Gapless quantum spin chains:
multiple dynamics and conformal wavefunctions
We study gapless quantum spin chains with spin 1/2 and 1: the Fredkin and Motzkin models. Their entangled groundstates are known exactly but not their excitation spectra. We first express the groundstates in the continuum which allows for the calculation of spin and entanglement properties in a unified fashion. Doing so, we uncover an emergent conformal-type symmetry, thus consolidating the connection to a widely studied family of Lifshitz quantum critical points in 2d. We then obtain the low lying excited states via large-scale DMRG simulations and find that the dynamical exponent is in both cases. Other excited states show a different , indicating that these models have multiple dynamics. Moreover, we modify the spin-1/2 model by adding a ferromagnetic Heisenberg term, which changes the entire spectrum. We track the resulting non-trivial evolution of the dynamical exponents using DMRG. Finally, we exploit an exact map from the quantum Hamiltonian to the non-equilibrium dynamics of a classical spin chain to shed light on the quantum dynamics.
August 6, 2019
- I Introduction
- II Fredkin and Motzkin spin chains
- III Continuum wavefunction and a parent Hamiltonian
- IV DMRG analysis
- V Conclusion and Outlook
- A Some combinatorics
- B DMRG calculations
Quantum critical systems display striking emergent phenomena such as universality in their static and dynamic properties.Sachdev (2011) Realistic models are however often very challenging to study due to the presence of strong interactions. This is why one-dimensional (1d) systems offer an ideal playground because they are often more tractable both numerically and analytically. For instance, many 1d quantum critical systems are described by conformal field theories (CFTs) at low energy. Such theories are highly constrained by symmetry alone.Di Francesco et al. (1999); Fradkin (2013) Even non-equilibrium dynamics following a quench can be studied in detail in 1d CFTs.Calabrese and Cardy (2006) A property that is useful to asses gapless quantum systems, both in and out of equilibrium, has come to the fore in recent years: the structure of entanglement. For instance the bi-partite entanglement entropy (EE) quantifies the amount of quantum entanglement between two subsystems. For 1d CFTs, the von Neumann EE has the form , where is the length of region (an interval embedded in the real line), and is the central charge of the CFT.Callan and Wilczek (1994); Holzhey et al. (1994); Calabrese and Cardy (2004) is a short-distance cutoff. In addition, a related but cutoff-independent quantity, the mutual information between two well-separated intervals has a power law dependence on the separation. The exponent depends on the scaling dimensions of local observables.Calabrese et al. (2009) These results demonstrate that the groundstates of CFTs are highly entangled.
However, many gapless systems cannot be described by relativistic CFTs at low energy. An example that we shall investigate in this work is the spin chain introduced by Bravyi et al: it describes spin exchange with bilinear and biquadratic interactions.Bravyi et al. (2012) It is called the Motzkin model because the groundstate is the equal weight superposition of so-called Motzkin paths (more on this below). Although the groundstate has a large EE, the dynamical exponent that describes the low energy gapless excitations obeys the bound ,Bravyi et al. (2012); Movassagh (2016) excluding the possibility that this model is described by a relativistic CFT. In addition, a recent large-scale DMRG study has shown that this model has different dynamical exponents for different excitations, thus exhibiting multiple dynamics at low energy.Chen et al. (2017a)
A similar spin model with was proposed in Ref. Dell’Anna et al., 2016. This model involves three-spin interaction and is related with Fredkin gates in the field of quantum computation, which explains why it is called the Fredkin model. Its groundstate EE also has logarithmic dependence for an interval embedded in a long chain. In this paper, we perform large-scale DMRG calculations to determine the excited states and show that this model also has , meaning that it cannot be described by CFT, and that it displays multiple dynamics just as the Motzkin spin chain.
