The mass spectrum of the Schwinger model with matrix product states

Abstract

We show the feasibility of tensor network solutions for lattice gauge theories in Hamiltonian formulation by applying matrix product states algorithms to the Schwinger model with zero and non-vanishing fermion mass. We introduce new techniques to compute excitations in a system with open boundary conditions, and to identify the states corresponding to low momentum and different quantum numbers in the continuum. For the ground state and both the vector and scalar mass gaps in the massive case, the MPS technique attains precisions comparable to the best results available from other techniques.

Published for SISSA by Springer

Received: August 15, 2013

Revised: October 29, 2013 Accepted: November 10, 2013

Published: November 20, 2013

The mass spectrum of the Schwinger model with matrix product states

M.C. Bauls,a K. Cichy,b,c J.I. Ciraca and K. Jansenb,d

aMax-Planck-Institut fr Quantenoptik,

Hans-Kopfermann-Str. 1, 85748 Garching, Germany

bNIC, DESY Zeuthen,

Platanenallee 6, 15738 Zeuthen, Germany

cFaculty of Physics, Adam Mickiewicz University,

Umultowska 85, 61-614 Poznan, Poland

dDepartment of Physics, University of Cyprus,

P.O. Box 20537, 1678 Nicosia, Cyprus

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We show the feasibility of tensor network solutions for lattice gauge theories in Hamiltonian formulation by applying matrix product states algorithms to the Schwinger model with zero and non-vanishing fermion mass. We introduce new techniques to compute excitations in a system with open boundary conditions, and to identify the states corresponding to low momentum and di erent quantum numbers in the continuum. For the ground state and both the vector and scalar mass gaps in the massive case, the MPS technique attains precisions comparable to the best results available from other techniques.

Keywords: Field Theories in Lower Dimensions, Lattice Gauge Field Theories

ArXiv ePrint: 1305.3765

Open Access doi:http://dx.doi.org/10.1007/JHEP11(2013)158

Web End =10.1007/JHEP11(2013)158

JHEP11(2013)158

Contents

1 Introduction 1

2 The model 2

3 Method 4

4 Results 8

5 Discussion 12

A Continuum limit and error analysis 16

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1 Introduction

In the last years, methods based on tensor network states (TNS) have revealed themselves as very promising tools for the numerical study of strongly correlated quantum many-body systems. They are ansatze for the quantum state, characterized by their entanglement content and well suited for lattice systems. The paradigmatic family of TNS is that of matrix product states (MPS) [16], which underlies the well-known density matrix renormalization group algorithm (DMRG) [7, 8]. The enormous success of DMRG for the study of one dimensional condensed matter problems has been better understood within the framework of TNS (see e.g. [9]). This has also enabled di erent extensions of the original algorithm, such as time-dependence [1012], so that MPS/DMRG methods constitute nowadays a quasi exact method for the study of ground states, low lying excitations and thermal equilibrium properties of quantum spin chains far beyond the reach of exact diagonalization. Being free from the sign problem which plagues quantum Monte Carlo (QMC) methods, higher dimensional TNS [1315] are seen as powerful candidates for the numerical exploration of long standing strongly correlated electron problems.

This success has motivated the application of TNS techniques to further problems, and a relatively new eld of application has been found in quantum eld theories. Conformal Field Theory inspires a generalization of MPS [16] useful for critical models. Furthermore, specic generalizations of TNS exist [1719] which are suitable for non-relativistic and relativistic QFT in the continuum. The discrete versions, on the other hand, are adequate for lattice eld theories, and can in particular be applied to models relevant to high energy physics problems. This route was rst explored using the original DMRG formulation [20, 21]. The deeper understanding of the technique achieved thanks to TNS has increased the power of these methods [22] and new results are now available for critical QFT [23, 24].

1

A particularly interesting study case is that of the Schwinger model [25, 26], or QED in one spatial dimension, the simplest gauge theory, which nevertheless exhibits features in common with more complex models (QCD) such as connement or a non-trivial vacuum, and has been adopted as a benchmark model where to explore lattice gauge theory techniques (see e.g. [2731] and references cited therein).

The application of MPS techniques to the Schwinger model was rst explored by Byrnes et al. [20] using the original DMRG formulation. The study improved by several orders of magnitude the precision attained by other Hamiltonian techniques for the ground state and the rst (vector) mass gap, for vanishing and non-vanishing fermion masses, using periodic boundary conditions (PBC). However, DMRG estimations for higher excited states lose precision fast, in particular for PBC, and the scalar mass gap was not explored in [20].

