Variational study of U(1) and SU(2) lattice gauge theories with Gaussian states in 1+1 dimensions
We introduce a method to investigate the static and dynamic properties of both Abelian and non-Abelian lattice gauge models in 1+1 dimensions. Specifically, we identify a set of transformations that disentangle different degrees of freedom, and apply a simple Gaussian variational ansatz to the resulting Hamiltonian. To demonstrate the suitability of the method, we analyze both static and dynamic aspects of string breaking for the U(1) and SU(2) gauge models. We benchmark our results against tensor network simulations and observe excellent agreement, although the number of variational parameters in the Gaussian ansatz is much smaller.
Gauge theories lie at the basis of our fundamental understanding of nature. Quantum electrodynamics (QED) describes the interactions of electrons and positrons with the electromagnetic field and is based on the U(1) gauge symmetry group. Quantum chromodynamics (QCD) accounts for the strong interaction between quarks and gluons and is based on the SU(3) gauge group. Those theories are roots of the Standard Model, that comprises our current understanding of particle physics. While in perturbative limits they are very well understood Weinberg (1996), this is not the case in general.
A very powerful framework to address nonperturbative regimes is lattice gauge theory (LGT) Wilson (1974); Kogut (1979) where the space (and time) is discretized. In such a theory the fermionic (matter) degrees of freedom reside in the sites of a cubic lattice, the bosonic (gauge) ones in the links, and they interact with each other in a gauge invariant way. The continuum limit is then recovered when the lattice constant is taken to zero, by properly renormalizing the coupling constants in the process. Monte Carlo methods Aoki et al. (2017); Fukushima and Hatsuda (2011); Dürr et al. (2008) have successfully been used in LGT to compute with a very high precision several physical properties in different models. This approach works extremely well as long as the so-called sign problem Troyer and Wiese (2005) is absent, which is the case for static (thermal equilibrium) problems in QED or QCD in the absence of a chemical potential. In dynamical scenarios or regimes in which Monte Carlo simulations suffer from the sign problem, one has to look for other techniques. Hence, there is an ongoing interest for alternative methods overcoming these limitations Cristoforetti et al. (2012, 2013); Hebenstreit et al. (2013); Scorzato (2015); Ammon et al. (2016); Gattringer and Langfeld (2016).
Recently, several research groups have addressed relatively simple models in LGT using tensor network techniques Orús (2014); Verstraete et al. (2008), motivated by the success of DMRG White (1992) and related methods to solve strongly correlated condensed matter systems in lattices. Those methods are based on variational ansätze over families of states. Most of the work Byrnes et al. (2002); Bañuls et al. (2013, 2017a); Buyens et al. (2016a, b); Bañuls et al. (2015, 2016, 2017b); Buyens et al. (2014, 2017a); Milsted (2016); Zapp and Orús (2017); Rico et al. (2014) so far has been concentrated in 1+1 dimensions, where such family corresponds to Matrix Product States (MPS). Despite their simplicity, those models contemplate many of the phenomena that are expected to occur in higher dimensions, such as confinement Coleman et al. (1975), string breaking Buyens et al. (2016b), etc. In fact, some studies have successfully analyzed models with the sign problem Pichler et al. (2016); Kühn et al. (2015); Brenes et al. (2018); Buyens et al. (2017b); Bañuls et al. (2017a), and thus raise the expectations about tensor network methods complementing Monte Carlo techniques. However, the extension of these methods to higher dimensions is still under development, and it is not clear if the methods will succeed in such cases (other methods suggesting to use projected entangled pair states for the study of lattice gauge theories have been proposed Zohar et al. (2015, 2016); Zohar and Cirac (2018); Zapp and Orús (2017)). Another important class of states that are commonly used for variational calculations are Gaussian states Bravyi (2005a); Kraus et al. (2009); Bravyi (2005b); Weedbrook et al. (2012). Those are defined for bosonic and fermionic theories, and comprise all states that can be generated by a Gaussian function of creation and annihilation operators acting on the vacuum. As they fulfill Wick’s theorem, one can compute expectation values very efficiently and thus use them for variational calculations Lieb (1981); Bach et al. (1994); Kraus and Cirac (2010); Shi et al. (2018). Unfortunately, in the case where both bosons and fermions are present, Gaussian states cannot accommodate any correlations between them beyond mean-field, and thus they are not useful for the description of lattice models with matter and gauge fields. Apart from that, their special form makes them unsuitable to study many complex phenomena.
In this paper we show how one can use Gaussian states as variational ansatz for LGT in 1+1 dimensions to study both ground state and dynamical properties. The main idea is to first apply transformations that disentangle the bosonic and fermionic degrees of freedom, followed by transformations that convert the Gaussian states in suitable ansätze for variational calculations. A similar procedure has been recently successfully applied to condensed matter models Shi et al. (2018); Ashida et al. (2017). We apply such method to the U(1) (Schwinger model) and SU(2) gauge groups, with special emphasis on the latter. We analyze the ground state, as well as the time dynamics in the presence of external charges. In order to benchmark the approach, we compare our results to those from either published MPS calculations Buyens et al. (2016b); Pichler et al. (2016) or by explicitly performing MPS simulations ourselves. Despite the fact that the number of variational parameters in the Gaussian ansatz is much lower than in MPS, we observe very good agreement, thereby showing the suitability of our approach to LGT simulations. Furthermore, the method presented here does not suffer from a violation of the area law, in contrast to MPS, which makes it suitable also for the study of real time dynamics after quenches.
