Entanglement in non-unitary quantum critical spin chains
Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to “non-unitary quantum mechanics”, which has seen growing interest from areas as diverse as open quantum systems, non-interacting electronic disordered systems, or non-unitary conformal field theory (CFT). We propose and investigate such an extension here, by focussing on the case of one-dimensional quantum group symmetric or supergroup symmetric spin chains. We show that the consideration of left and right eigenstates combined with appropriate definitions of the trace leads to a natural definition of Rényi entropies in a large variety of models. We interpret this definition geometrically in terms of related loop models and calculate the corresponding scaling in the conformal case. This allows us to distinguish the role of the central charge and effective central charge in rational minimal models of CFT, and to define an effective central charge in other, less well understood cases. The example of the alternating spin chain for percolation is discussed in detail.
pacs:05.70.Ln, 72.15.Qm, 74.40.Gh
The concept of entanglement entropy has profoundly affected our understanding of quantum systems, especially in the vicinity of critical points JPHYSAspecial (). A growing interest in non-unitary quantum mechanics (with non-hermitian “Hamiltonians”) stems from open quantum systems, where the reservoir coupling can be represented by hermiticity-breaking boundary terms Prozen (). Another motivation comes from disordered non-interacting electronic systems in dimensions (D) where phase transitions, such as the plateau transition in the integer quantum Hall effect (IQHE), can be investigated—using a supersymmetric formalism and dimensional reduction—via 1D non-hermitian quantum spin chains with supergroup symmetry (SUSY) Zirnbauer (). SUSY spin chains and quantum field theories with target space SUSY also appear in the AdS/CFT correspondence Beisert (); Volkerreview () and in critical geometrical systems such as polymers or percolation Parisi (). Quantum mechanics with non-hermitian but PT-symmetric “Hamiltonians” also gains increased interest Bender ().
Can entanglement entropy be meaningfully extended beyond ordinary quantum mechanics? We focus in this Letter on critical 1D spin chains and the associated 2D critical statistical systems and CFTs. This is the area where our understanding of the ordinary case is the deepest, and the one with most immediate applications.
For ordinary critical quantum chains (gapless, with linear dispersion relation), the best known result concerns the entanglement entropy (EE) of a subsystem of length with the (infinite) rest at temperature . Let denote the reduced density operator, where is the normalized ground state and . The (von Neumann) EE then reads . One has for , where is a lattice cutoff and the central charge of the associated CFT. For the XXZ chain, .
Statistical mechanics is ripe with non-hermitian critical spin chains: the Ising chain in an imaginary magnetic field (whose critical point is described by the Yang-Lee singularity), the alternating chain describing percolation hulls ReadSaleur01 (), or the alternating chain describing the IQHE plateau transition Zirnbauer (). The Ising chain is conceptually the simplest, as it corresponds to a rational non-unitary CFT. In this case, abstract arguments Doyon1 (); Doyon2 () suggest replacing the unitary result by
where is the effective central charge. For instance, for the Yang-Lee singularity, but ; in this case (1) was checked numerically Doyon1 (). It was also checked analytically for integrable realizations of the non-unitary minimal CFT. The superficial similarity with the result for the thermal entropy per unit length of the infinite chain at suggests that (1) is a simple extension of the scaling of the ground-state energy in non-unitary CFT ISZ (). But the situation is more subtle, as can be seen from the fact that the leading behavior of the EE is independent of the (low-energy) eigenstate in which it is computed Sierra ().
There are two crucial conditions in the derivation of (1): the left and right ground states must be identical, and the full operator content of the theory must be known. These conditions hold for minimal, rational CFT, but in the vast majority of systems the operator content depends on the boundary conditions (so it is unclear what is), and , begging the question of how exactly , and are defined.