In this paper we apply the more stable and numerically e cient MPS for open boundary conditions (OBC) to the same problem. We work in the subspace of physical states satisfying Gauss law, in which the model can be written as a spin Hamiltonian with a long-range interaction [32, 33]. We improve the existing techniques to nd higher excited states and devise a method to identify vector and scalar excitations in nite open chains, although the charge that distinguishes them is only a good quantum number in the continuum. Using these methods we compute the ground state and the vector and scalar mass gaps for vanishing and non-vanishing fermion masses, with enough precision to conduct the extrapolation to the continuum limit. Our study shows the feasibility of TNS solutions for similar LGT, in the Hamiltonian formulation.

The rest of the paper is organized as follows. In section 2 we briey introduce the Schwinger model, and review its formulation as a spin Hamiltonian. Section 3 presents the MPS formalism, with special emphasis on the new techniques used to overcome the challenge posed by this particular kind of problems. In section 4 we present our numerical results and conclude in section 5 with a discussion and outlook.

2 The model

The massive Schwinger model [25], or QED in two space-time dimensions, is a gauge theory describing the interaction of a single fermionic species with the photon eld, via the Lagrangian density,

L =

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14F F , (2.1)

with F = A A, being g the coupling constant and m the fermion mass. The

physics of the model is determined by the only dimensionless parameter m/g. The massless case, m = 0, can be solved analytically, via bosonization [25], and also the free case, g = 0, has exact solution, so that for very large or very small values of m/g a perturbative study is possible around one of the solvable limits, while in general a non-perturbative treatment may be needed. One of the features of the model is the existence of bound states, the two lowest ones of which, the vector and the scalar, are stable for any value of m/g [26, 34].

The spectrum of the Schwinger model has been studied with many techniques. The most accurate numerical estimations have been performed on the lattice. For the vector

2

g /

A m)

mass the best results were obtained with DMRG [20], while strong coupling expansion (SCE) got the most accurate prediction for the scalar mass [35, 36] and the most precise values in the massless case [37].

In the temporal gauge, A0 = 0, the Hamiltonian density reads

H = i

1(1 igA1) + m

+ 12E2. (2.2)

The electric eld, E =

A1, is xed by the additional constraint of Gauss law, 1E =

g

0 , up to a constant of integration, which can be interpreted as a background eld [26].

The model can be formulated on a lattice. Here we focus on the Kogut-Susskind staggered formulation [38],

H =

i 2a

Xn

nei nn+1 h.c.[parenrightBig]+ m

Xn(1)nnn +ag2 2

XnL2n, (2.3)

where a is the lattice spacing. We consider a lattice with open boundaries, with sites n = 0, . . . N 1. On each site there is a single-component fermion eld n, while n are

the gauge variables sitting on the links between n and n + 1, and connected to the vector potential via n = aqA1n. Ln is the corresponding conjugate variable, [n, Lm] = inm,

connected to the electric eld as gLn = En. We work in a compact formulation, where n becomes an angular variable, and Ln can adopt integer eigenvalues. Gauss law appears as the additional constraint

Ln Ln1 = nn

1

2 [1 (1)n] . (2.4)

Instead of the Hamiltonian (2.3), it is usual to work with the dimensionless operator W =

2ag2 H, and to dene parameters x =

1g2a2 and =

2mg2a . In this picture, fermions on even and odd sites respectively correspond to the upper and lower components of the spinor representing the fermionic eld in the continuum. Gauss law (2.4) determines the electric eld up to a constant which can be added to Ln and represents the background electric eld.

Using a Jordan-Wigner transformation, n =

Qk<n(izk)n, where = 12(x iy), the model (2.3) can be mapped to a spin Hamiltonian [32],

H = x

N2

Xn=0

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+nn+1 + n+n+1

[bracketrightbig]

+

2

N1

Xn=0[1 + (1)nzn] +

N2

Xn=0(Ln + )2 . (2.5)

Gauss law reads then Ln Ln1 = 12 [zn + (1)n], and can be used to eliminate the gauge degrees of freedom, leaving the Hamiltonian [39]

H =x

N2

Xn=0

+nn+1 + n+n+1

[bracketrightbig]

+

2

N1

Xn=0[1 + (1)nzn]

+

N2

Xn=0

" + 12n

Xk=0((1)k + zk)[bracketrightBigg]2, (2.6)

3

where is the boundary electric eld, on the link to the left of site 0, which can describe the background eld.

A useful basis for this problem is then

|i |i0i1 . . . iN2iN1i, (2.7)

with im = {0, 1} labeling the 1 eigenstates of zm (on site m). An even site in spin

state |1i corresponds to the presence of a fermion, while an odd site in state |0i is an

antifermion at the corresponding position. The integer valued = . . . , 1, 0, 1, . . . is the

only gauge degree of freedom left, but with OBC it is non-dynamical, as the Hamiltonian cannot connect states with di erent values of . Here we choose = 0 and omit it in the following, as we will be concerned with the case of zero background eld. Moreover, we are interested in states with zero total charge (

Pn zn = 0 in the spin language), so we consider

3 Method

In this work we use the MPS ansatz to approximate the ground and lowest excited states of the Hamiltonian (2.6). A MPS for a system of N d-dimensional sites has the form

| i =

where {|ii}d1i=0 are individual basis states for each site. Each Aik is a D-dimensional matrix

and the bond dimension, D, determines the number of parameters in the ansatz. MPS are known to provide good approximations to ground states of local Hamiltonians in the gapped phase, but have also been successfully used for more general models.