Another reason why these spin models are interesting is because their Hamiltonians can be written as a sum of local projectors. The groundstate is annihilated by all the projectors, and has an energy exactly equal to zero. As such, these spin chains share a similar structure to the quantum dimer model (QDM) defined in 2d, and have potential applications in quantum computation due to their “frustration-free” nature. The QDM was introduced by Rokhsar and Kivelson to describe the low-lying singlet excitations in quantum antiferromagnets.Rokhsar and Kivelson (1988) On the square lattice, the QDM model has a special parameter in its phase diagram, known as the Rokhsar-Kivelson (RK) point, where the critical groundstate is an equal weight superposition of all dimer configurations. This state admits an integer valued height representation defined mod 4, Baxter (1982); Nienhuis (1987) and in the continuum limit, the coarse-grained height field can be described by a free compact boson with a dispersion, and therefore the quantum wavefunction is conformally invariant in 2D space.Ardonne et al. (2004) The RK form of the QDM Hamiltonian ensures that one can map the quantum dynamics to a non-equilibrium classical system governed by a Markovian master equation. Under this mapping, the groundstate coincides with the classical equilibrium distribution and the low energy excited state corresponds to the classical relaxation modes. Relying on this quantum-classical connection, Henley used classical Monte Carlo simulations to numerically study the dynamics in QDM and found that the dynamical exponent is Henley (1997); Moessner et al. (2001). This result suggests that the QDM at the RK point can be effectively described by the quantum Lifshitz model Ardonne et al. (2004); Fradkin (2013).
It should be noted, however, that it is not true that all lattice models which can be expressed as a sum of local projection operator must necessarily have dynamics. A case to the point is the 2d quantum eight vertex model of Ref. Ardonne et al. (2004) which, as its classical analog, has two lines of fixed points where the 2D quantum system is critical. While along one of these critical lines, the six vertex line, the dynamics indeed has dynamics, along the so-called Ashkin-Teller line using classical Monte Carlo simulations, Ref. Isakov et al. (2011) found a continuously varying dynamical critical exponent. Here we will find a similar behavior in the Fredkin and Motzkin chains.
Keeping this picture in mind, we will study the groundstates for both Fredkin and Motzkin spin models in the continuum limit and construct possible effective field theory for them. In both spin models, the groundstates are in sector and have height representations which can be considered as the uniform superposition of Motzkin and Dyck paths, respectively.Bravyi et al. (2012); Dell’Anna et al. (2016) The paths are defined in the upper half-plane and can be understood as the trajectories for a classical discrete random walk. In the continuum limit, the groundstate can be described in terms of coarse-grained height field with the constraint . The difference between Dyck and Motzkin paths lies in the diffusion constant of the random walk, carries over to the quantum wavefunction via a dimensionless parameter.
We further construct a bosonic parent Hamiltonian for the continuum groundstate. However, in contrast to the 2d quantum Lifshitz model, the bosonic field theory is not the effective theory for the Motzkin and Fredkin spin chains. Our careful numerical DMRG analysis of the dynamical exponent in the Motzkin Chen et al. (2017a) and Fredkin models suggests that lies close to 3 rather than 2: . This large dynamical exponent in both spin models might be partially caused by the conservation of . Similar behavior was found previously in the Kawasaki non-equilibrium dynamics for a classical Ising spin chain at low temperature, where the relaxation mode is governed by subdiffusive spin motion and has .Kawasaki (1966); Cordery et al. (1981); Grynberg (2010)
We further study the lowest excited state for the Fredkin model in the sector and find a different dynamical exponent , indicating that this model has multiple dynamics. This behavior is common for example in metallic quantum critical points in higher dimensions, where various order parameter fluctuations can have different dispersions.Hertz (1976); Millis (1993); Sachdev (2011); Oganesyan et al. (2001); Meng et al. (2012); Lederer et al. (2016)
Finally, we study the stability of the Fredkin chain with respect to a ferromagnetic Heisenberg interaction; the strength of the new interaction is proportional to the coupling , which varies from 0 to 1. At , this model corresponds to Heisenberg and Fredkin models, respectively. We numerically explore the groundstate and dynamical exponents of our Fredkin-Heisenberg model. Away from the point , the analytical form of the groundstate is unknown. We use DMRG to study the groundstate and find that when , it is different from the Fredkin case at , suggesting that the Fredkin model is unstable in the presence of the Heisenberg. Away from , the dynamical exponent for the lowest excitation drops to a value smaller than 3 and approaches 3 as we decrease to zero. At , although the bulk is the ferromagnetic Heisenberg chain, the lowest excitation cannot be described by the diffusion mode due to the boundary condition being used, which favors up (down) spin on the left (right) boundary. Further study for the gapless low energy excitations in other spin sector demonstrates that the dynamical exponents can also be different in different spin sectors.
