Critical Coupling for Two-dimensional $\phi^4$ Theory in Discretized Light-Cone Quantization

We solve for the critical coupling in the symmetric phase of two-dimensional $\phi^4$ field theory using Discretized Light-Cone Quantization. We adopt periodic boundary conditions, neglect the zero mode, and obtain a critical coupling consistent with the critical coupling reported using conformal truncation in light-front quantization. We find a 17% dfference from the critical coupling reported with light-front quantization in a symmetric polynomial basis.


Introduction
It is well known that most of our intuition about spectra and wave functions in quantum mechanics come from solving Hamiltonians and a variety of methods exist for this purpose. In molecular, atomic and subatomic quantum systems, sophisticated and efficient numerical approximation techniques have been developed and refined over the years to solve quantum many-body Hamiltonians. One would like to make use of the intuition developed and experience gained in solving non-relativistic systems to devise methods to solve systems governed by relativistic quantum field theories. But Hamiltonian methods were rarely developed for this purpose. "This is a pity", as was noted sometime back [1]. There are major stumbling blocks to this path caused by the presence of infinitely many degrees of freedom and the mandatory need for renormalization.
Especially noteworthy is the severe divergence caused by vacuum processes which are addressed analytically in the perturbative framework. One has to learn how to handle them or at least how to side step them in the non-perturbative Hamiltonian framework.
It was Dirac [2] who first pointed out the advantages of light-front dynamics among different Hamiltonian formulations of relativistic many-body theories. Great interest in light-front dynamics arose because lightfront quantization provides the theoretical basis for the celebrated parton model [3,4] which facilitates an intuitive understanding of high energy processes. Many additional virtues of light-front quantization have been identified and studied since the work of Dirac, but a non-perturbative numerical approach was missing. Discretized Light-Cone Quantization (DLCQ) [5][6][7][8] was proposed in the mid 1980s as a novel, non-perturbative numerical technique to solve quantum field theories.
Two-dimensional interacting scalar field (φ 4 ) theory, in spite of being the simplest of interacting quantum field theories, has a rich structure and has been the subject of both rigorous mathematical analysis [9,10] and various non-perturbative numerical approaches. Not surprisingly, soon after the proposal of DLCQ, it was applied to the symmetric phase of two-dimensional φ 4 theory [11]. As a result, many strengths and a few weaknesses of the method were identified already in those early days. It was known that the system undergoes a phase transition from the symmetric phase to the symmetry broken phase at strong coupling as a result of quantum fluctuations. Among the observables calculated in Ref. [11] was the critical coupling for the vanishing mass gap in the odd particle sector as a function of the dimensionless total longitudinal momentum K. It was realized that the continuum limit is reached by taking K → ∞ which requires reliable extrapolation of finite K results. An extrapolation was attempted in Ref. [12] using finite K results with maximum K of 20. The limited data available was extrapolated to K = 100 and the corresponding critical coupling was found to be ≈ 33.
The critical coupling has been calculated since then, with various nonperturbative techniques within both Lagrangian and Hamiltonian frameworks. Within the light-front Hamiltonian formalism, two works have recently emerged which are of immediate relevance to DLCQ, namely Burkardt et al. [13] and Anand et al. [14].
In Burkardt et al., the Hamiltonian is expressed in the Fock space basis in the continuum. Here, using the positivity of the light-front longitudinal momentum, the terms in the Hamiltonian containing pure creation or annihilation operators are dropped. The state also is expanded in a Fock basis where the coefficients are multi-particle wave functions. Then the multi-particle wave functions are expanded in terms of symmetric polynomials. These polynomials span the entire range of the momentum fraction x from 0 to 1, including the end points. A finite dimensional Hamiltonian is obtained by truncating the maximum number of bosons and the maximum number of basis functions used.
In Anand et al., the light-cone conformal truncation is used. As explained in detail in Ref. [15], one starts from the conformal limit of the two-dimensional φ 4 theory, which is a free massless theory. A complete set of basis states of conformal quadratic Casimir C is constructed using primary operators. These states are used to express the Hamiltonian of the two-dimensional φ 4 theory in light-cone quantization. After truncating the basis to states with Casimir eigenvalue C below some threshold C max , the resulting finite dimensional Hamiltonian matrix is numerically diagonalized. The positivity of the light-front longitudinal momentum is employed in this work.
By the use of basis functions in the continuum in the above two references, sensitivity to the small longitudinal momentum region can be explored and handled very efficiently. On the other hand in the previous and current work on two-dimensional φ 4 theory with Periodic Boundary Conditions (PBC), exactly zero longitudinal momentum, which is a constrained (non-dynamical) mode in the symmetric phase, is dropped. Furthermore, DLCQ, in contrast with Refs [13,14], uses a uniform grid. Therefore, an arbitrarily small longitudinal momentum can be accessed only in the continuum limit. Thus, by approaching the continuum limit in DLCQ numerically and comparing the resulting value for the critical coupling with those of Refs [13,14], we can quantify the efficiency and legitimacy of the DLCQ program.
Motivated by the work of Rozowsky and Thorn [16], who argued that the exactly zero longitudinal momentum mode (hereafter referred to as the zero mode) is not necessary to describe spontaneous breaking of symmetry, the symmetry broken phase of two-dimensional φ 4 theory was investigated in DLCQ in detail with both PBC with the omission of the zero mode [17] and Anti-Periodic Boundary Conditions (APBC) [18,19] where the zero mode is naturally absent. Masses of low-lying states, their parton distributions and coordinate space (x − ) profiles were successfully calculated. These accumulated results provide another motivation to revisit the symmetric phase of two-dimensional φ 4 theory in DLCQ.
To analyze realistic quantum field theory problems in 3+1 dimensions, currently, the most efficient lightfront Hamiltonian approach is the Basis Light-Front Quantization (BLFQ) method [20,21], which frequently employs DLCQ in the light-front longitudinal direction and orthonormal basis functions in the transverse space. BLFQ utilizes the vast practical experience gained in solving strongly interacting nuclear many-body systems and employs the same methodology and techniques to diagonalize the Hamiltonians of enormous dimensionality. Thus the lessons learned in studying the convergence and extrapolation issues of DLCQ applied to strong coupling problems in 1+1 dimensions are also directly relevant for the BLFQ program.
Within the Hamiltonian-based approaches, in order to quantitatively assess the strengths and weaknesses of DLCQ, corresponding calculations in the Instant form with a Fock-space based Hamiltonian are a must.
This was lacking for a long time. Fortunately, results of such calculations in both the symmetric [22] and broken [23] phases of two-dimensional φ 4 theory have become recently available. We also note that, over the past two decades there have also been many studies of the critical coupling of two-dimensional φ 4 theory.
See, for example, Refs. [24][25][26][27][28][29][30]. However, with the exception of Refs. [13] and [14], a direct comparison of the value of critical coupling via DLCQ presented in this work with the other works cited is not possible because of the matching problem between equal-time and light-front methods [13,31].
Our primary focus in the present work is to obtain the critical coupling of two-dimensional φ 4 theory in the continuum limit of DLCQ for the transition from the symmetric phase to the broken phase. To be more precise, we extract three separate critical couplings that each produce a vanishing mass gap at finite resolution, K. These are the critical couplings for the lowest state of the odd sector, the lowest state of the even sector and the first excited state of the odd sector. We find agreement of all three critical couplings in the continuum limit (K → ∞), implying degeneracy at a vanishing mass gap, to within numerical uncertainties, which we interpret as the signal for spontaneous symmetry breaking.