The MPS approximation to the ground state can be found variationally by successively minimizing the energy, h |H| i h | i , with respect to each tensor Ak until convergence. Each

such optimization reduces to the eigenvalue problem of an e ective Hamiltonian that acts on site k and its adjacent virtual bonds. The basic algorithm, closely related to the original DMRG formulation [7, 8], was introduced in [5], and is nowadays widely used.

In DMRG it is possible to target several of the lowest eigenstates [41], or, in the case of a rst excited state with a distinct quantum number, to run the ground state search in di erent sectors. In the MPS formalism, it is natural to extend the variational method to determine excited states by performing a constrained minimization which imposes orthogonality with respect to the already computed states (ground and lower excited states), as proposed in [42] and recently extended in [43]. Here we introduce a simpler variation of the ground state algorithm, which allows us to compute MPS approximations to the lowest energy eigenstates at lower cost, without using any explicit symmetry. This is especially interesting for a nite system with open boundary conditions, where momentum is not a good quantum number. It is worth noticing that in the case of translationally invariant (TI) systems, either nite and periodic or innite, a very successful approach exists based on the construction of well-dened momentum MPS [44].

4

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chains with even N.

d

Xi0,...iN1=1tr(Ai00 . . . AiN1N1)|i0, . . . iN1i, (3.1)

For nite systems, the MPS algorithms for open boundary conditions are numerically more stable and in general more e cient, although improved methods have been recently proposed for periodic systems [45, 46]. However, nite-size e ects are much larger for OBC and therefore simulation of larger chains may be needed to reach the thermodynamic limit reliably.

After having found the ground state of the system, | 0i, we can construct the projector

onto the orthogonal subspace, 0 = 1| 0ih 0|. The projected Hamiltonian, 0H 0, has | 0i as eigenstate with zero eigenvalue, and the rst excited state as eigenstate with energy

E1. Given that E1 < 0, what we can always ensure by adding an appropriate constant to H, the rst excitation corresponds then to the state that minimizes the energy of the projected Hamiltonian,1

E1 = min | i

h | 0H 0| i

h | i

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= h | (H E0| 0ih 0|) | i h | i

. (3.2)

This minimization corresponds to nding the ground state of the e ective Hamiltonian He [1] = 0H 0. The procedure can be concatenated to nd subsequent energy levels, so that, to nd the M-th excited state, we will search for the ground state of the Hamiltonian

He [M] = M1 . . . 0H 0 . . . M1 = H

M1

Xk=0Ek| kih k|. (3.3)

Each of these ground state searches can be solved by applying the standard variational MPS algorithm to the corresponding e ective Hamiltonian (3.3), which can be constructed from the set of all previously computed MPS levels {| ki} (see gure 2). The expression

to be minimized at each step of the MPS iteration is then minAk A

kHkAk

A kNkAk , where Hk and Nk are the e ective Hamiltonian and norm matrix acting on site k and its two adjacent virtual bonds, so that each tensor can be found by solving a generalized eigenvalue problem

HkAk = NkAk. As illustrated in gure 1, Hk and Nk are computed by contracting all

tensors but Ak, and the computational cost can be optimized, as in the original algorithm, by storing and reusing the partial contractions that compose these e ective matrices. In general this is most easily done when all the terms in the problem Hamiltonian are expressed as a matrix product operator (MPO) [47]. In this particular case it is more convenient to keep separate temporary terms for each of the contractions h k| i, and to reconstruct from

them the e ective projectors on every site, before constructing the e ective Hamiltonian. In this way, the number of operations required for each such term scales as dD3, without increasing the leading cost of the algorithm. Due to the larger number of terms that need to be kept, however, the cost of nding the M-th level once all the lower ones are known will be about M times higher than that of the original ground state search,2 and thus the total cost for computing up to the M-th level will scale as M2D3.

The masses of the particles in the theory are given by the energies of the zero momentum excitations. In nite systems with OBC momentum is di cult to identify and we

1Alternatively, we can use a Hamiltonian H + | 0ih 0|, with > E1 E0.

2This refers only to the leading computational cost, as in some cases convergence of a particular excited states may result slower due to small gaps.

5

h |H| i

h | i

(a) Pictorial representation of the MPS, and Hamiltonian and norm contractions

Hke

+

| i

Hke [M]=

(b) E ective Hamiltonian for site k and excited level M.