The structure of the paper is as follows. We first review both Fredkin and Motzkin spin models in Sec. II. Then we discuss the ground in the continuum limit and compute various local observables and entanglement properties in Sec. III. In Sec. IV.1, we study the dynamical exponent for the Fredkin model, as well as its change in the presence of the Heisenberg interaction. We summarize and conclude in Sec. V. In Appendix A, we briefly explain some combinatorics used in Dyck and Motzkin path calculation. In Appendix B, we gives the detail for DMRG calculation in Fredkin-Heisenberg model.
Ii Fredkin and Motzkin spin chains
We introduce the Hamiltonians for the 2 quantum spin chains that are the focus of this paper. The first one is the Fredkin model Dell’Anna et al. (2016) and has spin 1/2, while the second is the spin 1 Motzkin model Bravyi et al. (2012). We describe the exact groundstates of the 2 spin chains.
where is the vector of Pauli matrices. We work with chains with an even number of sites . In the bulk Hamiltonian, can be expressed in terms of projectors:
where is the singlet built out of neighboring spins. We thus see that the bulk term involves 3-spin interactions leading to two spin exchange processes: and . These two moves are denoted as Fredkin gates in the field of quantum computation.
The Fredkin Hamiltonian is constructed in terms of projection operators which commute with the total operator. The model therefore has symmetry. The unique groundstate is known exactly and corresponds to an equal-weight superposition of states defined through Dyck paths.Dell’Anna et al. (2016); Salberger and Korepin (2016) For example,
A -step Dyck path in the upper half-plane is naturally represented in terms of height variables defined through
meaning that the spin is a discrete derivative of : . The height field is defined on the “dual” lattice so that its index runs from to . In the height representation, since we have spin , the state maps to the step , , while maps to the step , . The Dyck path is a succession of such steps; it starts at , ends at , and obeys the constraint that the height field does not cross below the -axis, i.e. . Examples are shown in Eq.(3) and Fig. 1 (a). As stated below, the groundstate is an equal-weight superposition of all allowed Dyck paths and satisfies and . Such a groundstate is a generalization of the Rokhsar-Kivelson (RK) type wavefunction to one dimension. With some elementary combinatorics (Appendix A), we can show that this groundstate has a large entanglement entropy for a bipartition into regions and . In the limit , the result is Dell’Anna et al. (2016)
When , the EE scales as , which takes a similar form as that for (1+1) dimensional CFTs with central charge .Calabrese and Cardy (2004) However, we will see that this model is not described by a CFT and is in fact less entangled. We mention in passing that the Fredkin model can be generalized to a half-integer spin model with , where the groundstate is equal weight superposition of colored Dyck path and has a square-root violation of the area law.Dell’Anna et al. (2016); Salberger and Korepin (2016); Salberger et al. (2016) This wavefunction can be further deformed into a weighted superposition of Dyck path with the groundstate properties and energy gap studied in Ref. Salberger et al., 2016; Udagawa and Katsura, 2017; Zhang and Klich, 2017.
Similarly, one can construct a spin RK type model with similar properties to the Fredkin chain. This model was actually introduced prior to the Fredkin one in Ref. Bravyi et al., 2012, and was further explored in Refs. Movassagh and Shor (2016); Movassagh (2017); Chen et al. (2017a). The Hamiltonian is Bravyi et al. (2012)
where corresponds to nearest neighbor exchange processes and is defined in terms of projectors,
and are eigenstates of . The boundary term is the projector
On the left boundary, it favors both and states, while on the right boundary, it favors and states. The bulk Hamiltonian is in fact of the anisotropic bilinear-biquadratic form:
where repeated spin indices are summed over. The coefficients are given in Ref. Chen et al., 2017a. The Motzkin Hamiltonian has a U(1) symmetry generated by . Movassagh (2017) Similarly to the Fredkin model, we can construct the groundstate of the Motzkin chain in the height representation with the identification . Since the Motzkin model has , can only jump by integers. The groundstate in the height representation (4) is a uniform superposition of all paths connecting and in the upper half-plane. These paths are formed by three types of moves: diagonal up , , diagonal down , and flat , which correspond to the three states , and in the basis, respectively. Such a type of path is called a Motzkin path; one example is shown in Fig. 1 (b). For example, when , the groundstate of Eq.(6) is
The Motzkin path wavefunction also has large EE and the leading term in EE is the same as that for Fredkin model shown in Eq.(5). Similarly, there is also a higher integer spin model with with the groundstate as the equal weight superposition of colored Motzkin path.Movassagh and Shor (2016); Zhang et al. (2016)
In both Dyck path and Motzkin path wavefunctions, for each height configuration, we have the constraint , which gives rise to the following groundstate expectation value Movassagh (2017)
where . In contrast, commutes with , and we find that the two-point correlation function takes a finite value, as we discuss in the next section.
