Entanglement Entropy of Local Operators in Quantum Lifshitz Theory
We study the growth of entanglement entropy(EE) of local operator excitation in the quantum Lifshitz model which has dynamic exponent . Specifically, we act a local vertex operator on the groundstate at a distance to the entanglement cut and calculate the EE as a function of time for the state’s subsequent time evolution. We find that the excess EE compared with the groundstate is a monotonically increasing function which is vanishingly small before the onset at and eventually saturates to a constant proportional to the scaling dimension of the vertex operator. The quasi-particle picture can interpret the final saturation as the exhaustion of the quasi-particle pairs, while the diffusive nature of the time scale replaces the common causality constraint in CFT calculation. To further understand this property, we compute the excess EE of a small disk probe far from the excitation point and find chromatography pattern in EE generated by quasi-particles of different propagation speeds.
In the past two decades, much effort was devoted to the study of the entanglement entropy(EE) in the quantum many-body systems. In the condensed matter systems in particular, EE serves as an valuable probe that can extract information such as topological datalevin_detecting_2006 (); kitaev_topological_2006 () and universal quantitiesvidal_entanglement_2003 (); fradkin_entanglement_2006 (); hsu_universal_2009 (); stephan_shannon_2009 () at critical points. For a review, see a list of literaturescalabrese_entanglement_2004 (); calabrese_entanglement_2009 (); calabrese_quantum_2016 (); fendley_topological_2007 (); nishioka_holographic_2009 () and references therein.
The dynamical behavior of the EE is one topic in this field. It provides a route to experimental measurement of EE in an extended system (see cardy_measuring_2011 () for a proposal of measuring Rényi entropy in an extended condensed matter system), which is still very difficult up to now (see islam_measuring_2015 () for a static measurement of second Reńyi entropy in an optical lattice with 4 cold atoms). The quench dynamics is also theoretically interesting on its own right. In a quench process, one prepares an initial state which is usually the groundstate of some Hamiltonian and then unitarily evolves the system with a different Hamiltonian . The EE is presumed to capture the spreading of excitation on the way to equilibrium.
There are generally two types of quench protocols:
Global quench. The global quench protocol evolves the ground state of with a globally different Hamiltonian . Consequently, the initial state has an extensive amount of extra energy compared to the ground state of and EE will grow tremendously afterwards. Computation in translational invariant 1+1d critical systems shows a linear growth of EE immediately after applying the new Hamiltonian. The EE ultimately saturates to a value that is proportional to the size of the subsystemcalabrese_entanglement_2007 ().
Local quench. In contrast, the local quench protocol only weakly perturbs the initial system. One example of this (usually called “local quench” or “cut and join” protocol in the literature) is to prepare two identical 1d ground states of and then join them together and evolve with the same form Hamiltonian of the doubled system. The only difference lies at the connecting points where the dynamics that used to be set by boundary conditions is now determined by a bulk term in the Hamiltonian of the doubled system. Again in the critical system, a CFT calculation reveals a logarithmic growth of EE whose coefficients is the 1/3 of the central charge of the CFT. This result is similar to the EE of a single interval on ground state of 1+1s CFT, in which case the length of the interval is in place of the time difference here.
What we will study in this paper is a local quench weaker than “cut and join”, which is called quench of local operator excitations. As the name suggests, we let a local operator act on the initial state and then evolve with the same Hamiltonian. Equivalently, one sets the new Hamiltonian as the sum of and a delta function pulse of local operators at the moment just before the quench. One meaningful measure here is the excess EE compared to that of the ground state, which is expected to reflect the strength and spreading of excitation created by the local operators. In critical systems described by a CFT, local primary field excitations are studied in he_quantum_2014 () and the excess EE increases for a long time to a limiting value equal to the logarithm of the quantum dimensionhe_quantum_2014 (). The growth of excess EE after the local operator excitation is constrained by causality: the excess EE is zero until the signal traveling at speed of light reaches the entanglement cut. The causality constraint and the saturation behaviors are kept in free boson systems in higher dimensionsnozaki_quantum_2014 (); the only change is the way of development of excess EE from zero to its maximal value.
