MPS, which is generalization of the AKLT state (in spin ladders),
where are the Clebsch-Gordan coefficients for two spin S/2 particles and is set of real parameters. The many-body ground state is . This MPS reproduces the spin AKLT ground state for the Kronecker delta Xu2008 () (see App. A). This corresponds to the generalization of having only triplets in the spin-half ladder case.
Considering a bipartition where we trace every other rung (Fig. 3a), the transfer matrix is
Note that each eigenvalue is degenerate due to symmetry. Using the properties of the Clebsch-Gordan coefficients, the transfer matrix can be brought to the spectral form Santosqdef ()
For future reference, we write also the transfer matrix a general bipartition, where we trace consecutive rungs,
From the previous results, it follows that the transfer matrix can be written in the form
Here, with the spin- operator (which acts in the auxiliary space of the bond indices for the MPS) and is the vertical projector operator onto the total spin channel
with and corresponding to consecutive auxiliary spaces. Note that this operator is a projector acting from vertically from bottom to top. Defining the ratio which defines the correlation length i.e. , so for gapped states 111Note that this approximation is increasingly good for bipartitions that contain many consecutive rungs, we can approximate obtained from the transfer matrix (Eq. (46)) up to first order in . In this approximation, takes the form
By performing the unitary transformation on every other site, we can simplify the previous expression using that , The effective Hamiltonian becomes
Eq. (55) corresponds to an effective Hamiltonian of a spin chain. At the VBS point (), for the bipartition where we trace every other rung, is given by
which corresponds to a ferromagnetic spin S/2 chain. For the other bipartition, where we trace every other two rungs, the BEH of the spin VBS becomes
up to an constant. This BEH is antiferromagnetic. This can be seen clearly in the limit where
with given in Appendix C. This even odd effect as a result of the partition appears due to the Neel-like order of VBS ground state.
The results for the generalized AKLT state can be summarized as follows. For an odd integer, corresponds to a ferromagnetic/antiferromagnetic Hamiltonian of a half integer spin, depending on the partition considered. In both cases the BES is gapless or possess a double degenerate ground state. For the antiferromagnetic case, this is proved by the Lieb-Mattis-Schultz theoremLieb1961 (). For an even integer, tracing every two other rungs of the ladder, the BEH is given by (57) which corresponds to an integer spin chain, with antiferromagnetic coupling. Generically, following Haldane’s analysis PhysRevLett.50.1153 (), the BES (for even S) for this partition is expected to have an entanglement gap.
iii.1 Degeneracy Preserving Deformations
In this section, we investigate deformations of the AKLT state. As we discussed in detail in for the ladder, when the eigenvalues of the transfer matrix coincide, can become critical. For the general spin case, we also expect that extra level crossings in the transfer matrix could change the low lying properties of . Usually, to argue that two states belong to the same phase, it is enough to show that during the interpolation the largest eigenvalue of the transfer matrix remains unique, to avoid a quantum phase transition (or more specifically, a divergent correlation length nakahara2012frontiers ()). This condition is clearly not enough to assure that the resulting transfer matrix faithfully represents the phase of the initial entanglement Hamiltonian. An extreme example corresponds to the interpolation between an arbitrary transfer matrix