Infinite matrix product states, boundary conformal field theory, and the open Haldane-Shastry model
We show that infinite Matrix Product States (MPS) constructed from conformal field theories can describe ground states of one-dimensional critical systems with open boundary conditions. To illustrate this, we consider a simple infinite MPS for a spin-1/2 chain and derive an inhomogeneous open Haldane-Shastry model. For the spin-1/2 open Haldane-Shastry model, we derive an exact expression for the two-point spin correlation function. We also provide an SU() generalization of the open Haldane-Shastry model and determine its twisted Yangian generators responsible for the highly degenerate multiplets in the energy spectrum.
pacs:11.25.Hf, 75.10.Pq, 02.30.Ik
Introduction.— For a long time, it has been known that the main curse of quantum many-body theory is the exponential growth of the Hilbert space dimension with respect to the number of constituting particles. In the last decades, the study of entanglement has significantly alleviated this curse, at least to some extent, by recognizing the fact that only a tiny corner of the Hilbert space, with small amount of entanglement, is pertinent for the low-energy sector of Hamiltonians with local interactions. This deep insight lies at the heart of tensor network states Verstraete08 (), a family of trial wave functions designed for efficiently representing the physically relevant states in the tiny corner. The best known instance among them is the Matrix Product States (MPS) in one spatial dimension, described in terms of local matrices with finite dimensions. Their entanglement entropies are bounded by the local matrix dimensions, which are nevertheless sufficient for accurately approximating gapped ground states of one-dimensional (1D) local Hamiltonians Verstraete06 (); Hastings07 (). This discovery not only provides a transparent theoretical picture for real-space renormalization group methods Wilson75 (); White92 (), but also leads to a recent complete classification of all possible 1D gapped phases Pollmann10 (); Chen11 (); Norbert2011 ().
For 1D critical systems, the low-energy physics is usually described by conformal field theories (CFT). Their ground-state entanglement entropies exhibit unbounded logarithmic growth Holzhey04 (); Vidal03 (); Calabrese04 () with respect to the subsystem size, indicating the deficiency of a usual MPS description. To overcome this difficulty, infinite MPS, whose local matrices are conformal fields living in an infinite-dimensional Hilbert space, have been introduced in Ref. Ignacio10 (). The lattice sites for the infinite MPS locate on a unit circle, embedded in a complex plane. This construction shares conceptual similarity to Moore and Read’s approach Moore91 () of writing 2D trial fractional quantum Hall states in terms of conformal blocks. For a variety of examples Ignacio10 (); Anne11 (); Tu13 (); Tu14a (); Tu14b (); Bondesan14 (); Ivan14 (); Benedikt15 (), the infinite MPS (as well as their parent Hamiltonians) have been shown to describe critical chains with periodic boundary conditions (PBC) and, furthermore, their critical behaviors are often related to the CFT whose fields are used for constructing the wave functions exception (). In this sense, the infinite MPS introduced in Ref. Ignacio10 () provide a systematic way of finding lattice discretizations of CFT.
In this Rapid Communication, we show that the infinite MPS ansatz can describe ground states of 1D critical systems with open boundary conditions (OBC), thus complementing the PBC case in Ref. Ignacio10 (). Unlike bulk CFT for periodic chains, open critical chains are instead described by boundary CFT. Taking a spin-1/2 chain as an example, we show how the infinite MPS with an image prescription allows us to derive an inhomogeneous open Haldane-Shastry model, including the original spin-1/2 open Haldane-Shastry models Simons94 (); Bernard95 () as special cases. Within the new formalism, an exact expression for the two-point spin correlator of the spin-1/2 open Haldane-Shastry model is obtained. This, together with numerical results for the entanglement entropy, is in perfect agreement with the theoretical predictions based on boundary CFT, which thus confirms that our infinite MPS with the image prescription is suitable for describing open critical chains. The open infinite MPS construction is readily applicable to any boundary CFT for finding their lattice discretizations. As a further example, we derive an SU() generalization of the open Haldane-Shastry model. We characterize its full spectrum and also determine the twisted Yangian generators responsible for the highly degenerate multiplets in the energy spectrum.
Infinite MPS and parent Hamiltonian.— Let us consider a spin-1/2 chain located on the upper unit circle in the complex plane, with lattice sites and complex lattice coordinates ( and ), see Fig. 1(a). We denote by () the spin-1/2 operators at site . The local spin basis is defined by , where (twice of the projection value). For each site, we introduce its mirror image in the lower unit circle, e.g., site has an image , with complex coordinate . Following Ref. Ignacio10 (), the wave function is written as a chiral correlator of CFT fields:
where ( denotes normal ordering) and (i.e., is the coordinate of the “barycenter” of and on the real axis). Here, is a chiral bosonic field from the free boson CFT, and for odd and even, respectively. Evaluating the chiral correlator in (1) yields a Jastrow wave function
where if and zero otherwise (note that must be even for ensuring a nonvanishing wave function). From the explicit form (2), it is transparent that the sign factor (originated from ) is the “Marshall sign”, since the Jastrow product in (2) is positive.