Using the transformed Hamiltonian, we study static as well as dynamic properties of string breaking. First, to probe the static aspects of the phenomenon, we compute the interacting vacuum of the theory in the presence of external charges. This allows us to determine the static potential, i.e. the excess energy compared to the vacuum without external charges as a function of the distance between the external charges Polyakov (1987). We demonstrate that the method reliably distinguishes between the regimes where a flux string is present in the ground state and string breaking occurs. Second, we simulate the real-time evolution of a flux string, compute local observables and monitor the spatially resolved flux profiles as well as correlation functions throughout the evolution. These studies allow us to clearly distinguish between the string and broken string cases in both the ground state and dynamical evolution. The ansatz captures the relevant features and it is possible to simulate the dynamics even in the scenario of a global quench.
The rest of the paper is structured as follows. In Sec. II we briefly review the Kogut-Susskind Hamiltonian lattice formulation Kogut and Susskind (1975) for a gauge theory with a compact symmetry group. Afterwards we show how the gauge field can be decoupled for systems with open boundary conditions (OBC) and discuss how to apply the variational method to the resulting formulation in Sec. III. Once the general framework has been established, we benchmark our approach for two specific cases. In Sec. IV we test the ansatz for describing static properties (Sec. IV.1) and real-time dynamics of string breaking (Sec. IV.2) for a U(1) LGT. For the former, we compare our results to those obtained in Ref. Buyens et al. (2016b) close to the continuum limit. In Sec. V we turn to the non-Abelian case of a SU(2) LGT. We introduce two additional unitary transformations which allow us to apply the Gaussian variational ansatz in the presence of external charges by decoupling dynamic and static fermions (Sec. V.1). Again, we characterize the ground state (Sec. V.2) and real-time dynamics (Sec. V.3) of the system and benchmark our results against MPS simulations. Finally, we conclude in Sec. VI.
The model we are studying is a (1+1)-dimensional LGT with compact gauge group. We adopt the Kogut-Susskind Hamiltonian formulation with staggered fermions Kogut and Susskind (1975) which reads
on a lattice with sites. For Eq. (1) to correspond to the discretization of the continuum theory as in Ref. Kogut and Susskind (1975), one identifies , and , where and are the bare coupling and mass, and is the lattice spacing. In general, for compact gauge groups the operators are unitaries with as many independent angular variables as generators of the Lie algebra associated to , . These generators satisfy
where are the structure constants, and labels a given representation. These angular variables are related to the gauge field on a link as . The fermionic field is a spinor in the same representation as and resides on site . The electric term in Eq. (1) can be written in terms of either the left or right electric fields and where each of them has components. They are related by a group element in the adjoint representation , , and fulfill the commutation relations
Moreover, as conjugate momenta of the gauge field, and fulfill the commutation relations
Physical states have to fulfill Gauss’ law, for all , where
In the expression above is the total charge which consists of the dynamical and static (external) charges at the site , .
In the case of the Abelian group U(1), is just a single component fermionic field and, since the structure constants vanish, . The link operators reduce to , where the phase represent an Abelian phase related to the gauge field as . In this case the commutation relations between the conjugate variables from Eq. (4) yield
The staggered charge is defined as , and is simply a real number. In the limit of strong coupling, , meaning that the hopping term in Eq. (1) can be neglected, the Hamiltonian can be solved analytically. The gauge invariant ground state in the sector of vanishing total charge is simply given by the odd sites occupied by a single fermion, empty even sites and the links carrying no electric flux
In the expression above the numbers in bold face indicate the fermionic occupation and indicates the total electric flux carried by gauge links. This state is the lattice analog of the Dirac sea or the bare vacuum of the theory.
For the gauge group SU(2), the fermionic fields in the fundamental representation are given by
taking into account the two colors components (“red”) and (“green”). The structure constants are given by the completely antisymmetric Levi-Civita symbol and the generators are represented by with the Pauli matrices. Therefore there are three independent angular variables on each link.
The total SU(2) color charge is then given by with three different components
for , where the external charges at a given site are nothing but spin operators. Note that when there is no static charge on site then .
Similar to the Abelian case of U(1), the ground state in the limit of large can be solved analytically yielding the bare vacuum
where the bold numbers now correspond to the occupation numbers of each color of fermions on a site and again indicates the gauge links carrying no color flux.
Unless stated otherwise, we fix for all the following the hopping amplitude to as unit of energy. Moreover, we also set .
iii.1 Decoupling the gauge field
Because of the absence of transversal directions in 1+1 dimensions, the gauge degrees of freedom are not truly independent. Hence, it is always possible to decouple the matter and the gauge fields by applying a unitary transformation to the Hamiltonian (1). Here we present a simple way of performing such a decoupling for systems with OBC. While similar transformations were carried in Bringoltz (2009); Lenz et al. (1994) for periodic boundary conditions, the resulting Hamiltonian takes a simpler form in our case. This unitary transformation works for any gauge symmetry given by a compact Lie group and a unitary representation. In the following we briefly summarize the main steps and show the full derivation in Appendix A.