In this Letter we explore this vast subject by concentrating on non-Hermitian models with SUSY or quantum group (QG) symmetry. We extend the general framework of Coulomb gas and loop model representations to EE calculations. We derive (1) for minimal non-unitary models, and define modified EE involving the true even in non-unitary cases. We finally introduce a natural, non-trivial EE in SUSY cases, even when the partition function .
EE and QG symmetry.
We first discuss the critical QG symmetric XXZ spin chain PS (). Let be Pauli matrices acting on space and define the nearest neighbor interaction
with , . The Hamiltonian with describes the ordinary critical XXZ chain on sites, but we add the hermiticity-breaking boundary term to ensure commutation with the QG (whose generators are given in the supplemental material (SM)).
Consider first 2 sites, that is . is not hermitian; its eigenvalues are real Staubin () but its left and right eigenstates differ. We restrict , so the lowest energy is (the other eigenenergy is ). The right ground state, defined as , is . We use the (standard) convention that complex numbers are conjugated when calculating the bra associated with a given ket; therefore . The density matrix
(in the basis ) is normalized, . Taking subsystem A (B) as the left (right) spin, the reduced density operator is , and therefore
This coincides with the well-known result for the symmetric (hermitian) XXX chain (). But since is non-hermitian, it is more correct to work with left and right eigenstates defined by and (or , since ). Restricting to the sector we have
where , denote the right eigenstates with energies . The left eigenstates , are obtained from (4)–(5) by . Normalizations are such that , and for . Since we need both L and R eigenstates to build a projector onto the ground state. We thus define
and . We justify the use of a modified trace shortly with both geometrical and QG considerations. Observe that is normalized for the modified trace (note the opposite power of ): . We now define the EE as
Entanglement and loops.
Eq. (7) admits an alternative interpretation in terms of loop models. Since obey the Temperley-Lieb (TL) relations,
their action can be represented in terms of diagrams: contracts neighboring lines, and multiplication means stacking diagrams vertically, giving weight to each closed loop. The ground state of is ( stands for loop). We check graphically that . With the scalar product ordinarily used in loop models (see SM), is correctly normalized. The density matrix is . The partial trace glues corresponding sites on top and bottom throughout (here site ). The resulting reduced density matrix acts only on (site ): . The gluing of creates a loop of weight , so . The agreement with (7) is of course no accident. Indeed, for any spin- Hamiltonian expressed in the TL algebra (and thus commuting with ), the EE—and in fact, the -replica Rényi (see below) entropies—obtained with the modified traces and with the loop construction coincide. We shall call these QG entropies, and denote them .
Coulomb gas calculation of the EE.
For the critical QG invariant XXZ chain with , the EE scales as expected in CFT, but with the true central charge (instead of ), where we parametrized . The simplest argument for this claim is field theoretical. We follow CardyCalabrese (), where the Rényi EE, , is computed from copies of the theory on a Riemann surface with two branch points a distance apart. As the density operator is obtained by imaginary time evolution, we must project, in the case of non-unitary CFT, onto in the “past” and on in the “future”, to obtain .
We calculate the QG Rényi EE using the loop model. The geometry of CardyCalabrese () leads to a simple generalization of well-known partition function calculations DFSZ (): an ensemble of dense loops now lives on sheets (with a cut of length ), and each loop has weight . Let denote the partition function. Crucially, there are now two types of loops: those which do not intersect the cut close after winding an angle , but those which do close after winding . To obtain the Rényi EE, we must find the dependence of on .
To this end we use the Coulomb gas (CG) mapping Nienhuis (); JesperReview (). The TL chain is associated with a model of oriented loops on the square lattice. Assign a phase to each left (right) turn. In the plane, the number of left minus the number of right turns is , so the weight results from summing over orientations. The oriented loops then provide a vertex model, hence a solid-on-solid model on the dual lattice. Dual height variables are defined by induction, with the (standard) convention that the heights across an oriented loop edge differ by . In CG theory, the large-distance dynamics of the heights is described by a Gaussian field with action and coupling .