Figure 1. Scheme of the algorithm for excited states. In 1(a) (left) we show the commonly used graphical representation of a MPS (see e.g. [40]). Each circle corresponds to one tensor (Ak) and

each of its legs represents one index, with lines that join two tensors representing a contracted index (as in a matrix multiplication). The open legs correspond to the physical indices on each site. A particular coe cient in the product basis corresponds to xing each of the open indices to a value (0, . . . , d 1). On the right, we show the representation of some usual contractions. We can

contract two MPS by joining their open (physical) indices to compute the norm (lower scheme) or insert an operator with a matrix product structure, as the Hamiltonian, to obtain the expectation value of the energy (above). In 1(b) we show the tensor network representation of Hk in the step

of the optimization where site k is computed. The term on the left is simply the contraction of the TN for hHi except for tensor Ak. Each term in the sum on the right is the TN for the expectation

value of one projector (h | mih m| i) leaving out the tensor at site k. The sum of all these terms

produces the e ective Hamiltonian at site k.

need to nd the excitations that will correspond to the lowest momentum in the continuum limit. In dynamical DMRG [48] momentum dependent quantities were extracted from nite DMRG calculations with open boundaries using quasimomentum states dened from the eigenstates of a free particle in a box. Here we use a di erent approach, based on the continuum momentum operator for the fermion eld, P =

[integraltext]

dx (x)ix (x). Its discretization yields, in the spin representation, and after rescaling by a factor 2

ag2 , the operator

P = ix

Xn[nzn+1+n+2 h.c.]. (3.4)

The expectation values of2P can be used to assign a pseudo-momentum to the spectral levels, in order to reconstruct the dispersion relation (gure 3).

6

Ak

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( m)ke

| mike

PM1m=0 Em

Input: Hamiltonian for an N-site chain, H; maximum bond dimension, D; tolerance, Output: MPS approximation to the ground state, | {Ak}iD, with energy E

{A0, . . . AN1} initial guess

E h |H| ih | i

E 1

sweeping direction right

while E > dok rst site for sweeping direction

while 0 k N 1 do

compute matrices Hke , N ke {contraction without tensor Ak (gure 1)}

solve generalized eigenvalue problem Hke A = minN ke A Ak A, Ek mink next site according to sweeping direction

end while E

[vextendsingle][vextendsingle][vextendsingle]EEkE

[vextendsingle][vextendsingle][vextendsingle]

E Ek

ip sweeping direction {left right}

end while

Figure 2. Schematic MPS variational algorithm for the ground state search. An e cient implementation requires imposing a canonical form of the MPS after each update, and reusing temporary calculations, among other optimizations (see [9, 40]).

2488 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

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2474

2382

2384

2476

2386

2478

2388

2480

2390

E m

E m

2482

2392

2394

2484

2396

2486

2398

2400 0.2 0.4 0.6 0.8 1

<OP2>

<OP2>

(b) m/g = 0.125

Figure 3. Dispersion relation (energy vs. hO2P i) obtained with MPS for x = 25, N = 160 and

m/g = 0 (left) and m/g = 0.125 (right). Shown are the states reconstructed with bond dimensions D = 40 (crosses), 80 (circles) and 100 (dots) (being on top of each other in the graph) until we have reached a scalar candidate. The appearance of the scalar is detected by the phase of the expectation value hSRi, with the scalar (indicated in blue in the plots) being the rst state with 0 above

the ground state (red) while states with || belong to the vector branch (green).

7

(a) m/g = 0

The model gives rise to stable particles, the lowest ones being, in the case of no background eld, the vector and scalar states. In the continuum model, these particles are distinguished by well-dened parity and charge conjugation quantum numbers [26], with the scalar living in the same sector as the ground state. On a staggered lattice with PBC it is possible to exploit the corresponding lattice symmetries to construct two orthogonal subspaces, one of them containing the vector, and the other the ground state and the scalar [49]. For a chain with OBC the number of surviving symmetries is even lower, with translational invariance lost. In practice this means that to calculate the scalar mass, we need to identify rst the momentum excitations of the vector, which appear at lower energy than the scalar. In [39] this was achieved by starting from the strong coupling limit (x 0), where vector and scalar states are known exactly, and smoothly changing the x

parameter while keeping a label on the scalar state. Instead, we use the expectation value of the spin transformation, SR = N1k=0x2k1T (1), where T (1) is the (cyclic) translation by

one spin site.3 In a system with PBC this operator basically describes the action of charge conjugation on the spins, with the translation exchanging the fermionic and antifermionic character of sites, and the x rotation accounting for the di erent meaning of a spin up on an even (fermion) site (empty) and on an odd one (occupied). Charge conjugation is a good quantum number and distinguishes the vector state (C = 1) from the sector containing

the scalar and the ground state (C = +1), even for nite systems. On the staggered lattice with OBC this is no longer true, and SR does not commute with the Hamiltonian.