The results of all these three protocols can be quantitatively explained by the physical picture of quasi-particles. The extra energy compared to the ground state of is assumed to be carried by coherent quasi-particle pairs. The subsequent time evolution separates the individual quasi-particles and EE is gained when the one of them in the pairs crosses the entanglement cut. In short, the generation of excess EE is ascribed to the proliferation and propagation of those quasi-particle pairs. We give a review about this framework in section II.
As far as we know, there is yet no analytic result of EE of local operator excitations in a non-relativistic system. Consequently, in this paper, we study the excess EE in such a system. The one we study is the quantum Lifshitz model whose dynamical exponent (while CFT has ) in the presence of the local vertex operator excitation. The model describes a critical line of the quantum eight-vertex model with one special point corresponding to the quantum dimer model on bipartite lattice. The scale invariance of the ground state wavefunction is what makes analytic calculation possible.
We take two subsystems, one the upper half plane, the other a disk and find that the excess EE will grow immediately after the local excitation and reach a limiting value of order the scaling dimension of the vertex operator. The typical time scale when the excess EE is considerable, i.e. of order of the maximal value, is the distance from excitation to the entanglement cut squared. This is when the quasi-particles diffuse to the entanglement cut, in consistent with . We also find small plateau structures in the short time dynamics of excess EE and conjecture that it reveals quasi-particle density of states and possible dispersion of different species during the propagation. In summary, the quasi-particle picture can still qualitatively interpret the results with a slight modification that replaces causality constraint with a diffusive behavior.
The structure of this paper is as follows. In section II, we review the quasi-particle picture of interpreting the dynamics of EE in the two quench protocols. Then in section III, we introduce the quantum Lifshitz model and the vertex operator excitation we will be focusing on. We define the excess EE in section IV and evaluate it for the upper half plane and disk in section V. We finally summarize our results in section VI.
Ii Review of the quasi-particle picture
Quasi-particle picture is a heuristic way of understanding the phenomenology of EE change in the quench problem. It is exact in the CFT calculation, and is also believed to be valid beyond CFT. Here we review the quasi-particle interpretation of the two quench protocols in order.
In the global quench protocol, the excess energy compared with the true ground state of the Hamiltonian is distributed across the system. For a translational invariant Hamiltonian, same types of quasi-particles are radiated on each point of the system. The number of entangled pair between region A and B is proportional to the area of the green region in Fig. 2, which grows linearly and saturates to the maximal value after ( is the length of subsystem A). This explains the linear growth and extensive saturation valuecalabrese_evolution_2005 (); calabrese_quantum_2016 ().
In the “cut-and-join” protocol of local quench, the extra energy is only distributed in the vicinity of the joint points. If we choose the region A to be a single interval that has distance to the joint point, then the EE will keep the ground state value of until the time when the quasi-particle traveling at the speed of light arrives the entanglement cutcalabrese_entanglement_2007 (); calabrese_quantum_2016 (). It will grow logarithmically afterwards. This picture naturally gives rise to the horizon effect.
For the quench of local operator excitation, the quasi-particle is created exactly at the point of the operator insertion. Again if the excitation point has a distance to the entanglement cut, causality constraint forces the entanglement to be unchanged until the arrival of quasi-particles, see Fig. 2. Here the EE will not be extensive since the local excitation only add a very small amount of single particle energy to the ground state. Its strength can be quantified by the quantum dimension, which represents the degrees of freedom of the quasi-particles. We see that the saturated value of excess EE is indeed proportional to this strengthhe_quantum_2014 (). This is another example of extracting topological data from EE.