Two-dimensional φ 4 theory in Discretized Light-Cone Quantization
We seek to solve a Hamiltonian (H) eigenvalue problem expressed in a suitable basis to obtain the low-lying mass spectroscopy. In this work, we solve for and present the eigenvalues of the mass-squared operator proportional to H. In principle, once the eigenvectors are obtained, we could evaluate matrix elements of additional observables O but we defer those efforts to a future project. For the present work, we aim to obtain that spectroscopy in the region of critical coupling producing a vanishing mass gap at a sequence of basis space cutoffs K. We then perform detailed extrapolations of these results to the continuum limit (infinite matrix dimension) accompanied by an uncertainty analysis.
The Hamiltonian for two-dimensional φ 4 theory in DLCQ was first presented in Ref. [11]. We start from the two-dimensional Lagrangian density with the light-front variables defined by x ± = x 0 ± x 1 .
The Hamiltonian density defines the Hamiltonian where L defines our compact domain −L ≤ x − ≤ +L. Throughout this work we address the energy spectrum of H.
The longitudinal momentum operator is where K is the dimensionless longitudinal momentum operator. The mass-squared operator M 2 = P + P − = KH whose eigenvalues we present in this work.
With PBC we can write where φ 0 is the zero mode operator and Φ contains non-zero modes.
By integrating the equation of motion ∂ + ∂ − φ + µ 2 φ + λ 3! φ 3 = 0 over the longitudinal space, one can show that (1) the zero mode operator vanishes in the free field theory, and (2) in the interacting theory, the zero mode operator is constrained and obeys a nonlinear operator equation. Thus incorporating the zero mode in DLCQ is a nontrivial problem. To the best of our knowledge, this problem has not yet been solved in a satisfactory manner. When the constraint equation is solved perturbatively, it leads to additional interaction terms in the Hamiltonian. For some processes studied, contributions of these new terms, however were found to vanish [32,33] in the infinite volume limit (L → ∞).
For the non-zero mode part Φ we use the solution of the equation of motion for the free massive scalar field in the continuum theory, which provides a convenient Fock space basis: Note that zero mode is absent in this expression. In DLCQ, the corresponding field expansion is with n = 1, 2, 3, . . . with PBC and n = 1 2 , 3 2 , 5 2 . . . with APBC. The normal ordered dimensionless longitudinal momentum operator The normal ordered Hamiltonian is given by klmn a † k a † l a n a m δ k+l,m+n klmn a † k a l a m a n + a † n a † m a † l a k δ k,l+m+n (9) with N lmn = 1, l = m = n, and

Symmetries and Vanishing Mass Gaps
Since the Hamiltonian exhibits the φ → −φ symmetry, the even and odd particle sectors of the theory are decoupled. In the "symmetric phase" of the theory (positive bare mass-squared µ 2 ) which we investigate here, the solutions at weak positive values of λ have simple structure. In the odd sector, the lowest solution is dominated by a single boson carrying all the light-front momentum K. In the even sector, the lowest solution is dominated by two bosons each carrying K/2.
We do not invoke mass renormalization so that, with increasing coupling, these two lowest states will each decrease towards zero mass. A separate value of the critical coupling at fixed K is obtained when each of these mass gaps vanishes. Mass gaps for higher states in each sector can also vanish at fixed K but at successively larger values of the critical coupling. In line with expectations, we find that, as a function of increased coupling, the lowest state of the odd sector vanishes before the lowest state of the even sector at each value of K.
For the present work, we are interested in calculating a set of critical couplings that produce corresponding vanishing mass gaps over a range of computationally accessible values of K. With these results, we then carry out extrapolations of these critical couplings to their continuum limits (K → ∞). We perform these calculations for the two lowest states of the odd sector and the lowest state of the even sector. We raise and answer the question, to within numerical precision, whether these critical couplings are the same or distinct in the continuum limit. We find that they are the same and we compare our continuum limit result for the critical coupling with results obtained with other light-front methods [13,14].
As mentioned above, many interesting questions can be addressed with detailed studies of the light-front wave functions at the critical coupling and as a function of K. In addition, one can examine the sensitivity to choice of boundary conditions, for example, by performing corresponding studies with APBC. We defer these and other valuable topics to future research projects.