However the phase of SR keeps memory of this distinction, and allows us to tag the levels accordingly, with the ground state and the scalar branch of the dispersion relation having phase (hSRi) 0 and the vector branch (hSRi) , as illustrated in gure 3.

4 Results

We have applied the methods described in section 3 to the Hamiltonian (2.6) to determine the ground state, and the vector and scalar mass gaps for various values of the fermion mass. In order to benchmark our results with existing data, we have studied di erent masses m/g = 0, 0.125, 0.25 and 0.5, for which reference values can be found in the literature.

Our goal is to compute, for each of these masses, the ground state energy density, and the vector and scalar mass gaps in the thermodynamic limit (N ) for di erent

nite values of x, and to extrapolate them to the continuum limit, corresponding to the lattice spacing a 0, or x . We have used values of x [5, 600]. In order to

extract the thermodynamic limit for each case, we need to simulate di erent system sizes, N, and apply nite-size scaling. The system sizes, N, cannot be chosen independently of the value of x, as the same number of sites N corresponds to di erent physical volumes Lphys = Na for di erent values of x 1/a2. From numerical simulations, we estimated

that N 20x ensures small enough nite volume e ects. We thus study for each value

of x several di erent system sizes, large enough for the condition above to be satised (for instance, for x = 5 we take N [50, 82] while for x = 600, N [540, 834]).

3The choice of a cyclic translation ensures unitarity of the operator, and is irrelevant in the thermodynamic limit.

8

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We have run the MPS algorithms described in the previous section for each set of parameters, (m/g, x, N), using bond dimensions D [20, 140], to nd the ground state

and excited levels at least until a candidate scalar state is found. We are interested in the subspace of null total charge, which corresponds to

PSz = 0 and can easily be imposed by adding a penalty term to the Hamiltonian. For every level, the MPS iteration stops when the relative change in the energy after one full sweep over the chain is below a certain tolerance, . This value has to be small to ensure a good precision after the extrapolations, but a smaller tolerance translates in more di cult convergence. Therefore we xed = 1012 for the ground state and the vector calculations, and = 107 for the longer computations required by the scalar states.

For every set (m/g, x, N) and for each of the levels we are interested in, we extrapolate our results to 1/D 0, as illustrated in gure 4. In the case of the ground state and the

vector state, almost all the bond dimensions are converged, while for the scalar mass gap, depending on a larger number of intermediate excited states, some of the larger systems are only computed with smaller bond dimension. If the bond dimension reached is large enough, we extrapolate linearly in 1/D. Otherwise we take the value corresponding to the largest D as our estimation for the energy, and the error as the di erence between this value and the one for the immediately smaller D.

The results of the extrapolation described above provide accurate estimators for the energy levels for various nite chains. We then proceed to scale these results with the nite system size to extract the ground state energy density, E0/(2xN), and the mass gaps, (E1(2) E0)/(2x) in the thermodynamic limit for each pair of parameters (m/g, x).

Finite-size corrections to the ground state energy density are linear in 1/N, while for the energy gaps, the leading corrections arise from a kinetic energy term O(/N2) [39]. Hence

the bulk limit is extracted by tting the ground state energy density as

E02xN 0 +

and the energy gaps as

. (4.2)

Figure 5 illustrates this step for x = 100 and masses m/g = 0 and 0.125.

Finally, the values in the continuum limit can be extracted by tting the results for each value of the mass to a polynomial in 1

x . We include only those values of x for which the thermodynamic limit could be extracted accurately (i.e. the corresponding level for the various system sizes was computed for a large enough D). The interval [xmin, xmax]

and the degree of the polynomial used for the t introduce the largest uncertainty in the nal estimators. We have thus performed a systematic error analysis in which the di erent possible ts are taken into account to estimate the continuum values and their errors from our data. The detailed method is described in the appendix A.

Figures 6 (ground state), 7 (vector) and 8 (scalar) show all the values in the thermodynamic limit as a function of 1/x, for each of the mass values, and illustrate the ts that produce the continuum limit.