The decoupling is achieved with the unitary transformation
where the superscript means that the product must be ordered from left to right with increasing site index . Applying this transformation to the Hamiltonian, we obtain
where the electric energy term in the sector of vanishing total charge exhibits a long-range Coulomb interaction
between the charges with . Moreover, in the sector of vanishing total charge Gauss’ law is transformed to acting on the physical space.
A few comments are in order. The transformation shown above is completely general and does not rely on the Gaussian variational approach. For the case of U(1) the resulting Hamiltonian is equivalent to the one used in previous numerical studies Hamer et al. (1997); Bañuls et al. (2013, 2017a) and recently realized in a quantum simulation experiment Martinez et al. (2016). Hence, the transformed Hamiltonian from Eq. (12) might be suitable for both the design of future quantum simulators as well as for other numerical methods.
iii.2 Variational approach
In order to solve the transformed Hamiltonian , we apply a time-dependent variational method following the ideas from Ref. Shi et al. (2018). Our variational ansatz in the untransformed frame corresponding to the Hamiltonian (1) is given by
where is a general fermionic Gaussian state, is the unitary transformation from Eq. (11), and is another unitary transformation which decouples dynamic and static fermions (see Sec. V.1 for more details). In particular, we see that the transformation is not Gaussian. As a result, although is a Gaussian state, the ansatz in the original frame is not.
Our goal is to study the evolution of under the Hamiltonian (1) in either imaginary or real time to obtain, respectively, the ground state or the dynamic properties. Equivalently, we can study the evolution of under the rotated Hamiltonian . As shown for example in Refs. Bach et al. (1994); Kraus and Cirac (2010); Shi et al. (2018), every fermionic Gaussian state is completely characterized by its covariance matrix
which collects all two-point correlation functions since , stand for the vectors collecting all annihilation and creation operators respectively, i.e., with the number of modes. For the Schwinger model , whereas for the SU(2) LGT we have twice as many modes, , due to the two colors of fermions. Thus, in order to compute the evolution of , we have to determine the evolution of in imaginary and real time which is given by the equations Shi et al. (2018)
In the expression above is the (effective) single-particle Hamiltonian in the -rotated frame (see Appendices A and B for details). Eq. (16) yields the ground state in the limit , while Eq. (17) describes real-time dynamics in the family of fermionic Gaussian states.
Iv U(1) gauge theory
Let us first consider the simple Abelian case of the Schwinger model. Since for the U(1) gauge group the external charges are merely complex numbers, and hence , the variational ansatz reads . We will first analyze the (interacting) ground state (subsection A), and then the real-time dynamics in the presence of two static charges (subsection B). In both cases we will consider the decoupled Hamiltonian from Eq. (12) that contains the matter fields only. In order to benchmark the Gaussian variational approach, we compare our results to the MPS computations carried out in Refs. Buyens et al. (2016b); Pichler et al. (2016).
iv.1 Static properties
In this section, we study the static potential between external charges for both the massless and the massive Schwinger model. First we compute the ground state energy by evolving the bare vacuum (Dirac sea) from Eq. (7) according to Eq. (16). In order to determine the ground state energy in the presence of static charges, we repeat the calculation but this time placing on top of the vacuum a pair of external charges separated by a distance and connected by a string of electric flux(to fulfill Gauss’ law).
In Fig. 1, we compare our results for the static potential, , to those obtained via a gauge-invariant MPS simulation in Ref. Buyens et al. (2016b) where the gauge field was not eliminated. To compare our results to those in Ref. Buyens et al. (2016b), we identify and . Then the continuum limit of the model is obtained as . One can introduce the dimensionless parameter , such that the continuum limit corresponds to .
In Fig. 1(a) we show numerical results from MPS and Gaussian ansätze in the case of two small but finite lattice spacing values, for the quantity
where is the exact analytical value of when in the continuum limit Iso and Murayama (1990). We observe that both ansätze behave very similarly, and the relative error between both is below considering MPS as the reference result.
For the massive Schwinger model, we choose in order to compare the potential with that in Ref. Buyens et al. (2016b). Although this case is not exactly solvable, it is well-known Adam (1997); Buyens et al. (2016b) that as long as the static charges are integer multiples of the fundamental charge , these will be completely screened by the particle-antiparticle pairs created out of the vacuum in the broken string case. As soon as reaches the critical distance and this happens, the potential saturates to a constant value, where the excited pairs of dynamical fermions screen the static charges creating two isolated color singlets (also called mesons Pichler et al. (2016)). In Fig. 1(b) the static potential for and different values of shows that the smaller , the easier it is to break the string, i.e., smaller values for . The accuracy of the Gaussian variational method can be validated in comparison to MPS with relative error bounded by even for close to .
In Fig. 1(c) we consider noninteger values of for . In this case the static charges cannot be completely screened via production of pairs, which can only screen their integer part. This fact gives rise to the appearance of several string breaking processes with a remaining flux connecting the charges. In this case the relative error between the two ansätze is bounded by in comparison with MPS results in Ref. Buyens et al. (2016b).