Methods of Solving for the Low-Lying Spectroscopy at Each K
Since the interacting theory is a function of a single dimensionless variable λ µ 2 , we adopt µ 2 = 1 for convenience. Thus, our results for functions of λ µ 2 are the same as our results for functions of λ. Results presented here were obtained using Cori, a supercomputer at the National Energy Research Computing Center (NERSC) [34], with the Many Fermion Dynamics (MFD) code adapted to bosons [17][18][19]35]. The Lanczos diagonalization method is used in a highly scalable algorithm that allows us to proceed to high enough values of K for smooth K → ∞ extrapolations.
While the low-lying eigenvalues provided by MFD were obtained to about seven significant figures (singleprecision with 32-bit words), the eigenvalues from lfphi4MH-00 and LFHC phi4 SJ were obtained to 14 significant figures (double precision or 64-bit accuracy). All three codes produced the same eigenvalues to within their respective precisions. The two new codes are not yet optimized to run on supercomputers so their use is limited in the present work. Our presented results are obtained from diagonalizations for K ≤ 42 in double-precision with lfphi4MH-00 and LFHC phi4 SJ while diagonalization results for K > 42 are obtained in single-precision with MFD.
We have performed calculations on meshes of values of K for the even and the odd sectors as detailed below. For each sector and each K we perform a set of calculations over a small range in λ sufficient to determine the critical coupling for the vanishing mass gap. In order to gain an impression of the computational effort, we present the dimensions of H in DLCQ with periodic boundary conditions at representative values of K, both without and with boson number truncation in Table 1. (8)  16  113  118  113  118  87  99  32  4163  4186  3774  3891  1426  1893  48  73593  73680  54486  58054  9616  15083  64  870677  870953  488759  542632  41171  75092  80  7897846  7898630  3186613  3696386  133295  278203  88  22053415  22054694  7428056  8810476  222513  494131  96  59056128  59058176  16465206  19970504  356993  840816   Table 1: Representative matrix dimensions for scalar φ 4 in DLCQ with periodic boundary conditions omitting the zero mode. Numbers in parenthesis in column headings signify the maximum number of bosons allowed which results in a truncated basis.

Matrix Dimension K Odd sector Even sector Odd sector (16) Even sector (16) Odd sector (8) Even sector
It is noteworthy that calculations near a vanishing mass gap involve strong coupling and increased level density (compared to weak coupling at the same K) near the lowest state of the system. Both of these features induce the need for considerable care in attaining and assuring the stability of numerical precision.
One can note, for example, that thousands of Lanczos iterations are required for the larger dimensional matrices solved for the lowest-lying solutions in the present effort. As K increases, the number of Lanczos iterations increases for eigenvalues near the critical coupling and the numerical precision begins to erode.
Future efforts will focus on improving the efficiency of the double-precision codes in order to achieve greater accuracy at higher K and to improve the results for the critical coupling in the continuum limit.

Numerical Results, Polynomial Fits and Extrapolations
As mentioned above, we focus our attention on the lowest two mass eigenstates of the odd sector and the lowest mass eigenstate of the even sector. For these three states we obtain the critical coupling where the : Critical couplings λ c /µ 2 for vanishing mass gaps as a function of 1/ √ K obtained with DLCQ using periodic boundary conditions. Solid (open) symbols represent DLCQ results for 3 states, according to the legend, that are used (not used) in the fits. The smooth curves are results for f 1 (blue), f 2 (red) and f 3 (green) resulting from procedures using polynomial fits as discussed in the text. The extrapolated value is the intercept on the vertical axis of 22.64 ± 0.17. The inset provides an expanded view of the region approaching the continuum limit. A vertical black bar represents our approximate systematic uncertainty which dominates our overall uncertainty (see text). The inset also presents, for comparison, a vertical purple bar (magenta bar) on the vertical axis representing the critical coupling result of Ref. [13] ( [14]) which is 26.39 ± 0.63 (23.12 ± 0.38). eigenvalue vanishes at each K. While details will be introduced and discussed below, we present here a brief overview of our main results. Fig. 1, displays the critical couplings for the vanishing mass gaps of these three low-lying states as functions of 1/ √ K. The symbols represent calculations using DLCQ for each of the three states indicated in the legend and the curves represent polynomial fits using these calculated results according to procedures discussed below.
We obtain the continuum limit by taking our functional fits to the limit K → ∞ as indicated by their intercepts with the vertical axis in Fig. 1. The inset provides an enlarged view of these functions as they approach the continuum limit. We obtain a result of 22.64 ± 0.17 for the critical coupling for these three states. Our result can be compared with the critical coupling result of Ref. [13] ( [14]) which is 26.39 ±0.63 (23.12 ±0.38). Hence, we are consistent with the result of Ref. [14] within the respective quoted uncertainties. Our result is presented as a vertical black band on the vertical axis of the inset to Fig. 1 while those of Ref. [13] and Ref. [14] appear as vertical purple and vertical magenta bands respectively.