9

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N +

N2 , (4.1)

E1(2) E02x 1(2) +

1(2)

N2 + O

[parenleftbigg]

1

N3

0.3

0.2

0.25

E2=18933.95992

E2=18571.93023

0.15

0.2

0.15

0.1

EE Dmax

EE Dmax

0.1

0.05

0.05

0

0

0.05 0 0.01 0.02 0.03

0.05 0 0.005 0.01 0.015 0.02 0.025

1/D

1/D

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0.04

0.03

E1=18945.73936

0.025

E1=18585.52621

0.03

0.02

0.02

0.015

EE Dmax

EE Dmax

0.01

0.01

0.005

0

0

0 0.005 0.01 0.015 0.02 0.025 0.03

0.005 0 0.01 0.02 0.03

1/D

1/D

6 x 104

20 x 105

5

E0=18957.65026

E0=18601.89956

15

4

EE Dmax

3

10

EE Dmax

2

5

1

0

0

1 0 0.005 0.01 0.015 0.02 0.025 0.03

5 0 0.005 0.01 0.015 0.02 0.025 0.03

1/D

1/D

m/g = 0

m/g = 0.125

Figure 4. Convergence of the energy levels with the bond dimension, D, for x = 100 and N = 300. In the left column, we show the results of the scalar (uppermost plot, blue), vector (middle plot, green) and ground state (lowest plot, red) for the massless case, m/g = 0, while the right column contains the corresponding results for the case m/g = 0.125. Plotted is the di erence between the computed energy at a certain bond dimension and the one for the maximum Dmax. The error bars

(sometimes smaller than the marker size) indicate the convergence criterion of the MPS algorithm, ( = 107 for the scalar and 1012 for the rest). Dashed lines show the extrapolation in 1/D, with the nal value displayed as a star (numerical value inside each panel).

As our nal result we obtain for the ground state energy density in the massless case the value 0 = 0.318338(22)(24), to be compared to the exact value, 1/ 0.3183099.

The rst error includes the propagated error from innite D and N extrapolations, as well as the systematic error from choosing the tting interval, while the second error is the di erence between results from cubic and quartic ts in 1/x. Note that the latter error

10

1.19

1.502

1.185

1.5

1.498

1.18

1.496

1.175

1.494

1.17

1.492

1.165

1.49

0 0.2 0.4 0.6 0.8 1 1.2

x 105

0 0.2 0.4 0.6 0.8 1 1.2

x 105

1/N 2

1/N 2

0.6

0.822

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0.598

0.82

0.596

0.594

0.818

0.592

0.816

0.59

0.814

0.588

0.586

0.812

0 0.2 0.4 0.6 0.8 1 1.2

x 105

0 0.2 0.4 0.6 0.8 1 1.2

x 105

1/N 2

1/N 2

0.3154

0.3096

0.3156

0.3098

0.3158

0.31

0.316

0.3102

0.3162

0.3104

0.3164

0.3106

0.3166

0.3108

0 1 2 3 4 5

x 103

0 1 2 3 4 5

x 103

1/N

m/g = 0.125

Figure 5. Finite size scaling for x = 100 and mass m/g = 0 (left column) and m/g = 0.125 (right column). The two uppermost rows show as a function of 1/N2 the values obtained for nite systems for the scalar (blue) and vector (green) mass gaps,

E2(1)E0

1/N

m/g = 0

2px . The dashed line corresponds to the t (4.2), with the resulting value indicated on the axis. The lowest row corresponds to the ground state energy density, E0

2xN , as a function of 1/N, and is tted according to (4.1). Notice that the lattice sizes are already close to the thermodynamic limit. The error bars are extracted from the extrapolation in D, shown in gure 4.

is important only in the case of the ground state energy for the scalar and vector mass gaps it is negligible with respect to the former error, as the data are very well described by polynomials only quadratic in 1/x. Our result for the ground state energy in the massive cases is compatible with the massless one, and independent of m/g, as expected [35].

The results for the vector binding energies, MVg := 1 2m/g, are shown in table 1 for

each value of m/g studied, together with the DMRG estimates [20] for comparison. In the massless case, the exact value is also displayed.

11

Vector binding energym/g MPS with OBC DMRG result [20] exact

0 0.56421(9) 0.5642(2) 0.56418950.125 0.53953(5) 0.53950(7) -0.25 0.51922(5) 0.51918(5) -0.5 0.48749(3) 0.48747(2) -

Table 1. Vector binding energies for di erent fermion masses obtained in this work. The second column shows the best DMRG estimates for the same masses.

Scalar binding energym/g MPS with OBC SCE result [36] exact

0 1.1279(12) 1.11(3) 1.128380.125 1.2155(28) 1.22(2) -0.25 1.2239(22) 1.24(3) -0.5 1.1998(17) 1.20(3) -

Table 2. Scalar binding energies for di erent fermion masses, compared to the most precise published results, from SCE.

For the scalar, the most precise results found in the literature for the binding energy,

show them together with our best t and the exact value for the massless case in table 2.

One of the factors explaining the lower precision attained in the scalar calculation as compared to the vector results is the longer time required to reach the scalar state with the MPS algorithm for excited states. As discussed in section 3 the computational cost scales like the square of the required number of levels. Since in many instances the rst scalar candidate appears at level 7 8 or above, this represents a cost over 50 times larger than

in the case of the ground state. We have thus opted for a trade-o between precision and e ciency, and applied for the scalar a less demanding convergence criterion than for the vector or the ground state, what also translates in less precision in the nal results. For the same reason, the largest bond dimensions are in some cases not available in the scalar calculations, which leads to somewhat less precise nite-size scaling.