As shown in Ref. Ignacio10 (), the infinite MPS (1) with coordinate choice , i.e., the case of equidistantly distributed spins on the whole unit circle, yields the ground state of the SU(2) Haldane-Shastry model Haldane88 (); Shastry88 (), which is a paradigmatic spin-1/2 chain with PBC.
Now we demonstrate that our infinite MPS (1) with the image prescription, , describes a spin-1/2 chain with OBC. Let us first derive a parent Hamiltonian for which (2) is the exact ground state. Based on the CFT null field techniques, it was shown Anne11 () that the decoupling equations satisfied by (1) lead to a set of operators annihilating the wave function (2), , where and is the Levi-Civita symbol [we assume summation over repeated indices and use the convention that is the sum over , whereas is the sum over both and ]. When adapting to our present OBC setup, we consider the operators , where and which also annihilate the wave function , since . The parent Hamiltonian for (2) is then defined as , where is the total spin operator and . After some algebra Supp (), we arrive at a long-range Heisenberg model
with ground-state energy , where .
Three choices of the lattice coordinates deserve special attention (see Fig. 1): (i) type-I: ; (ii) type-II: ; (iii) type-III: . For these three cases (termed as uniform cases afterwards), one obtains , , and , respectively. Accordingly, the parent Hamiltonians, after removing the (unimportant) total spin operator and constant terms in (3), have purely inverse-square exchange interactions (between the spins and also their images), which coincide with the open Haldane-Shastry models first introduced in Refs. Simons94 (); Bernard95 (). These uniform models are integrable and have highly degenerate multiplets in their energy spectrum Simons94 (); Bernard95 (), similar to their periodic counterpart Haldane92 (), see Fig. 2 for the full spectrum of the open and periodic Haldane-Shastry models with . We postpone the discussion of this degeneracy until presenting the SU() generalization of these models, where a unified treatment is possible. The Hamiltonian (3) with lattice coordinates other than the three uniform cases is an inhomogeneous generalization of the open Haldane-Shastry models and does not exhibit the huge degeneracy in the spectrum.
Spin correlator.— A nontrivial application of the infinite MPS formulation is that, for the wave function (2), the spin correlation functions can be computed easily. Since , one has and , which lead to a set of linear equations relating two-point correlators Anne11 (), where . These equations are sufficient for computing the two-point spin correlators for arbitrary choices of (both inhomogeneous and uniform cases). The generalization to arbitrary higher-order spin correlators is rather straightforward.
Most remarkably, for the type-I uniform case, these linear equations allow us to find an analytical expression for the two-point spin correlator Supp ()
In Fig. 3 various correlators from (4) are compared with the theoretical predictions Eggert92 () based on the SU(2) Wess-Zumino-Witten (WZW) model with free boundary condition. When two spins at sites and are both far from the boundary, one expects that the correlator recovers the result for PBC Gebhard87 (), for large , where is a constant. However, if one of the two spins (say, the one at site ) is very close to the boundary, the theory developed in Ref. Eggert92 () predicts (: nonuniversal constant) with a boundary critical exponent that differs from in the bulk. For the correlator between nearest neighbors, it was predicted Ng96 (); Laflorencie06 () that , where is the Luttinger parameter, , and are constants. We treat the nonuniversal constants as fitting parameters and find excellent agreement between the exact result (4) and the SU(2) WZW predictions (see Fig. 3).
Entanglement entropy.— To provide further support that the wave function (2) is relevant for open critical chains, we numerically compute the Rényi entropy via Monte Carlo method Ignacio10 (); Hastings10 (), where is the reduced density matrix of the first spins. In Fig. 4 we plot for the wave function (2) in the type-I uniform case with . For open spin-1/2 chains described by the SU(2) WZW model with free boundary condition, one expects the Rényi entropy to be Laflorencie06 ()
with central charge , Luttinger parameter , and nonuniversal constants. Fixing and and treating as fitting parameters, the numerical results are in good agreement with the theoretical prediction (see Fig. 4). For the type-II and type-III uniform cases, we have verified via Monte Carlo simulations that their Rényi entropies also agree with (6), suggesting that they all belong to the SU(2) WZW model with free boundary condition.