The reason why the Gaussian ansatz turns out to describe the model so accurately, can be understood as follows. In the limit , the Hamiltonian (12) becomes quadratic and then it can be exactly solved by the Gaussian ansatz. On the other hand, for , the hopping term vanishes and the ground state is given by the bare vacuum (7). In fact, this state turns out to be a Gaussian state as well and therefore both limits are accurately captured by the Gaussian ansatz. This analysis together with the results for the charge and flux distribution profiles shows that the Gaussian ansatz accurately captures the equilibrium properties of the U(1) Schwinger model. In the following sections we return to our original parameter convention fixing as unit of energy similar to the discrete models considered in Refs. Pichler et al. (2016); Kühn et al. (2015).
iv.2 Real-time dynamics
In this section we probe the validity of the Gaussian ansatz to describe dynamical properties of the Schwinger model by solving Eq. (17) for certain initial conditions. In particular we are interested in the dynamics of string breaking. We consider two different scenarios: (i) an initial dynamical string configuration whose ends at and can freely propagate, which we call the free string, given by the non-Gaussian state for odd distance; (ii) a string with two static charges on its ends, referred to as a static string. Moreover, time is measured in units of .
For the static string created on top of the interacting vacuum with , one expects that the string will dynamically break for the length which exceeds the critical distance. In Fig. 2, the real-time evolution of the electric flux configuration in different parameter regimes for a static string is shown.
In Fig. 2(a) we show the electric flux for the noninteracting massless case . This case can be analytically solved with the Gaussian ansatz where the initial state is the interacting vacuum with half filling i.e., for all sites . The calculation shows that a line of electric flux connecting the static charges does not increase the energy of the system and therefore the string state is stationary, as can be seen in the figure.
In the intermediate regime, shown in Fig. 2(b), a more intriguing feature emerges. We observe the formation of anti-string configurations, i.e., a string with opposite orientation of charges at its ends in comparison to the original string, in the center where the production of particle-antiparticle pairs takes place Pichler et al. (2016). This effect is called Schwinger mechanism where the creation of these pairs in a uniform electric field is viewed as the quantum process in which virtual pairs can be separated to become real pairs once they gain the binding energy of twice the rest mass energy Hebenstreit et al. (2015); Kasper et al. (2016). We will further study this phenomenon in the following section.
Fig. 2(c) shows the result for the strong coupling regime , . In this case, the interacting vacuum in the outer region is stable due to the high cost in energy of creating pairs out of the vacuum. Nevertheless an oscillation between a string and a broken string is realized in the center.
For the free string whose ends can freely propagate, we consider the string created on top of the interacting vacuum and the bare vacuum, i.e., the ground state of . The result for the bare vacuum is used to qualitatively compare our results with those obtained from the DMRG calculations in Ref. Pichler et al. (2016). We note that in Ref. Pichler et al. (2016), the gauge field was not eliminated and an effective Quantum Link model with spin-1 operators on the link was considered. Moreover, in order to qualitatively compare to the existing results, we plot our results for the electric field within the range .
The real-time evolution of the electric flux configuration for the initial free string is shown in Fig. 3. For the noninteracting massless Schwinger case, which is exactly solvable with a Gaussian state, fermionic excitations can be created out of the vacuum at no energy cost which gives rise to large fluctuations of the electric field. Notice the difference in the evolution of the electric field in Figs. 3(a) and 3(d) for a string created on top of the interacting and the bare vacuum, respectively. In the former case, Fig. 3(a) shows that only the initial string propagates on the lattice. For the string imposed on top of the bare vacuum, the evolution corresponds to a global quench, which results in an interference of two wave fronts in the bulk as can be seen in Fig. 3(d). The difference of our results in Fig. 3(d) with respect to a similar study with a spin-1 Quantum Link model in Ref. Pichler et al. (2016) can be explained due to the large fluctuations of the electric field, which lead to a maximal value at during the evolution. As a result, the quantum link model in this parameter regime with is not equivalent to the Schwinger model considered in this paper Kasper et al. (2016).
In Figs. 3(b) and 3(e), we choose , . In this intermediate regime the string created on top of the interacting vacuum is breaking due to the creation of particle-antiparticle pairs, as Fig. 3(b) reveals. At the beginning, string/anti-string configurations emerge alternatingly and finally disappear at a later time for which a steady state is reached. Fig. 3(e) clearly shows that for a string imposed on top of the bare vacuum, we again observe an interference between the electric field wave fronts coming from the center and the boundary due to the quench dynamics.
In the regime of strong coupling and large mass, , , the Hamiltonian (12) is dominated by the mass and long-range interaction terms rendering the dynamical fermions outside the string region essentially static. As a result, the bare and interacting vacua are very similar. Thus, as shown in Figs. 3(c) and 3(f), the time evolution of the electric field for a string imposed on the interacting vacuum is almost identical to the one imposed on the bare vacuum. In both cases the string is not completely broken. Because of the small fluctuations of the electric field in this regime, our results from Figs. 2(c), 3(c) and 3(f) are in very good agreement with those obtained with a Quantum Link model in Ref. Pichler et al. (2016).
In general, we observe that the Gaussian ansatz captures the relevant features of the static and dynamical aspects of string breaking and correctly describes the production and propagation of particle-antiparticle pairs.