5 Discussion

We have successfully employed open boundary MPS to compute the ground state and several excited levels of the lattice Schwinger model, using a staggered formulation, and for various values of the fermion mass. Although in the physical subspace in which we work the model contains long range interactions which do not decay with the distance, we have found that MPS produce a very good approximation not only to the ground state, but also to higher excited levels, and we are able to reach precisions comparable to those available from other techniques, or in some cases even slightly better. Additionally, the precision

12

JHEP11(2013)158

MSg := 22m/g in the massive case, correspond to the strong coupling expansion [36]. We

0.31

0.285

0.29

0.312

0.295

0.314

0.3

0.316

0.305

0.31

0.318

0.315

0.32 0 0.05 0.1 0.15 0.2 0.25

0.32 0 0.05 0.1 0.15 0.2 0.25

1/px

(a) m/g = 0

1/px

(b) m/g = 0.25

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0.29

0.26

0.295

0.27

0.3

0.28

0.305

0.29

0.31

0.3

0.315

0.31

0.32 0 0.05 0.1 0.15 0.2 0.25

0.32 0 0.05 0.1 0.15 0.2 0.25

1/px

(c) m/g = 0.125

1/px

(d) m/g = 0.5

Figure 6. Ground state energy density as a function of 1/x. The error of each point (smaller than the markers) reect the uncertainties of the linear ts (4.1), and the star on the vertical axes indicates the (exact) solution for the massless theory, 0 = 1/ 0.31831. The dashed line

corresponds to the t of the whole computed range, x [15, 600], to a cubic function in 1/x.

Excluding small values x < 50 produces a better t, as shown by the solid line, for the interval x [100, 600]. Our nal estimate (see appendix A) for the massless case is 0 = 0.318338(22)(24),

and for all other masses it is compatible with this value.

we reach is not extremely sensitive to the value of the mass, what further points to the usefulness of TN techniques for the non-perturbative parts of the parameter space.

In order to determine the mass gaps, we have also extended the MPS tools to approximate excited states, and we have proposed a modied algorithm that is e cient and allows us to approximate more than ten excited states in chains of hundreds of sites. These techniques provide an ansatz for the targeted states, so that not only the energies, but also other observables can be precisely determined [50].

Our results validate the applicability of tensor network techniques, in particular MPS, for lattice gauge theory problems. We have obtained very precise results, even though the open boundary MPS cannot represent states with well-dened momentum, a problem that a TI ansatz might likely overcome.

13

0.63

1.08

0.62

1.07

0.61

1.06

0.6

1.05

0.59

1.04

0.58

1.03

0.57

1.02

0 0.05 0.1 0.15 0.2 0.25

1/px

(a) m/g = 0, MVg = 0.56421(9)

1.01 0 0.05 0.1 0.15 0.2 0.25

1/px

(b) m/g = 0.25, MVg = 0.51922(5)

JHEP11(2013)158

0.85

1.55

0.84

1.54

0.83

1.53

0.82

1.52

0.81

1.51

0.8

1.5

0.79

1.49

0.78 0 0.05 0.1 0.15 0.2

1/px

(c) m/g = 0.125, MVg = 0.53953(5)

1.48 0 0.05 0.1 0.15 0.2

1/px

(d) m/g = 0.5, MVg = 0.48749(3)

Figure 7. Energy gap for the vector state, 1, as a function of 1/x. The uncertainties of the quadratic ts (4.2) from nite-size scaling are smaller than the marker size. The exact solution for the m/g = 0 case, 1(m/g = 0) = 1/ 0.5641895 is indicated with a star. Displayed are the

ts for the whole interval x [20, 500] (dashed line), including up to quadratic terms in 1/x, and

the same t for [30, 300] (solid line), but they are practically indistinguishable at the scale of the plots. The nal values for the binding energies, MVg , and their errors (see appendix A) are displayed under the corresponding plots.

The long term goal would be to generalize these studies to higher dimensional systems, getting in this way closer to the target of lattice QCD studies. The MPS ansatz employed in this work is one-dimensional, and its applicability to higher dimensional problems is limited, a fact well understood in terms of its entanglement content. Nevertheless, the entanglement structure in the MPS has a natural generalization to two and higher dimensions in the PEPS ansatz [13, 40]. Algorithms exist for the ground state search and for simulating the evolution also in the case of PEPS, and the ansatz can directly be applied to fermionic systems [5153]. Although the higher computational costs limit the nowadays feasible system sizes and bond dimensions, PEPS are good candidates to study higher dimensional lattice gauge problems. Our MPS study is a rst step to assess the feasibility of TNS to describe the relevant physics in this kind of problems.