Lowest state of the odd sector
We now present details underlying the results displayed in Fig. 1. For the lowest state of the odd sector, we fit a 5th-degree polynomial in 1/ √ K to the DLCQ results for the coupling that provides a vanishing mass gap for the lowest mass-squared eigenvalue at each K. We determined our preferred functional form and variable by testing many functional forms and variables 1 . Our selected function and variable provide robust extrapolations reasonably independent of the range of DLCQ results included in the fit (see Sec. 4).
To be more specific, we determine the function where z = 1/ √ K, the subscript "1" specifies the function for the lowest state of the odd sector and j runs from zero to 5. All fits presented in this work are performed by minimizing the mean square deviation between the adopted function and the DLCQ results. The coefficients of the fits are tabulated in Table 2.
We will discuss the residuals of our fits after we introduce the other fits.
In order to limit the role of finite K artifacts that play a larger role at lower values of K, we selected 1 For example, we tried various polynomial fits as functions of 1/K and 1/K 3/2 . We also tried exponential fits as well as fits in the form A * K B + C. We ultimately settled on a polynomial as a function of 1/ √ K.
our 52 DLCQ results in the range 12 ≤ K ≤ 88 to determine the coefficients in Eq. 12. This range reaches the maximum for which we obtained DLCQ results for the lowest state of the odd sector. In particular, we employed 31 data points with unit increments over 12 ≤ K ≤ 42, 19 data points with increments of two in K over 42 < K ≤ 80, and the points at K = 84 and K = 88. The choice of mesh in K is arbitrary but aimed to provide a significant number of points in the range where available computational resources were sufficient and a robust fit could be obtained. We display the resulting fit as a solid blue line in Fig. 1. We compare the resulting fit with 7 additional points in the range 5 ≤ K < 12 to illustrate that the fit function also performs reasonably well in this range.
We examined the light-front wave functions for the lowest state of the odd sector just below and just above the critical couplings at each K. We found that they remained dominantly single-boson in character.
By proceeding further beyond the critical coupling we find states with more complex character crossing this single-particle dominated state to become the lowest-lying state. The details of these transitions in the region beyond critical coupling will be presented in a separate work [42].

Lowest state of the even sector
We anticipate that the results for the lowest state of the even sector will likely follow a pattern related to results for the lowest state in the odd sector since φ 4 in 1+1 dimensions is known to have no even bound states [43]. Thus, we suppose that the mass of the lowest state of the even sector at K, which is expected [11] and found to be dominated by 2-boson configurations, will be close to twice the mass of the lowest state in the odd sector at K/2, which is dominated by the 1-boson configuration. We define f 2 (z) to be the function describing the vanishing mass gap of the lowest state of the even sector and we introduce "R 2/1 " for the ratio ("R") of the mass of the lowest state of the even sector ("2") to the mass of the lowest state of the odd sector ("1"). We formalize what we anticipate by suggesting that will produce a smooth function with reduced finite basis artifacts. To examine this supposition, we plot the DLCQ results for the ratio R 2/1 in Fig. 2 as a function of z 2 = 1/K. We explored different variables for the horizontal axis of this plot and found that z 2 = 1/K produced results that we could then fit with good precision by a 4th-degree polynomial in 1/K while other variable choices required higher-degree polynomials for fits of comparable quality. In other words, we define the fit function for the ratio R 2/1 through where j runs from zero to 8 in increments of 2. For consistency in the treatment of finite K artifacts, the lower limit in K for the even sector DLCQ data is chosen so as to correspond with the chosen lower limit DLCQ results: K = 24 to 80 DLCQ results: K < 24 not used in the fit 4th degree polynomial fit Figure 2: Ratio of critical coupling for the vanishing mass gaps of the lowest state at K of the even sector and the lowest state of the odd sector at K/2 versus 1/K. Symbols represent the results from DLCQ and the smooth curve is a fourth degree polynomial in z 2 = 1/K fit to the 29 results represented by the solid symbols. The coefficients of the fit B j (see Eq. 14) are presented in Table 2. The vertical dashed line signifies the boundary between regions where finite basis artifacts are strong (to the right for lower K) yet tending to cancel in this ratio as observed by the smooth behavior over all K presented. K = 12 used for the DLCQ data adopted for the fit for f 1 . Therefore, the DLCQ data for the lowest state of the even sector included in the fit are the 29 points ranging over 24 ≤ K ≤ 80 with increments of 2 (solid symbols in Fig. 2). The best fit produces the smooth curve in Fig. 2 and the residuals for this fit are presented below in Fig. 6.
We tabulate the coefficients B j of the fit in Eq. 14 in Table 2. Note that the coefficient B 0 ≈ 1.000267 provides the ratio of our extrapolations for f 2 to f 1 in the continuum limit. The difference of B 0 from unity fits within our overall uncertainty for the critical coupling in the continuum limit (see below).
Based on the polynomial fits for R 2/1 (z) and f 1 (z), we can obtain f 2 (z) using Eq. 13: We present the curve for f 2 (z) from this approach in Fig. 1 along with the DLCQ results for the lowest state of the even sector. We note that, in order to facilitate comparisons, the results in Fig. 1 are presented as Since the DLCQ data for the ratio R 2/1 (including the data excluded from the fit and represented as open symbols in Fig. 2) appear to be a smooth function of 1/K, we can conclude that finite basis artifacts appear to be reasonably cancelled when forming this ratio. This cancellation facilitates a good fit with a 4th-degree polynomial over the range in z 2 = 1/K that covers 24 ≤ K ≤ 80. Furthermore, the resulting fit reasonably describes the DLCQ results below K = 24 that are not included in the fit. For ease of visualization, the K = 24 boundary is represented by a vertical dashed line in Fig. 2. We note that this K = 24 boundary occurs near the point of inflection in R 2/1 but we do not find this inflection point to be physically significant.
We test the finite basis artifact cancellations further by comparing the fit displayed in Fig. 2 Table 2) to ≈ 1.0000843 when all points in Fig. 2 are included in the fit.
The results shown in Fig. 2 are predominantly below unity but approach unity in the continuum limit.
This indicates that, using our connection between the lowest state of the even sector and the lowest state of the odd sector, the effect of finite K is to produce a bound state in the lowest state of the even sector at the critical coupling for its vanishing mass gap. Thus, at finite K, the lowest even state, when massless, cannot decay to two of the lowest massless states at K/2 of the odd sector. We checked the light-front wave functions for the lowest solution of the even sector near critical coupling at each K and confirmed that they are overwhelmingly dominated by the 2-boson configuration.
It is important to note that our results in Fig. 2 do indicate that degeneracy of the even and odd sector, a condition for spontaneous symmetry breaking, occurs in the continuum limit at the vanishing mass gaps of both sectors -i.e. the ratio of critical couplings approaches unity as K → ∞. In the continuum limit a mix of lowest state solutions from the even and odd sector becomes a solution. In addition, as we see in Sec.
3.3 additional degeneracies will occur and will lead to solutions with well-mixed particle content.