With replicas, we get in this way bosonic fields . The crux of the matter is the cut: a loop winding times around one of its ends should still have weight , whilst, since on the Riemann surface, it gets instead . We repair this by placing electric charges at the two ends (labelled ) of the cut, and , where will be determined shortly. More precisely, we must insert the vertex operators before computing . This choice leaves unchanged the weight of loops which do not encircle nor intersect the cut. A loop that surround both ends (and thus, lives on a single sheet) gathers from the turns, and from the vertex operators (since the loop increases the height of points and by ). The two contributions give in the end , summing up to as required. Finally, for a loop encircling only one end we get phases , so the correct weight is obtained setting .
To evaluate the we implement the sewing conditions on the surface, with mod , by forming combinations of the fields that obey twisted boundary conditions along the cut. For instance, with , we form and . While does not see the cut, is now twisted: . For arbitrary , the field does not see the cut, while the others are twisted by angles with . Using that the dimension of the twist fields in a complex bosonic theory is Martinec () we find that the twisted contribution to the partition function is with . Meanwhile, the field , which would not contribute to the EE for a free boson theory (here ), now yields a non-trivial term due to the vertex operators with : with . Assembling everything we get . Inserting and gives the Rényi entropies
( is obtained for ), hence proving our claim.
We emphasize that the spin chain differs from the usual one simply by the boundary terms . These are not expected to affect the ordinary EE, and the central charge obtained via the density operator (with , but normalized as in our introduction) will be .
Entanglement in non-unitary minimal models.
We now discuss the restricted solid-on-solid (RSOS) lattice models, which provide the nicest regularization of non-unitary CFTs. In these models, the variables are “heights” on an Dynkin diagram, with Boltzmann weights that provide yet another representation of the TL algebra (8), with parameter and . The case is Hermitian, while leads to negative weights, and hence a non-unitary CFT. One has , and, for , the effective central charge—determined by the state of lowest conformal weight ISZ () through —is . The case gives the Yang-Lee singularity universality class discussed in the introduction.
Defining the EE for RSOS models is not obvious, since their Hilbert space (we use this term even in the non-unitary case) is not a tensor product like for spin chains. Most recent numerical and analytical work however neglected this fact, and EE was defined using a straightforward partial trace, summing over all heights in compatible with those in . In this case, it was argued and checked numerically that in the unitary case, and in the non-unitary case. Note that matches that of the loop model based on the same TL algebra, with . For details on the QG EE in the RSOS case, see the SM.
The RSOS partition functions can be expressed in terms of loop model ones, . In the plane, the equivalence Pasquier () replaces equal-height clusters by their surrounding loops, which get the usual weight through an appropriate choice of weights on . With periodic boundary conditions, the correspondence is more intricate due to non-contractible clusters/loops. On the torus DFSZ1 (), is defined by giving each loop (contractible or not) weight , whereas for the RSOS model contractible loops still have weight , but one sums over sectors where each non-contractible loop gets the weights for any . The same sum occurs (see SM for details) when computing of the Riemann surface with replicas: non-contractible loops are here those winding one end of the cut. Note also that for RSOS models, so the imaginary-time definition of in unambiguous Doyon1 (); Doyon2 ().
Crucially, the sum over is dominated (in the scaling limit) by the sector with the largest , that is and . In the non-unitary case (), , and the EE is found by extending the above computation. We have still , but now . To normalize at , one must divide by to the power , with the same charges:
EE in the SUSY chain.
Percolation and other problems with SUSY (see the introduction) have , hence , and the EE scales trivially. Having a non-trivial quantity that distinguishes the many universality classes would be very useful. We now show that, by carefully distinguishing left and right eigenstates, and using traces instead of supertraces, one can modify the definition of EE to build such a quantity.