Iii Introduction to Quantum Lifshitz Model and Its Dynamics
iii.1 Quantum Hamiltonian
The Quantum Lifshitz model is a compact boson theory that describes the critical behavior of the quantum eight-vertex modelardonne_topological_2004 (); fradkin_field_2013 (). The quantum Lifshitz model has Hamiltonian
where is a compact boson field and is its conjugate momentum. Due to the absence of the regular stiffness term , this theory does not have Lorentz symmetry. The dynamic exponent is .
By varying values of , the Hamiltonian in equation (1) can in general model a critical line of the quantum eight vertex modelardonne_topological_2004 (). What we have in mind however, is the Rokhsar-Kivelson(RK) critical point of the square lattice quantum dimer modelmoessner_quantum_2011 (); fradkin_entanglement_2006 (); henley_relaxation_1997 (); ardonne_topological_2004 () which is at . There, the compact boson field is naturally identified as the coarse grained height field on square latticeardonne_topological_2004 (); moessner_quantum_2011 ().
It is generally believed that the Hamiltonian (1) gives the correct time evolution of the quantum dimer model. We here present two heuristic ways to justify this point.
One of the them is a Ginzburg-Landau type argument that keeps lowest order possible terms that consistent with the required symmetryfradkin_field_2013 (). In the dimer problem, translational and rotational symmetries enforce the Hamiltonian to have the following form
When , the system will flow to a phase that pin the field to fixed value, which is identified to be the columnar phase away from the RK point. On the other hand, corresponds to an unstable Hamiltonian that is not semi-positive definite. At , reduces to the Hamiltonian (1) and can be diagonalized using
as a series of harmonic oscillators (normal ordered), such that
The ground state wavefunction is thus annihilated by and has a Gaussian form
where is partition function of the free compact Boson
This reproduces the fact that at Rokhsar-Kivelson critical point, the dimer density operator (derivative of the boson)kasteleyn_statistics_1961 (); fisher_statistical_1961 () has a power law correlation function.
Another independent derivation is proposed by Nienhuisnienhuis_two_1987 () and Henleyhenley_relaxation_1997 () who map the quantum Hamiltonian of “flipping” dynamics to a Monte Carlo process. The classical Monte Carlo process gives exactly the same probability distribution as the ground state wavefunction on the dimer basis. And the relaxation to the equilibrium is the imaginary time quantum evolution. To obtain a continuous description, the stochastic process is heuristically written as a Langevin equation with a Gaussian noise. The Hamiltonian (4) which governs the corresponding master equation is then identified as the effective Hamiltonian.
The vertex operator () is a sensible set local operators in the compact boson theory which create a Boson coherent state. It is also the electric operator in the quantum dimer model contextfradkin_field_2013 (). We consider the excess EE generated by this local excitation. Specifically, we act the vertex operator on the ground state and evolve for time to reach a state
The excess EE is defined to be the difference of EE between and . As a function of time, it should reflect the spreading of the local excitation.
iii.2 Time Evolution of the Vertex Operator
In this subsection, we solve the time evolution equation and express the in terms of the ground state boson operators. For clarity, in this subsection we temporarily turn to the hatted notation to denote the boson operator and reserve the unhatted for the eigenvalue of the basis.
Since the ground state is annihilated by , we rewrite the state as
where the time dependent boson operator is the solution of Heisenberg equation
The sign in indicates another equivalent convention used in he_quantum_2014 () that interprets the as the time of local excitation and as the time of measurement.
The and are the non-standard creation/annihilation operators; they have the commutation relation
We can then solve their Heisenberg equations
which gives the time evolution of the boson operator
Now consider acting the operator on the ground state. We have
One should interpret as a Schödinger time evolution. We add a small real positive constant to the imaginary time
to control the ultraviolet divergence; this is the damping term that screens out the high energy modecalabrese_evolution_2005 (). The operator obeys
whose solution in free space is given by
where is the standard heat kernel in 2d
The time evolved vertex operator is thus
on the ground state.
Iv Excess Entanglement Entropy
In this section, we define and derive the replica formula for the excess EE.