V SU(2) gauge theory
In the previous section we demonstrated the suitability of the ansatz to study Abelian gauge theories. However the Gaussian variational method is also adequate for studying non-Abelian lattice gauge models. To illustrate that, we investigate an SU(2) LGT, which exhibits a richer Hilbert space structure compared to the Schwinger model. As before, we will consider the decoupled Hamiltonian (12) that only contains the fermionic matter fields. This allows us to correctly estimate vacuum energies in the absence of static charges via the variational ansatz . The situation is different if we consider static charges. Unlike the U(1) case, now they are noncommuting operators. As a result the non-trivial color entanglement between the external charges and dynamical fermions prevents the description with a Gaussian state even after applying the transformation .
To overcome this difficulty, we introduce two additional non-Gaussian unitary transformations, and , which efficiently disentangle the static and the dynamical degrees of freedom. In the new frame, the static charges and the dynamical fermions are decoupled and the former appear as classical variables in the rotated Hamiltonian. While a transformation decoupling a single static charge from the dynamical fermions has been already used in the context of the Kondo model Ashida et al. (2017), here we generalize this approach and introduce a new transformation to decouple the second static charge. Moreover, we also introduce the correlation function between the static charges and between static charges and dynamical fermions. Using the variational ansatz , we analyze again ground state properties in the presence of two static charges as well as real-time dynamics.
v.1 Decoupling the static charges
To study the string-breaking phenomenon we insert a pair of static charges at two sites and . This corresponds to the static charge distribution
When the string is present, the pair of static charges forms the spin singlet state , and when the string is broken, each static charge forms a singlet state with the surrounding dynamical fermions. As a result, neither of them is a Gaussian state.
However, if we are able to decouple the static charges from the dynamical fermions with a unitary transformation , the transformed Hamiltonian (conditioned on the spin state of static charges) only contains operators acting on the dynamical fermions which can be studied with the Gaussian state approach. In fact, this decoupling is possible if the Hamiltonian has certain parity symmetries.
The problem we are trying to tackle here resembles a two-impurity problem described by the Hamiltonian (12), where the impurities are two static charges described by the Pauli matrices . The Hamiltonian has the parity symmetry
where the operator and is defined to be
This symmetry corresponds to the rotational invariance of the entire system along the -direction by .
Similar to the single-impurity Kondo model Ashida et al. (2017), we can construct the unitary transformation
which transforms into the operator of the first impurity, . Since is a symmetry of the Hamiltonian, is conserved in the new frame, i.e.,
and can be considered as a “classical” variable, where . The explicit form of is
where the electric term reads
To decouple the second static charge located at , we notice that the Hamiltonian is rotationally invariant along the -direction for an even number of dynamical fermions. Thus, is the parity symmetry of the Hamiltonian , where
A second unitary transformation
allows us to rotate the parity operator for the second charge, as . Thus, in the new frame, commutes with the Hamiltonian . The final form of the transformed Hamiltonian is given by
which only depends on the two commutating operators for the static charges, where the operator is defined as
and the relation
has been used assuming an even number of dynamical fermions.
A remarkable feature of this Hamiltonian is that the operators for static charges become classical variables. Indeed, a general state in the rotated frame can be written as , where and are eigenstates of and , respectively. The evolution of the state is governed by the Hamiltonian of dynamical fermions and will be approximated by the Gaussian state .
To show the decoupling procedure and the corresponding symmetries in a compact way, we can rewrite the symmetry operators as
in the original frame before applying and , where and are the total (dynamical plus external) SU(2)-charge operators along the and the directions. Apart from the prefactor determined by the fixed dynamical fermion number, these two symmetries correspond to the rotational invariance of the original Hamiltonian along the and the directions by angle . The analysis above implies that a single unitary transformation can be applied directly to transform both symmetry operators to and of static charges, respectively. In the new frame, and commute with the Hamiltonian , and, thus, become the classical variables and . Correspondingly the symmetry operators and take the values and , respectively. As will be shown in Sec. V.2, the relation of , , and provides us a clear picture of the ground state configuration from the symmetry analysis.
We emphasize that and are two non-Gaussian unitary transformations entangling static charges and dynamical fermions. This entanglement induced by and can be seen from the variational state
in the original frame, which correctly captures the physics for the appropriate choice of the parameters and the Gaussian state. Note that if the Gaussian state is taken to be the Dirac sea (Eq. (10)), it is a common eigenstate of all operators with the same eigenvalue (either or ) and Eq. (34) becomes the singlet state between static charges for , while it becomes a triplet state for any other choice of and . However, unlike the Dirac sea, a general Gaussian state does not preserve the rotational symmetry of the Hamiltonian, and gives rise to entanglement between static and dynamical fermions. We will explore this entanglement structure in greater detail in our studies of the ground state and the real-time dynamics.
To explicitly characterize the entanglement between static charges and dynamical fermions, we introduce two gauge invariant correlation functions
between the static charges, and
between the static charge at and the dynamical fermion at . Note that in the gauge-field-free frame, these correlation functions become
In the limit where the string dominates (e.g. in the initial state, when the string is superimposed on the bare vacuum), the two static charges form a singlet state through the flux string connecting them, hence approaches . In the opposite regime, when the string is broken, a color singlet is formed between each static charge and the dynamical fermions, and the correlation vanishes. Hence, this correlation function indicates the occurrence of string breaking
In the latter case, the entanglement between the static charge at and the screening “cloud” of dynamical fermions can be characterized by . Furthermore, the correlation functions enable us to give a precise description of entanglement structure between the static charges and the dynamical fermions for in and out-of-equilibrium dynamics.