First excited state of the odd sector
Returning to the odd sector, we consider the critical coupling for the vanishing mass gap of the first excited state as a function of z which we define as f 3 (z   Table 2. The vertical dashed line signifies the boundary between regions where finite basis artifacts are strong (to the right for lower K) yet tending to cancel in this ratio as observed by the smooth behavior over all K presented.
are presented in Table 1.
Since we anticipate and find that this state is dominated by 3-boson configurations, we extend the logic used above to now consider the ratio of the vanishing mass gap for this state to the vanishing mass gap for the lowest state of the odd sector. Hence, we define this ratio as We plot the DLCQ results for this ratio in Fig. 3 as a function of z 2 = 1/K. Again, this choice of variable produces a sequence of results that is well-described by a 4th-degree polynomial where j runs from zero to 8 in increments of 2 with the resulting smooth curve displayed in Fig. 3. Our fit includes 19 DLCQ points over the range 36 ≤ K ≤ 90 in increments of 3 displayed as solid symbols in Fig. 3. Note that we attain K = 90 for the first excited state of the odd sector owing to the boson number truncation discussed above. The choice of the lower limit in K for this state follows a similar line of reasoning for consistent treatment of finite K artifacts that we invoked for the lowest state of the even sector in Sec. 3.2.
We tabulate the coefficients C j of the fit in Eq. 17 in Table 2. Note that the coefficient C 0 ≈ 1.000463 provides the ratio of our extrapolations for f 3 to f 1 in the continuum limit. The difference of C 0 from unity also fits within our overall uncertainty for the critical coupling in the continuum limit (see Sec. 4).
We now offer an observation about finite basis artifacts analogous to our previous observation concerning Fig. 2. That is, it is interesting to note that finite basis artifacts appear again to be well-suppressed in the ratio R 3/1 since the 4th-degree polynomial represents DLCQ results accurately over a range in z 2 = 1/K that extends well below the K = 36 boundary in Fig. 3 that separates the results employed in the fit (solid symbols) from those excluded (open symbols).
We are now in a position to obtain f 3 (z), based on our pair of polynomial fits We present the curve for f 3 (z) from this approach in Fig. 1 along with the DLCQ results for the first excited state of the odd sector.
Following our exploration of the finite basis artifact cancellations for R 2/1 we compare the fit displayed in Fig. 3 for R 3/1 with a fit that includes all the points presented in the Fig. 3. Such an expanded fit goes smoothly through all the points and is nearly indistinguishable from the curve in Fig. 3 for the entire range of 1/K displayed. For comparison, the K → ∞ point changes from ≈ 1.0004631 (B 0 in Table 2) to ≈ Fig. 3 are included in the fit.

when all points in
Unlike the ratio R 2/1 , the ratio R 3/1 remains everywhere above unity at finite K while approaching unity smoothly and monotonically as K → ∞. This indicates that the first excited state of the odd sector at finite K, when approaching a vanishing mass, is able to decay to three separated interacting bosons each carrying K/3 units of the longitudinal momentum of the parent state. Unity in the asymptote of R 3/1 signals degeneracy of the 3-boson dominated state with the 1-boson dominated state at a vanishing mass gap for both in the continuum limit.
Up to this point, as summarized in Fig. 1, the DLCQ results indicate the anticipated degeneracy of the even and odd sectors and the lowest and first excited states of the odd sector in the continuum limit K → ∞ we are consistent with the results of Ref. [14] as depicted in the inset to Fig. 1.

Residuals and Uncertainty Quantification
There are a number of sources of uncertainties in our implementation of DLCQ. An abbreviated list includes: 1. Hamiltonian mass eigenvalue calculations 2. Interpolation for the vanishing mass gap 3. Selection of functions and variables for fitting critical couplings as a function of K.

Role of residual finite K artifacts
Readers less interested in the details of the uncertainty analysis may wish to either scan subsection 4.3 which details the largest source of our uncertainty or skip this section entirely.