The time dependent EE after the local excitation at is the Von Neumann entropy
w.r.t the time dependent density matrix
associated with the state .
The excess EE is defined to be
where the symbol in this paper always denote the difference between time and time 0.
The way to compute EE is to take the analytic continuation of the Rényi EE
at . In field theory setting, the quantity has replicated fields properly glued togethercalabrese_entanglement_2009 (), thus the name “replica trick”.
We evaluate this time dependent EE by using replica trick at each time slice. In fact, the EE of static ground state EE has been evaluated and refined by many groups fradkin_entanglement_2006 (); oshikawa_boundary_2010 (); zaletel_logarithmic_2011 (); stephan_shannon_2009 (); zhou_entanglement_2016 (). Here we extend the trick used in fradkin_entanglement_2006 (); zhou_entanglement_2016 () to an excited state.
The extension is different from the replica trick commonly used in CFT calculationcalabrese_entanglement_2009 (), so we give a brief review of its derivation. We use discrete notation to derive . Denoting and as sets of complete orthonormal basis on subsystem A and B, for the ground state density matrix , we have
This expression has copies of fields, while the delta functions enforce the constraint that the fields of odd indices are created by concatenating parts from adjacent fields of even indices. It is then equivalent to remove those odd fields; meanwhile duplicate the even fields and require them to have same value on the cut. The later condition on cut ensures the possibility of stitching two parts from two even fields to create the odd field between them. This procedure is depicted in 3
The derivation for the excited state is almost the same except for the insertion of operators and in front of the partition function.
So relabeling the surviving even field from to , we have
where the point function in the numerator is evaluated on the manifold in Figure 3. This formula has similar structure as the CFT calculation in he_quantum_2014 (), where the 2 point function in the numerator is evaluated on the -sheeted Riemann surface.
We use target space rotation to deal with the gluing condition. First we separate the field into classical and quantum part
such that and , then the quantum fluctuation has Dirichlet boundary condition and the action separates
Notice that the classical part does not evolve with time
so in the average on the glued manifold , the real valued classical field on the vertex operator cancels,
where the summation is subject to the boundary condition
Since the field satisfies Dirichlet boundary condition on the entanglement cut, we have
as a result
The summation of the classical modes is the same as the ground state, so that rotational trick used there works in the same way. We rotate these classical fields by the following unitary matrix
such that after the rotation
It is noted that the -th field is the center of mass mode that is the only one dependent on the value on the cut
while the rest of the fields decouple
The degrees of freedom on the entanglement cut determine , while the rest determines the Dirichlet two point function, hence if we combine the two, we should collect all the degrees of freedom in this region and end up with a free two point function
In fact, we can do this with the identification of the field
This free two point function will cancel one of the two point functions in the denominator. We therefore have
Note that the sum over is time independent and as a result will be canceled in the excess EE. The excess Rényi entropy is therefore
where now denotes the non-compact free boson and denotes the difference between time and .
iv.1 Green Function
The two point function of the vertex operators ultimately will be reduced to the Green function of the boson field. In this subsection, we define and calculate the free space Green function in space and time directions.
The equal time Green function on the ground state
satisfies Laplace equation
In free space( plane), the solution is well known
We define the equal space Green function, which is useful later, to be the Green function evaluated at the same position but different imaginary time
By taking the derivative and use the property of the heat kernel,
Convergence of the integral requires , which is satisfied by adding a damping parameter to both imaginary time and . Up to a constant
The rather than factor is a manifestation of . These two limits of the Green function agree with the general Green function expression in fradkin_field_2013 ().