14

1.3

1.84

1.82

1.25

1.8

1.78

1.2

1.76

1.15

1.74

0 0.1 0.2 0.3 0.4 0.5

1/px

(a) m/g = 0, MSg = 1.1279(12)

1.72 0 0.1 0.2 0.3 0.4 0.5

1/px

(b) m/g = 0.25, MSg = 1.2239(22)

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1.6

1.58

2.26

1.56

1.54

2.24

1.52

2.22

1.5

1.48

2.2

1.46 0 0.1 0.2 0.3 0.4 0.5

1/px

(c) m/g = 0.125, MSg = 1.2155(28)

0 0.1 0.2 0.3 0.4 0.5

1/px

(d) m/g = 0.5, MSg = 1.1998(17)

Figure 8. Energy gap for the scalar state, 2, as a function of 1/x. The error bars correspond to the uncertainties of the nite-size scaling ts (4.2). The star on the vertical axes in the m/g = 0 plot shows the exact solution 2(m/g = 0) = 2/ 1.12838. Fits for the whole range of x (cubic)

and for a smaller interval (x [15, 150]) are shown (respectively dashed and solid lines). The errors

for large values of x are in this case much more signicant, because they require solving larger systems, for some of which the maximum bond dimension reached was not large enough. The nal values for the binding energies, MS

g , and their errors (computed as described in appendix A) are displayed under the corresponding plots.

One of the main advantages of the MPS and other TNS methods is that they easily allow us to attack other problems which are more complicated for standard lattice techniques. In particular, it is straightforward to include a chemical potential term in the Hamiltonian, which is interesting in many-avour Schwinger models. Additionally, we could perform a similar study in a more complicated lattice theory, in particular with a non-Abelian symmetry. Thermal equilibrium properties can be easily studied with TNS, and it is also possible to simulate real time evolution, which opens the door to out-of-equilibrium questions. Thus even if tensor networks are presently not competitive with standard Monte Carlo techniques for 4D quantum eld theories, they might allow us to address problems that are not amenable for customary lattice eld theory techniques.

15

Acknowledgments

This work was partially funded by the EU through SIQS grant (FP7 600645). K.C. was supported by Foundation for Polish Science fellowship Kolumb. This work was supported in part by the DFG Sonderforschungsbereich/Transregio SFB/TR9. K.J. was supported in part by the Cyprus Research Promotion Foundation under contract PO E KY H/EM EIPO /0311/16.

A Continuum limit and error analysis

As we have described in the main body of the text, we have a set of data corresponding to a given bond dimension D, lattice size N and coupling x. Having performed extrapolations to innite D and N, we are left with the dependence of our observables on x and we want to take the continuum limit x . The leading order cut-o e ects are O(1/x), but we are

sensitive also to higher-order corrections (O(1/x), O(1/x3/2), . . .). Therefore, in general,

our continuum extrapolated result depends on the chosen tting interval in x. To take this systematic uncertainty into account, we perform the following error analysis procedure, similar to those employed in data analysis for Lattice QCD (see e.g. refs. [54, 55]).

Let us assume that, for a xed value of m/g, we have a set of data for the chosen observable, O: {(xi, Oi, Oi)}ni=1, for n values of x, such that Oi is the observable eval

uated at a nite lattice spacing corresponding to coupling xi, and Oi is its respective error (originating both from extrapolations to innite D and N). We choose some minimal number of points (nmin) and we t every possible subset of p consecutive data points, with p nmin, to a given polynomial function g(x). Each such t gives us an estimate of the

continuum value of the chosen observable, Oc we denote these continuum values by Oc , where = 1, . . . , Mts. Each t has its respective value of 2, denoted by 2 :

2 =

Xi: xit (g(xi) Oi)2( Oi)2 (A.1)

where the sum runs over all data points entering the t labelled by .

Our preferred value for the observable in the continuum limit is then dened as the median of the distribution of estimators Oc weighted by exp(2 /Nd.o.f.), where Nd.o.f. is

the number of degrees of freedom.To estimate this quantity, the ts are reordered such that for all , O < O +1. We

then dene the cumulative distribution function f ( = 1, . . . , Mts):

f =

JHEP11(2013)158

P =1 exp(2 /Nd.o.f.)

PMts =1 exp(2 /Nd.o.f.)

. (A.2)

Our preferred value for the considered observable in the continuum limit is then dened as the value of O for which is the smallest number that satises f 0.5. The cor

responding error (the systematic error from the choice of the tting range) is computed as 0.5(O+ O), where O+ is the value of O corresponding to the smallest such

that f 0.8415 and O is the value of O corresponding to the smallest such that

16

f 0.1585. This denition of the error is motivated by the fact that in the limit of an

innite number of ts, the weighted distribution will be Gaussian and this denition of the error will correspond to the width of such Gaussian distribution.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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