Analogously to Sec. IV for the U(1) gauge group, we make use of the evolution Eqs. (16) and (17) for to study the string-breaking phenomenon in the ground state and its real-time dynamics. The effective mean-field Hamiltonian appearing in the Eqs. (16) and (17) is derived in Appendix B. Unlike the Abelian case, the Hamiltonian contains exponential functions of the creation and annihilation operators, hence the derivation of the mean-field Hamiltonian follows a much more sophisticated procedure Shi et al. (2018).
v.2 Static properties
Let us first apply the Gaussian ansatz in combination with the unitary transformations , and to study the static aspects of the string-breaking phenomenon in the ground state of the SU(2) LGT. Additionally, to benchmark the Gaussian variational ansatz, we also perform MPS calculations (see Appendix C for details) and compare the results of both simulations. We observe that with our choice of parameters in both optimization algorithms, numerical artifacts are negligible for the effects we want to observe. Note that again we measure energy in units of .
First of all, we compare the values for the ground state energies in the absence of static charges i.e., the vacuum energies . In Table 1, considering MPS as the reference result, we show the relative difference
between Gaussian and MPS ansätze for . As expected, the MPS simulation yields lower values for the vacuum energy in all cases but even in the strong coupling regime (with and ) the relative error is bounded below . Moreover we observe that the larger the coupling becomes with respect to the mass and the hopping , the more relevant the (non-quadratic) electric energy term becomes, and, thus, resulting in larger errors.
After comparing the interacting vacuum energies, we proceed to study the system in the presence of two static charges. Due to the parity symmetries, we conclude that the Hamiltonian has four sectors labeled by the eigenvalues and of and . Fixing , we can compute the ground states in all of these four sectors with Eq. (16). The global ground state is then the one with the lowest energy. In Fig. 4 we present our results for , (Fig. 4(a)), , (Fig. 4(b)), and , (Fig. 4(c)).
In Fig. 4(a) the blue asterisks correspond to the ground state energy in the sector . As expected, before the string breaks, the ground state energy (potential energy) grows linearly with the distance of the static charges. In the broken string case, the ground state energy in the sector shows an oscillatory behavior between odd and even distances. Computing the ground states in all four sectors, we find that for even distances the global ground state in the broken string regime is in the sector , and has the same energy as for odd distances, as shown by the red triangles in Fig. 4(a). The energy of the global ground state (potential energy) obtained from the non-Gaussian ansatz forms the expected plateau once the string has broken and quantitatively agrees with that from the MPS calculations (black circles). Computing the expectation value of and in the ground state obtained from the MPS calculations we observe the same values for and in the different regimes. As shown by the green squares and the pink crosses the MPS results confirm that the global ground state is in the sector in the string and string-breaking cases for odd distances, while it is in the sector and for even distances in the broken string state. To understand why takes different values for even and odd distances in the string-broken state, one can inspect the total SU(2) charges (including both static charges and dynamical fermions) and the total number of dynamical fermions for even and odd distances. It turns out that the ground state for odd distances is half-filled, and the total SU(2) charge is zero, which leads to for the system size . However, for even distances, the ground state of the system in the broken string case always prefers to add or reduce two dynamical fermions and as a result, is kept the same and changes the sign.
In Figs. 4(b) and 4(c), the potential energy , is plotted for , and , . As decreases, i.e., for relatively stronger interactions, the description given by the non-Gaussian ansatz still agrees with the MPS result very well, with a relative error smaller than .
The presence or not of the string in the ground state can also be distinguished in the electric flux profiles. These are shown in the second and third rows of Fig. 4 for two sets of distances corresponding to the string and the string-breaking regimes, with the first static charge always located at . In the string case, the two static charges are confined by the flux string. In the string-breaking regime, each of the static charges is completely screened by the surrounding dynamical fermions, and the flux string is broken. Because of the global SU(2) symmetry each group component contributes equally to . The flux distribution is also in good agreement with the one obtained from the MPS calculations, as shown by the red solid curves.
In the last row of Fig. 4, the correlation function between the static charges as a function of the distance is shown. The breaking of the string takes place at some distance . For much shorter distances () is close to indicating that the static charges are confined by the electric flux string and form a singlet state. In contrast, for much longer distances () is essentially , thus implying that the static charges are screened by the surrounding dynamical fermions and the string is broken. Close to the distance at which the breaking takes place, the correlation function calculated from both the non-Gaussian ansatz (blue asterisks) and the MPS results (black circles) displays a sudden change for , (Fig. 4(e)) and a smooth transition for , and , (Figs. 4(f) and 4(g)).
The correlation function between the first static charge and the dynamical fermions also displays different patterns in the string and string-breaking regimes. As shown in Fig. 5, the static charges are hardly entangled with the dynamical fermions when the string is present, hence resulting in small values for . In contrast, reveals that a symmetric cloud of dynamical fermions screens the static charges when the string is broken. In particular, due to the global SU(2) symmetry of the Hamiltonian each group component contributes equally, we see that is close to . Hence, the external charges are forming approximately a SU(2) singlet with the surrounding dynamical fermions.