Iii Continuum wavefunction and a parent Hamiltonian
Although the groundstate for the Fredkin and Motzkin lattice models are known exactly (as given above), it will prove convenient to write down the wavefunctions in the continuum. This will allow for a simple and physical derivation of many results, and will make manifest the emergent symmetries of the models. In particular, we will see that spatial conformal invariance emerges in the bulk of the chain.
iii.1 Continuum version of the Motzkin and Fredkin groundstate
In the continuum limit, the groundstate for both Motzkin and Fredkin models can be written as Chen et al. (2017a)
where the bosonic field represents the coarse-grained height variable introduced above. In the lattice model, the height is set to zero at either end of the chain, hence we impose the following Dirichlet boundary condition on the quantum field
The normalization factor takes the form of a (0+1)-dimensional partition function:
where is the Heaviside function that enforces to be non-negative to match the constraint of the lattice Dyck/Motzkin paths that appear in the groundstate (see Fig. 1).
Eq.(13) gives a unified picture for the groundstate of both the Fredkin and Motzkin models. The difference between the two lattice wavefunctions is encoded in the parameter , and can be understood in terms of a random walk problem. The Dyck and Motzkin paths describe a one dimensional random walk on the non-negative half-integers and integers, respectively, with the horizontal axis of the path as the “time” direction, Fig. 1. The random walks are constrained to start and finish at the origin , which is called a Brownian excursion. The wavefunction (13) is then probability of a given random path, and its specific form follows from the Legendre equation obeyed by Chen et al. (2017a). The diffusion constant of the random walk is , and takes different values for the Dyck and Motzkin chains. For the Dyck type random walk (Fredkin model), the variance at a typical step is , while for the Motzkin type random walk, the variance at a typical step is . Since the diffusion constant is given by , we have in the Fredkin and Motzkin groundstates, respectively.
We can now compute various groundstate properties in the continuum limit. Let us start with the expectation value of . By virtue of Eq.(4), in the continuum limit we have
We thus have . The expectation value of the height field is easily computed by mapping the calculation to that of an elementary quantum mechanics problem Chen et al. (2017a), which offers yet a different perspective on the groundstate. We map the height variable to the position of the quantum particle, , and the position to imaginary time, . The expectation value then maps to Feynman path integral
with the constraint that the particle starts at the origin and ends there at time . is the Euclidean action of the particle:
where the potential is simply a hard wall that prevents the particle from penetrating the region . We can thus rewrite (17) as
where we have used the eigenstates of in the Heisenberg representation. is the probability distribution of finding the particle at position at time , and is thus normalized . We have restricted the integrals to positive values of due to the potential. After evaluating the propagator Chen et al. (2017a), we find
where we have reverted back to the field theory formulation in terms of . We thus get
As expected, we find that vanishes at the boundaries and takes its maximal value at the middle point, with the characteristic square root dependence of Brownian motion. The expectation value Eq.(23) goes from positive at to negative at , which is a consequence of the boundary conditions in the lattice models Eqs.(1,9) that favor the up (down) spin on the left (right) boundary. The expectation value rapidly approaches zero deep in the bulk. Letting and considering the limit , we find that , as shown in Table 1. This result matches the calculation of Refs. Movassagh, 2017; Dell’Anna et al., 2016 for the groundstate of the Motzkin model.
Using a similar method, we can calculate the joint probability distribution function for a path to have its height equal to at and at (Fig. 3 (b)):
Considering the limit deep inside the bulk with