Uncertainties in eigenvalues
We have mentioned above that we have developed, tested and employed three independent codes. Two of these codes, "lfphi4MH-00" [36] and "lfphi4SJ-00" [37], are special purpose codes that have produced our mass eigenvalues in double-precision (64-bit accuracy) for K ≤ 42. They have been benchmarked with each other to verify their eigenvalues agree within double-precision.
A third code (MFD) [17][18][19]35] is a Fortran77/Fortran90 code which is MPI parallelized. This code, which has been enabled to solve boson systems, employs the Lanczos algorithm to obtain the lowest masssquared eigenvalues in single-precision (32-bit accuracy). We note that MFD employs double-precision for key components of the Lanczos iteration process in order to maintain overall single-precision accuracy.
Benchmarking the three codes for the region K ≤ 42 where all three run successfully, confirms MFD produces results with single-precision accuracy. MFD alone is employed to obtain our results in the range K > 42.
In the region where the mass gap vanishes, the mass-squared eigenvalues are typically less than unity.
In this region we find that MFD's lowest eigenvalue has an uncertainty of about unity in the fifth decimal place. Support for the assertion of MFD's accuracy at larger values of K, which most strongly influences our extrapolations, will be seen below in our presentation of detailed results at K = 75 in Sec. 4.2. The open symbols present 1000 times the difference between the corresponding solid symbol and fit. The inset provides an expanded view of the two quadratic fits as they cross the vanishing mass gap. As expected, the coupling must be larger for the truncated calculation relative to the full calculation to produce a vanishing mass gap.

Interpolation method and uncertainties
In order to obtain the critical coupling for a vanishing mass gap at a specific value of K, we perform a sequence of calculations at values of the coupling adjacent to the vanishing mass gap. We perform these calculations over a narrow range of coupling separately for each of our three states since different values of the critical coupling will emerge for each state at each value of K. This is evident from taking any vertical line through the final results presented in Fig. 1 and projecting the intercepts to the vertical axis.
For K ≤ 42 we use "lfphi4MH-00" to search for the value of the critical coupling to eight significant figures producing a vanishing mass gap. For K > 42 we use MFD to produce results on a mesh of values of the coupling that span the vanishing mass gap and obtain the critical coupling by interpolation as described below.
We choose the K = 75 example to present our procedure in more detail. In addition, we choose the most challenging state to converge (requiring the largest number of Lanczos iterations), the first excited state of the odd sector for this detailed study. In fact, we already experienced Lanczos convergence difficulties when attempting full calculations for this first excited state in the odd sector at a value of the coupling close to the critical coupling (see below). This led us to consider implementing a boson number truncation at 16 bosons for DLCQ basis states in the range K > 75.
To be specific, the DLCQ eigenvalues for the full calculation of the first excited state in the odd sector at K = 75 in Fig. 1 were obtained with 7500 Lanczos iterations. Beyond 7500 Lanczos iterations we experienced numerical difficulties for this first excited state. Note that even more iterations would be needed at higher K to achieve convergence. By contrast, 1600 Lanczos iterations are sufficient to converge the lowest state with a full basis in the odd sector at couplings near the vanishing mass gap even at K = 88, the highest K we achieved with a full basis.
We present the DLCQ K = 75 results for the first excited state in the odd sector as solid symbols in Fig. 4 on an evenly spaced mesh that spans the vanishing mass gap. We chose 11 points about evenly distributed above and below what we estimated as the critical coupling with increments of 0.20. We present two cases in We then fit the results of each calculation with a second-degree polynomial which we found suitable for extracting the critical coupling by interpolation. Fig. 4 then presents the residuals between each of the fits and the corresponding calculated points as open symbols with the corresponding shape and color. In order to render the displacement of the residuals from zero more visible, we multiply the residuals by a factor of 1000.
The residuals appear to be reasonably distributed about zero which suggests the adequacy of our choice of a second degree polynomial as the function for fits and interpolations. Furthermore, as may be anticipated from the reduction in the Hamiltonian matrix size by more than a factor of 2 from 4,059,416 for the full calculation to 1,824,575 for the matrix truncated to the limit of 16-boson configurations, the residuals are noticeably reduced with that truncation. The resulting rms deviation between calculation and fit is 4.56x10 −5 (5.61x10 −6 ) for the full calculation with 10 points (16-boson truncation with 11 points). The resulting uncertainty in the critical λ at K = 75 is obtained by incorporating information from the slope of the quadratic fit function. With this we arrive at an uncertainty in the critical λ at K = 75 of 1.57x10 −4 (1.83x10 −5 ) for the full (16-boson truncation) calculation.
In order to examine the difference in the interpolated critical coupling, we provide an inset to Fig. 4 with an expanded scale. The inset shows the two fit functions provide vanishing mass gaps that differ by about 0.00002 in their critical couplings. As expected, a larger coupling is needed with the truncated matrix to produce a vanishing mass gap. This uncertainty is small compared to other uncertainties that we will assess below.
Referring back to the discussion in Sec. 4.1, we can also use these deviations presented in Fig. 4 to infer an eigenvalue uncertainty. That is, taking the deviations between calculation and fits as a gauge, we estimate that eigenvalues at the extrema in this figure, on average, are accurate to about 4 (5) decimal places for the full (truncated) calculation.
We conclude this discussion by noting that we explored changing the range of the couplings spanning the vanishing mass gap. We also explored increasing the number of points calculated for the fitting and interpolation process. This led us to estimate that our uncertainty in the deduced critical coupling at each fixed K and for each of the three states studied could conservatively be assigned a value of unity (two units) in the sixth (fourth) decimal place for K ≤ 42 (K > 42). We note that, for K > 42 this error estimate is the same as the systematic shift in the critical coupling shown to arise from truncation to 16boson configurations in Fig. 4. Hence, at this stage of the uncertainty analysis, our dominant uncertainty in deduced critical couplings is less than five units in the fourth decimal place.  Table 3: Extrapolations of λ c for the lowest state of the odd sector as a function of the degree of the polynomial adopted and as a function of the Max K included in the data set. The overall range is K = 12 − 88 for a maximum of 52 DLCQ points. The extrapolations of the 5 th degree polynomial exhibit the least sensitivity to the Max K over the range of Max K values displayed in the table.