The Green function in this case is associated with the operator . In the following we will instead calculate in terms of the Green function of the standard Laplacian operator , and relate it to the two point function via
iv.2 Excess Entanglement Entropy in terms of Green Function
In equation (41), we need the two point function of the diffused vertex operator
The exponent is a source term in the Gaussian path integral
Here is a real operator because
we thus have the standard results for Gaussian integral
The integral consists of four Green functions in time direction
Define the cross Green function
the excess Rényi entropy is
V Results and Discussion
We have obtained the free space Green function in subsection IV.1, where the equal space Green function is evolved from the equal time Green function. When imposing Dirichlet boundary condition on the cut, we can solve Dirichlet problems in A (and similarly in B)
and then construct the Green function on the whole plane as
The step function in Equation (59) implement the fact that the Dirichlet boundary condition destroys the correlation between two regions while modifying it within each region through “boundary charge”.
Then the equal space Green function is constructed from the equal time Green function through
In electrostatic language, the free space is the potential energy between two Gaussian charge distributions (albeit being imaginary), while is the same thing in the presence of induced boundary charge on the entanglement cut. Hence the difference, about which the EE is concerned, only depends on the boundary charge.
In appendix B, we showed that (the double derivative of) the Dirichlet Green function is solely determined by a boundary integral,
We can integrate the heat kernel once to define
and recognize that
Using this, the cross Green function becomes
where is the kernel on the boundary
which is singular when is approaching . In general we can regularize it as
The excess Reńyi EE is therefore
In the following, we analyze two cases where the Green function can be easily figured out by methods of images.
v.1 Infinite Plane
v.1.1 Excess EE
We consider the geometry of infinite plane with entanglement cut on the x-axis. In complex coordinate, the equal time Green function in region A can be easily written down
The integral of the heat kernel
is given by sin integral function
By equation (67), the excess EE is
We can also write the integral in Fourier space through the standard Hilbert transform,
Appendix C gives an alternative route for the calculation and reaches a simpler expression
where and are the Fresnel cos/sin integrals
v.1.2 Quasi-Particle Interpretation
Despite some minor modifications, the quasi-particle picture is still able to interpret the growth of EE in this non-relativistic model.
First of all, the excess EE is a monotonically increasing function of time and eventually saturates to a constant value. The maximal value is proportional to the scaling dimension of the vertex operator, which can be regarded as a dimensionless measure of the strength of the operator. The saturation indicates the exhaustion of quasi-particles, in other words, almost all the downward travelling quasi-particles are in the lower half plane. It is comforting to confirm the fact that the local vertex operator excitation only inject small energies to the system.
The causality constraint in CFT is superficially violated. The excess EE grows almost immediately after . In fact, the horizon effect is only visible in the regime where , where is speed of light (or the equivalent threshold speed in the condensed matter system). While in this non-relativistic theory, the quasi-particle speed is far less than the speed of light, so that we can essentially take the limit, squeezing the zoom to empty. Instead, the typical diffusive behavior is taking place of the causality constraint. The excess EE grows to at the time scale , when the majority of quasi-particle diffuses to the entanglement cut.
It is interesting to zoom out the small time regime shown in Fig.6. A linear fit in the log-log plot shows the slope to be and hence there is a linear increase of excess EE in the short time regime. There are several staircase-like plateau of increasing sizes appear in the growing region, with the last one stacked on top saturating to a limiting value. It is tempting to assume that the quasi-particles disperse: phenomenologically, we see those separated groups of particles arriving sequentially on the entanglement cut. We examine this idea by using a disk probe in the next section.
v.2.1 Excitation in the Center
If the vertex operator is placed in the center, then the boundary integral kernel function will only be a function of the angle . On the other hand, , which is the integral of Gaussian, is only a function of the radius. Thus the regularized boundary integral (66) is identically zero.
The vanishing of excess EE in this geometry indicates that the quasi-particle are distributed and travelling with spherical symmetry. Points of excitation away from the center is thus a possible way to probe and decompose the quasi-particle distribution.
v.2.2 Using Small Disk as a Probe
Now we place a disk of radius centered at the origin, and the excitation at distance away from the center. The equal time Green function becomes
The kernel on the circle is
where is the Hilbert transform on circle, and
Rescaling , we have