Therefore, our results for the potential energy, the electric flux profiles, and the correlation functions provide a comprehensive description of the string-breaking phenomenon. In the string case, the two static charges connected by the electric flux string form a SU(2) singlet state, and are only weakly entangle with the half-filled dynamical fermions. As the distance increases, the flux string becomes longer and longer, which leads to the linear increase in the potential energy of static charges. At distances larger than a certain threshold value , each static charge forms a color singlet with the dynamical fermions in the screening cloud, and the string is broken. For odd distances, the static charges are on sites of different staggered mass. This term favors double occupancy in one of them, and vacancy in the other. On the contrary, for even distances both static charges are located at sites favoring double occupancy with dynamical fermions (vacancy). Thus two fermions are annihilated (created) to screen the static charges. The screening results in a single dynamical fermion being localized on each of the static charges. In both cases the potential energy reaches a plateau indicating the occurrence of the string breaking.
All in all, the non-Gaussian ansatz accurately describes the string-breaking phenomenology in the (1+1)-dimensional SU(2) LGT, where the entanglement between the static charges and the dynamical fermions is entirely encoded by the transformations and . It also allows us to effectively determine the relevant observables such as the potential energy, the electric flux profiles and the correlation functions.
v.3 Real-time dynamics
In this subsection, we study the dynamical aspects of the string-breaking phenomenon using the non-Gaussian ansatz from Eq. (34). Since the transformations and can only characterize the entanglement between static charges and dynamical fermions, but not the entanglement between different dynamical fermions, we restrict ourselves to study the dynamical features of static strings.
In Fig. 6, the real-time evolution of the flux distribution and the correlation function are depicted for the static string imposed on top of the interacting vacuum for various combinations of and .
Figs. 6(a)-6(c) shows that the interacting string is stable at short time scales, before it eventually breaks. Similar to the U(1) case, the time scale of the breaking process depends on the length of the initial string, the mass and the coupling constant (recall we have fixed ). Moreover, the string-breaking behavior is also captured by the correlation function between the two static charges, as shown in the second row of Fig. 6.
At initial times, the static charges connected by the flux string are in a singlet state, thus . For short time scales, the dynamical fermions start to accumulate around the static charges and partially break the string, which corresponds to a damped oscillation in . Eventually, at the static charges are completely screened by the surrounding dynamical fermions, the flux string is completely broken and oscillates around .
Comparing Figs. 6(d) and 6(e) to Fig. 6(f), we find that the larger the initial length is, the longer it takes for the string to break. This can be explained as follows. For larger distances , the system requires a longer time to relax from the initial interacting string state to the string-breaking state by transporting a dynamical fermion between the two sites occupied by the static charges. Moreover, for the same distance between static charges, to decrease the mass and coupling constant , effectively implies to increase the hopping parameter , which speeds up the screening process. Therefore, the string breaks faster for smaller values of and , as can be seen in Figs. 6(d) and 6(e).
Vi Conclusion and Outlook
In this paper, we introduce a new family of variational ansätze which is suitable to study lattice gauge models. Our method relies on three unitary transformations that rotate the original Hamiltonian (1). First, we derive which decouples the matter and the gauge degrees of freedom. The resulting rotated Hamiltonian (12) is completely general and can be addressed with any numerical or analytical technique. For the U(1) LGT, the Gaussian ansatz can describe the interacting vacuum state and the in and out of equilibrium transition between the string and broken string cases. However, for the SU(2) LGT, while the ansatz is able to describe the interacting vacuum, it cannot be directly applied to study the string breaking in the presence of two static charges. Thus, we further introduce a non-Gaussian ansatz with two unitary transformations and that characterize the entanglement between static charges and dynamical fermions.
Using the Gaussian ansatz, we investigate the static and dynamical aspects of string breaking in the Schwinger model. First, we compute the interacting vacuum of the theory in the presence of two external charges and determine the static potential. For small distances between the external charges, we clearly see a linear increase in the static potential, thus indicating a flux string is present in the ground state. For larger distances beyond a critical one, it is energetically favorable to break the flux string resulting in a flattening of the potential. In general, we observe excellent agreement with previous tensor networks studies of the model Buyens et al. (2016b) and we are able to precisely determine the regimes in which string breaking occurs. Second, we simulate the real-time dynamics of a flux string between two external charges. Computing the site resolved flux profiles we can also clearly distinguish between a string state and the breaking case in the out-of-equilibrium scenario. Even if we perform a global quench on the initial string state, we are able to simulate the dynamics with the Gaussian variational method.
This variational ansatz is not limited to the Abelian case and we can also explore static as well as dynamical properties for a SU(2) LGT in the presence of external charges. Since in this case the Gaussian ansatz cannot be directly applied to the rotated Hamiltonian (12), we show how to overcome this limitation with the two additional unitary transformations and . These two transformations, which decouple the static charges and the dynamical fermions, not only allow us to address the resulting Hamiltonian with fermionic Gaussian states, but also shed light on the entanglement structure between the dynamical fermions and the static charges. In the presence of a string we observe that the external charges are correlated among themselves. In contrast, in the breaking case correlations between the dynamical fermions and the static charges develop. Together with the static potential, the site resolved color-flux profiles and the correlation functions between charges, this allows us to give a comprehensive description of the string-breaking phenomenon. As for the Schwinger model, our results are in very good agreement with those obtained from MPS simulations. Furthermore, we also simulate the evolution of a color-flux string in real-time. Again, the flux profiles as well as the real-time evolution of the correlation function , clearly indicate whether the initial color flux string is breaking.