SU(n) generalization.— As a further application we generalize the above SU(2) example to the SU() case. For the SU() WZW model, the infinite MPS have been proposed in Refs. Tu14b (); Bondesan14 (). Here we take in all sites SU() spins transforming under fundamental representations, with local basis denoted by (). Following Ref. Tu14b (), the CFT fields for defining the infinite MPS (1) are given by , where is a -component vector denoting the fundamental weight of (e.g., and for SU(3), see Tu14b ()), is a vector of chiral bosonic fields, and is a Klein factor, commuting with vertex operators and satisfying . Evaluating the CFT correlator (1), the SU() wave function takes a simple Jastrow form, (sgn: signature of a permutation), where (), for a given configuration , is the position of the th spin in the state .
Following a procedure similar to the SU(2) case Supp (), we obtain a two-body parent Hamiltonian for , , where () are SU() generators in the fundamental representation, normalized as . The three uniform choices of , very much the same as the SU(2) cases, bring the parent Hamiltonian into SU() open Haldane-Shastry models
with purely inverse-square interactions.
Motivated by the SU(2) result Bernard95 (), we have numerically observed that the full spectrum of the SU() open Haldane-Shastry model (7) is described by the formula, where and (, , and for the three uniform cases, respectively), is an integer satisfying , and are distinct integer/half-integer rapidities (, , and for each individual uniform case), satisfying the generalized Pauli principle which is the same as that for the SU() Haldane-Shastry model with PBC Kawakami92 (); Ha92 (): only those sets without or more consecutive integers/half-integers are allowed Haldane92 ().
Twisted Yangian.— Our numerical results also indicate that the “supermultiplet” structure in the spectrum, which already shows up in the SU(2) case (see Fig. 2), persists in the SU() open Haldane-Shastry models (7). To explain this degeneracy, we slightly generalize the monodromy matrix found for the spin-1/2 open Haldane-Shastry models Bernard95 () to the SU() case. Through a third-order expansion of the monodromy matrix Supp (), we obtain the nontrivial conserved charge responsible for the SU() open Haldane-Shastry models (7)
where , swaps the spin states at site and (more explicitly, ) and and are given by (i) type-I: , ; (ii) type-II: , ; (iii) type III: , , respectively. The conserved charge and the total spin both commute with (7), but does not commute with the SU() Casimir operator . This explains the appearance of degenerate eigenstates with different SU() representations. As the monodromy matrix relevant for these models (with open boundaries) satisfies the reflection equation Sklyanin88 (), the algebraic structure of the SU() open Haldane-Shastry models (7) is the twisted Yangian Olshanski92 (). Thus, the conserved charges and form the lowest twisted Yangian generators.
Conclusions.— In this Rapid Communication, we have shown that infinite MPS with the image prescription are relevant for 1D critical chains with OBC, by presenting a spin-1/2 example, as well as its SU() generalization. We have constructed inhomogeneous open Haldane-Shastry models as their parent Hamiltonians, including the three open Haldane-Shastry models as special uniform cases. For the type-I spin-1/2 open Haldane-Shastry model, an exact expression for the two-point spin correlator has been derived and compared with theoretical predictions, supporting that the low-energy effective theory is the SU(2) WZW model with free boundary condition. We also characterize the full spectrum of the SU() open Haldane-Shastry models and determine the twisted Yangian generators responsible for the highly degenerate multiplets in the energy spectrum. The present infinite MPS with open boundaries is readily applicable to any boundary CFT for finding their lattice discretizations. As an outlook, we expect that the infinite MPS with OBC could be very useful for proposing trial wave functions for single-impurity Kondo problems, where boundary CFT are known Affleck90 (); Affleck91 () to play an important role.
Acknowledgment.— We acknowledge J. I. Cirac and A. E. B. Nielsen for helpful discussions. This work has been supported by the EU project SIQS, FIS2012-33642, QUITEMAD (CAM), the Severo Ochoa Program, and the Fulbright grant PRX14/00352.
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Appendix A Inhomogeneous open Haldane-Shastry models
In this Section, we provide details on the derivation of the spin-1/2 inhomogeneous open Haldane-Shastry model and its SU() generalization.
To construct the spin-1/2 inhomogeneous open Haldane-Shastry model, we use the operators annihilating the spin-1/2 open infinite MPS
to build a positive semidefinite operator
where we have used , , , and . Then, we obtain
The following cyclic identity is the key for simplifying (3):
By using this identity, we obtain
where we have defined and have used (the latter can be easily proved by using the cyclic identity ).
Then, the spin-1/2 inhomogeneous open Haldane-Shastry model is defined by
whose ground-state energy is given by .
The derivation of the SU() inhomogeneous open Haldane-Shastry model follows the similar steps for the spin-1/2 case. The operators annihilating the SU() infinite MPS are given by suppTu14b (); suppBondesan14 ()
where and are the SU() totally symmetry tensor and the totally antisymmetric structure constant, respectively. Similar to the spin-1/2 case, we consider the positive semidefinite operator
where we have extensively used the identities listed in the Appendix A in Ref. suppTu14b (). Notice that
where . Together with (5), we obtain