Selection of polynomials for fitting
We first consider the DLCQ results for f 1 (z) shown in Fig. 1 and represented by Eq. 12. The choice of variable z = 1/ √ K emerged after attempts with z to a variable power revealed that a robust low-degree polynomial fit could be obtained with a polynomial in z itself. The meaning of "robust" will become more apparent shortly.
Among several methods of partitioning our DLCQ results for fitting and extrapolating, we present in Table 3 an example of extrapolations to the continuum limit (K → ∞ or z → 0) using a sequence of polynomial degrees. In particular, we extrapolate the critical coupling for the lowest eigenstate of the odd sector to the continuum limit using a range of polynomial degrees (column labels) and a set of cutoffs in K   Table 4: Extrapolations of λ c to the continuum limit (K → ∞) from 5 th degree polynomial fits (f 1 of Eq. 12) using ranges of DLCQ results specified by the upper value (Max K) and the lower value (Min K) of the range. The last row presents the result when all 52 DLCQ results over the range 12 ≤ K ≤ 88 are employed.
By sliding the upper cutoff down from Max K = 88 we find comparable extrapolations to those shown already in Table 3  We estimate that the extrapolations using Max K ≥ 64 and Min K ≤ 36, i.e. those below the double line in Table 4, indicate that the fifth degree polynomial is a robust choice. The mean value of these 13 extrapolations (using the entry on the bottom row just once) is 22.618 with a maximum deviation ≈ 0.072.
We will observe that this is close to our final result.
Based on these studies we see that the systematic uncertainties from the choice of the degree of the polynomial (Table 3) and to the choice of the range of data included in the 5 th degree polynomial fit (Table 4) dominate our overall uncertainties compared to those estimated above. To be conservative, we quote our critical coupling and its associated uncertainty so as to encompass the results in the last row of Table 3 as well as all those in Table 4 below the double line. This provides our final result for the critical coupling: 22.64 ± 0.17. By electing the single overall uncertainty to encompass these selected results in Tables 3 and 4, we estimate that our smaller uncertainties, discussed above, are well covered.

Residual finite K artifacts
Having obtained the functions representing the DLCQ results, we turn our attention to discussing the residuals, the differences between fit functions and DLCQ calculations and ratios of DLCQ calculations.
We present the residuals for the functions f 1 , f 2 and f 3 in In light of these relationships of f 2 and f 3 to f 1 , our DLCQ results for f 1 extend further towards the continuum limit than our results for the other two functions. Hence, we focus on the residuals for f 1 with the understanding that their narratives, with appropriate changes in their arguments, relate strongly to the narrative for f 1 .
Viewing the residuals for f 1 in Fig Since DLCQ results for f 1 with a rescaled argument are employed in the determination of ratios of DLCQ  Figure 6: Residuals, plotted as functions of (K/2, K/3) for (R 2/1 , R 3/1 ) respectively, between polynomial fits and calculated ratios of mass-squared eigenvalues. The residuals are scaled, as indicated by the scaling factors in the legend. The ratio R 2/1 (R 3/1 ) is defined in the text by Eq. 13 (Eq. 16) and displayed in Fig. 2  (Fig. 3). Filled (open) symbols correspond to DLCQ results employed (not employed) in the fits. Connecting straight line segments are included to help guide the eye. The horizontal axis X is defined for each function according to the points from f 1 employed in its evaluation (see text for details).
are correlated resulting in reduced f 3 residuals extending to the limits of our current calculations.
To be complete, we also present residuals for R 2/1 and R 3/1 in Fig. 6 scaled by the large factors indicated in the legend. We observe that the patterns of the larger residuals for R 2/1 and R 3/1 follow the larger residuals for f 2 and f 3 respectively in Fig. 5. We can trace this approximate repeating pattern back to the results shown for f 1 in Fig. 5. In particular, we observe that the residuals for f 1 are about an order of magnitude smaller for K ≥ 42 than those for f 2 and f 3 in their corresponding domains of K ≥ 24 and K ≥ 36 respectively (note the scale factors in the legends). Hence, the residuals for R 2/1 and R 3/1 are dominated by the numerical deviations of calculated points for f 2 and f 3 from their polynomial fits.