On the one hand, the unitary transformation is not limited to the two specific cases we address and can be used with arbitrary SU() gauge groups. More generally, it might be possible to derive a similar transformation for certain discrete gauge groups extensively used in the context of condensed matter physics. On the other hand, and are also valid in higher spatial dimensions and could be generalized to decouple the static charges for SU() gauge groups by for example, the embedding of SU(2) into SU(). This would allow us to study more general gauge models such as (1+1)-dimensional QCD with the Gaussian ansatz. Additionally, these two transformations for the SU(2) case might have also applications in condensed matter physics. For example they could potentially be applied to study problems such as RKKY interactions between two Kondo impurities induced by the fermionic bath.
Although here we combine the rotated Hamiltonian with the use of non-Gaussian states, we expect the decoupled formulation itself to be useful for a variety of other approaches. On the one hand, as our MPS results for the SU(2) case show, this formulation can be directly addressed with tensor networks. Compared to previous MPS studies of SU(2) gauge models Kühn et al. (2015); Bañuls et al. (2017b) we do not have to truncate the gauge Hilbert spaces to finite dimension and our formulation can be readily extended to arbitrary gauge groups SU(). On the other hand, it could have potential applications for the design of future quantum simulators for LGT. In the quantum simulation of the SU(2) LGT, two spins 1/2 can be considered on each site to realize the spin Hamiltonian 111The spin Hamiltonian can be found in Appendix C. via a Jordan-Wigner transformation Kühn et al. (2015). This formulation generalizes the one recently implemented in a quantum simulator for the Schwinger model Martinez et al. (2016) to non-Abelian gauge groups. Since the gauge field can be completely eliminated, this might allow for simpler experimental realizations compared to previous proposals. The rotated Hamiltonian might also be interesting for studying the large limit of gauge theories ’t Hooft (2012). Since the gauge degrees of freedom are absent in the rotated Hamiltonian the effort for addressing larger values of only grows moderately.
Acknowledgements.We would like to thank the authors of Ref. Buyens et al. (2016b) for kindly sharing with us their results in order to test the Gaussian ansatz in comparison to MPS results for the U(1) lattice gauge theory. We are also very grateful to Erez Zohar for enlightening discussions and comments. Moreover, PS acknowledges financial support from “la Caixa” fellowship grant for post-graduate studies. TS acknowledges the Thousand-Youth-Talent Program of China. SK acknowledges financial support from Perimeter Institute for Theoretical Physics. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. ED acknowledges support from Harvard-MIT CUA, NSF Grant No. DMR-1308435, AFOSR Quantum Simulation MURI, AFOSR-MURI: Photonic Quantum Matter, award FA95501610323. JIC acknowledges ERC Advanced Grant QENOCOBA under the EU Horizon 2020 program (grant agreement 742102).
Appendix A Eliminating the gauge field
Here we explicitly derive the unitary transformation which allows for eliminating the gauge field in Hamiltonian (1) for any non-Abelian gauge group with unitary representations as it is for example the case of SU() with . Different to derivation in Ref. Bringoltz (2009); Lenz et al. (1994), we choose to work with open boundary conditions. Moreover, for the ease of notation, we use the Einstein summation convention for group indices throughout this Appendix.
a.1 Transformation to decouple the gauge field
In order to eliminate the gauge field appearing in the hopping term of the Hamiltonian (1) we look for a unitary transformation given by a fermionic operator that leaves the matrix components invariant. Let us recall how and transform under gauge transformations Zohar and Burrello (2015)
where with and is the representation given by
From now on the representation index will be omitted as long as there is no need for a particular one. Given these transformation laws, the hopping term changes as follows
where we denote the gauge transformation applied to the fermionic field living on the site by . Thus, in order to remove the gauge field in the hopping term with this transformation we see that the condition has to be fulfilled, or equivalently
To solve the set of equations (44), we fix the left open boundary conditions to and for both gauge and fermionic fields, giving as a result 222Note that for periodic boundary conditions the th link corresponds to the th link. Hence, in that case we can use this condition to fix the boundary values.. Solving for all links we obtain
where we did not make reference to any specific representation and the index has been omitted. In fact we see that the unitary transformation which generates the linear transformation (45) is given by
where the link variables describing the gauge field were discussed bellow Eq. (2) and the superscript “” means that the operators must be ordered from left to right with increasing index .
a.2 Rotated Hamiltonian
So far we have only considered the hopping part of the Hamiltonian. To complete the transformation of the Hamiltonian, we still have to compute the mass term and the color-electric energy term. Using Eq. (41), it is easy to see that the mass term is invariant under . To see how the electric field transforms under , we use the commutation relation between the conjugate variables from Eq. (4) and then since , we see that has the same matrix structure as because the fermionic charges are just complex numbers on the gauge Hilbert space. Therefore we obtain
Thus, omitting the index, transforms under as
where we have applied the transformation law
for and otherwise, i.e., transforms as a 3-vector under color rotations. Thus in the rotated frame Gauss’ law takes the easy form
Now using the relation between and given by