We illustrate this by the alternating chain ReadSaleur01 () which describes percolation hulls. This chain represents the TL algebra (8) with , and involves the fundamental () and its conjugate () on alternating sites, with . The 2-site Hamiltonian, , restricted to the subspace (where are bosonic and is fermionic), reads
The eigenvectors are and ; note that conjugation is supergroup invariant (i.e., ). Hence, despite the misleading expression, is not unitary. The density operator is and satisfies . The reduced density operator . If we define the Rényi EE also with the supertrace, we get for all . It is more interesting (and natural) to take instead the normal trace of ; this requires a renormalization factor to ensure . We obtain then and thus . This equals the QG Rényi EE with .
This calculation carries over to arbitrary size. One finds that with weight , provided non-contractible loops winding around one cut end in the replica calculation get the modified weight instead of . We can then use the CG framework developed in the context of the non-unitary minimal models to calculate the scaling behavior. We use (10), with for percolation (), and . It follows that is purely imaginary, and that with .
All these results were checked numerically. As an illustration, we discuss only the case , for which the RSOS and loop models have , while for the RSOS model. In the corresponding chain, we measured the (ordinary) EE as in (3), the QG Rényi EE as in (7), and the QG Rényi EE for the modified loop model where non-contractible loops have fugacity (instead of ). This, recall, should coincide asymptotically with the Rényi EE for the RSOS model. Results (see figure 2) fully agree with our predictions.
While we have mostly discussed the critical case, we stress that the QG EE can be defined also away from criticality. An interesting example is the alternating chain, for which staggering makes the theory massive (this corresponds to shifting the topological angle away from in the sigma-model representation). Properties of the QG Rényi EE along this (and other) RG flows will be reported elsewhere.
To summarize, we believe that our analysis completes our understanding of EE in 1D by providing a natural extension to non-unitary models in their critical or near-critical regimes. There are clearly many situations (such as phenomenological “Hamiltonians” for open systems) where things will be very different, but we hope our work will provide the first step in the right direction. Our approach also provides a long awaited “Coulomb gas” handle on the correspondence between lattice models and quantum information quantities. In the SM we apply this to show that, in the case of non-compact theories, the well-known term will be corrected by terms (with, most likely, a non-universal amplitude), in agreement with recent independent work BenjaminOlalla ().
Acknowledgments: The work of HS and JLJ was supported by the ERC Advanced Grant NuQFT. The work of HS was also supported by the US Department of Energy (grant number DE-FG03-01ER45908). We thank B. Doyon and O. Castro-Alvaredo for inspiring discussions and comments.
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Appendix A Supplementary Material
In these notes we provide additional details for some results of the main text. We first provide additional motivation for our definition of the entanglement entropy (EE) from the perspective of the quantum group (QG) symmetry, and we prove that . Next, we give more examples of the computation of the QG EE for larger larger systems in various representations. We elaborate on the construction in the RSOS case, detailing in particular the mapping between the RSOS and loop model representations. Finally we discuss the emergence of a term in the non-compact case.
a.1 symmetry for the reduced density operator
Our definition of the EE relies on using a modified trace, known as a Jones trace, in which a factor of the type is inserted under the usual trace symbol. To ensure that the resulting reduced density operator makes sense in the QG formalism, we must ensure that it commutes with the generators of .
We therefore consider the XXZ spin- chain, with boundary terms as described in the main text. The Hamiltonian commutes with the following generators:
Since the generators commute with the Hamiltonian, they share the same right and left eigenvectors. As a consequence they commute with the density operator
We split the spin chain in two parts , and define the reduced density operator using a Jones trace over the part . We consider the case where is in the middle of the chain between and , so that and . Thus
Let us check that the generators of on the subsystem commute with the reduced density operator . We have the following relations:
Consider first :
Obviously , , and commute. Since also commutes with :
For the two last terms we performed a cyclic permutation under the trace. We can now sum all terms and this proves . Next we do the same for :
The first term of the right-hand side reads
thanks to the cyclic permutation under the trace and the commutation of and . We then deal with the second term involving :