Summary and Outlook
The main result of the present work is that DLCQ is capable of producing the critical coupling for the vanishing mass gap in two-dimensional φ 4 theory, with accuracy competitive with other methods. This is very important for several reasons. The earlier work on this problem [11] was done in an era when both DLCQ and supercomputing were in their infancy. There were many lingering doubts about the importance of zero longitudinal momentum modes and the reliability of DLCQ which is conceptually simple and employs a uniform grid in momentum space. When the exactly zero longitudinal momentum mode is dropped in this theory, a question is raised about the potential role it could have played near the critical coupling which is associated with the onset of boson condensation. Our results show that working with Periodic Boundary Condition (PBC) and ignoring the zero mode appears to be a valuable and reliable approach and produces results consistent with an alternative light-cone approach that uses basis functions that span the entire region of momentum fraction from 0 to 1 in the continuum [14].
Nevertheless, a constrained longitudinal zero mode does exist when PBCs are used. Hence, in spite of the excellent agreement with another light front method which does not employ DLCQ, one may still ask how its inclusion will affect the present result for the critical coupling. As we already mentioned in Sec. II, incorporating the zero mode in DLCQ is a nontrivial problem and to the best of our knowledge, this problem has not yet been solved in a satisfactory manner. A proposal on how to address this issue is presented, for example, in Ref. [44]. Further in-depth investigations are needed to settle this issue quantitatively.
Since the critical coupling in this theory is associated with a second order phase transition from the symmetric phase to the spontaneously broken symmetry phase, one crucial issue is whether DLCQ can detect the clear signal of spontaneous symmetry breaking (SSB). In the symmetric phase the eigenstates of the Hamiltonian share the symmetry of the Hamiltonian which is invariant under φ → −φ. But in the symmetry broken phase, while the Hamiltonian retains the symmetry, one can find eigenstates that do not share this symmetry. As we show here, DLCQ predicts, to within numerical precision, the degeneracy of the lowest state of the odd sector and the lowest state of the even sector in the continuum limit at the vanishing mass gap -a key feature of spontaneous symmetry breaking. By the linear superposition of these two states, it now becomes possible to construct an eigenstate of the Hamiltonian which does not share the symmetry of the Hamiltonian -a clear signature of SSB.
Spectrum degeneracy between odd and even sectors is a feature of the entire SSB phase and hence is manifested also for couplings above the critical coupling. However, the critical point itself has other characteristic features. A continuous spectrum is a key feature of the critical point. Thus the spacing between eigenvalues should also vanish [14]. In the critical region governed by strong interactions, it becomes numerically more challenging to access the masses of the excited states. But, using DLCQ, we are able to establish the vanishing mass gap also between the lowest two excitations in the odd sector in the continuum limit, a result previously achieved by Ref. [14]. We also observe that, away from the critical point, this degeneracy is lifted. We note that the critical coupling of Ref. [13] employing a polynomial basis, was obtained after a truncation in the maximum of bosons allowed in the configurations retained (7 for the odd and 8 for the even sectors respectively). In addition they also implemented independent tuning of resolutions in each Fock sector. We performed a set of DLCQ calculations with an 8-boson truncation for f 1 and followed the same fitting and extrapolation procedures presented above. With our 8-boson truncation we obtain a critical coupling of 23.85 ± 0.17. Thus, it appears that the difference in results for the critical coupling between Ref. [13] and DLCQ is not due to an 8-boson truncation in DLCQ alone.
There are many open questions which require further investigation. We discuss a few salient examples.
The results presented in this work are made possible with advances both in code development (software) and access to state of the art supercomputers (hardware). By additional investments in code development and by securing additional computational resources, can we further refine our prediction for the critical coupling?
Can one demonstrate the vanishing of the wave function renormalization constant at the critical point [45,46] in the continuum limit? Can one compute the critical exponents? They have the virtue that they are universal, independent of the regularization and the renormalization schemes. On the other hand except for the case of the mass gap, they are notoriously difficult to calculate.
How would results change if we adopt Anti-Periodic Boundary Conditions (APBC) where both the conceptual and technical problems associated with the constrained zero mode operator are absent? Unlike the SSB phase, in the symmetric phase we expect the spectrum to be independent of the boundary conditions as the continuum limit is approached.
Can we reveal the detailed nature of the phase transition by proceeding to stronger coupling with PBC?
From our finite but large K results we have observed that in the symmetric phase there is no level mixing in the low-lying spectrum. That is, invariant masses of the lowest two states in the odd sector and of the lowest state in the even sector go to zero at the critical coupling without crossing each other. This is in agreement with the result of Ref. [14]. This implies that for couplings less than the critical coupling, the lowest state is predominantly a one-boson configuration with its parton distribution exhibiting a peak at momentum fraction x = 1.
Our initial investigation beyond the critical coupling has revealed multiple level crossings at stronger couplings. As the coupling steadily increases, the dominant peak of the parton distribution of the lowest excitation shifts to lower x values indicating the dominance of multi-parton configurations. Thus further investigations beyond the critical coupling could reveal the detailed map of how a condensate of bosons steadily builds up. We plan to study this phenomena in detail and extrapolate to the continuum if possible, despite the numerical challenges one faces at strong coupling. In other words, can we enhance the details of kink formation with the higher resolution (higher K) capabilities achievable with modern supercomputers?
Can we return to the broken phase, which we have investigated previously with both the PBC and APBC, extract the critical coupling with enhanced precision and relate the critical couplings of the symmetric and the broken phases? The weak/strong duality of this theory was discussed long ago by S.-J. Chang [47] and is studied recently in the equal-time Hamiltonian Fock space method in Ref. [23].
Can we make the connection with the results from equal-time calculations more precise following the insights provided by Refs. [13,31]? In this connection, a fundamental assumption is the equivalence between Light Front and Instant form field theories, which has been studied for a long time. Of primary concern is the fate of vacuum [48,49] and self energy (tadpole) contributions. Here one analyzes Feynman diagrams expressed in light-cone variables in order to clarify the potential importance of the zero longitudinal momentum mode. For more recent works see Refs. [50][51][52]. Collins [53] has given a pedagogical treatment of the subject where a cure is also provided for circumventing the "zero mode" problem and for obtaining correct results in perturbation theory. For an application of this proposal within DLCQ see Ref. [54].
Coming back to the matching of results from the Instant form and the Front form theories, the problem arises mainly from the difference in mass corrections. This leads to a difference in the effective dimensionless coupling λ/µ 2 . How to resolve this issue non-perturbatively is presented in Ref. [13]. A more ambitious program is presented in Ref. [31]. These corrections are addressable within DLCQ [55] which require additional calculations as defined in Ref. [31]. A detailed investigation of this matching using DLCQ requires a separate future effort.