Finite density condensation and scattering data - a study in $\phi^4$ lattice field theory

We study the quantum field theory of a charged $\phi^4$ field in lattice regularization at finite density and low temperature in 2 and 4 dimensions with the goal of analyzing the connection of condensation phenomena to scattering data in a non-perturbative way. The sign problem of the theory at non-zero chemical potential $\mu$ is overcome by using a worldline representation for the Monte Carlo simulation. At low temperature we study the particle number as a function of $\mu$ and observe the steps for $\mbox{1-,}$ 2- and 3-particle condensation. We determine the corresponding critical values $\mu_n^{crit}, \, n = 1,2,3$ and analyze their dependence on the spatial extent $L$ of the lattice. Linear combinations of the $\mu_n^{crit}$ give the interaction energies in the 2- and 3-particle sectors and their dependence on $L$ is related to scattering data by L\"uscher's formula and its generalizations to three particles. For 2-$d$ we determine the scattering phase shift and for 4-$d$ the scattering length. We cross-check our results with a determination of the mass and the 2- and 3-particle energies from conventional 2-, 4-, and 6-point correlators at zero chemical potential. The letter demonstrates that the physics of condensation at finite density and low temperature is closely related to scattering data of a quantum field theory.


INTRODUCTION
It is well known that phenomena in low energy physics can be described in terms of a few low energy parameters, which in the context of this letter was, e.g., discussed in the seminal paper [1]. As a particular instance of this relation the condensation of particles at low temperature and non-zero chemical potential can be related to scattering data of the corresponding quantum field theory. More specifically one may show [2] that the critical values of the chemical potential where one observes condensation steps at low temperature and small volume are related to the finite volume many particle energies, which in turn are related to the scattering length [1].
For studying low energy properties non-perturbative methods need to be applied, e.g., lattice simulations. However, for many finite density lattice field theories the action is complex. The Boltzmann factor has a complex phase and cannot be used as a probability in a lattice Monte Carlo study. This "sign problem" has recently been overcome for several theories by exactly mapping them to a worldline representation with only real and positive weights (see [3][4][5][6][7] for the model considered here).
In this letter we study the complex φ 4 field at finite chemical potential using a worldline representation. We compute the particle number as a function of the chemical potential at low temperature and determine the condensation steps for the 1-, 2-and 3-particle sectors. Analyzing their volume dependence we show nonperturbatively that the condensation steps are indeed related to the scattering data of the theory.

WORLDLINE REPRESENTATION
The dynamical degrees of freedom of the charged φ 4 field are the complex valued fields φ x assigned to the sites x of a d-dimensional lattice with periodic boundary conditions. Here we consider d = 2 and d = 4, i.e., we work on lattices with volumes V = N s ×N t and V = N 3 s × N t , where N s is the spatial extent in lattice units. N t is the extent in Euclidean time direction (the d direction), which also equals the inverse temperature β in lattice units. The grand canonical partition sum is given by the The lattice action is given by The bare mass m b enters via the parameter η ≡ 2d + m 2 b , λ denotes the coupling of the quartic self-interaction and µ is the chemical potential.
For nonzero µ the action (1) is complex and the model has a sign problem in the conventional formulation. The sign problem can be solved by exactly mapping the theory to a worldline representation where also at finite µ all weights are real and positive, such that a simulation is possible directly in terms of the worldlines [3][4][5][6][7]. In the worldline form the partition sum is given by The sum is over all configurations of the worldline variables k x,ν ∈ Z assigned to the links of the lattice. The worldline variables k x,ν obey a zero divergence constraint, which we write as a product over all lattice sites and at each site x a Kronecker delta δ(j) ≡ δ j,0 enforces vanishing divergence ∇ · k x ≡ ν (k x,ν − k x−ν,ν ) = 0. Consequently the worldline variables k x,ν must form closed loops of conserved flux.

arXiv:1804.01580v2 [hep-lat] 23 May 2018
The chemical potential µ couples to the temporal winding number ω[k] of the conserved flux: By comparing the µ-dependent term in (2) to the standard form e µβN for the µ-dependence of the grand canonical partition sum, we conclude that the net-particle number N is given by the temporal net winding number ω[k] of the worldlines. Finally each configuration comes with a weight factor with The weight factor is itself a sum over configurations {a} of auxiliary link variables a x,ν ∈ N 0 . The integrals I(s x ) come from integrating out the radial degrees of freedom of the original field variables at site x. The argument s x is a non-negative integer combination of the auxiliary variables and the moduli of all k-fluxes that run through x, defined as s x = ν |k x,ν | + |k x−ν | + 2(a x,ν + a x−ν ) . For the numerical simulation the integrals I(s x ) are precalculated and stored for sufficiently many values of the arguments s x ∈ N 0 . In a Monte Carlo simulation of the worldline form the variables k x,ν and the auxiliary variables a x,ν need to be updated, such that the k x,ν obey all constraints. We apply a strategy based on the worm algorithm [8] with the details discussed in [9].
For the 4-d simulations we use lattices with N t = 320 and 640, and N s between 3 and 10 at coupling values of η = 7.44 and λ = 1.0. The statistics is 10 5 to 2 × 10 5 configurations. For 2-d we use N t = 400 and N s between 2 and 16 with η = 2.6, λ = 1.0 and a statistics of 4 × 10 5 .
We conclude this section with an estimate of discretization effects for the 4-d case. As we will see below, at the couplings we use the infinite volume mass m ∞ of the lowest excitation is m ∞ ∼ 0.168/a, where a is the lattice constant. For the momentum cutoff p c = π/a we thus find p cut /m ∞ = π/0.168 ∼ 18.9. The largest energy we use in our study is the 3-particle energy W 3 for N s = 4, which at W 3 (4) ∼ 0.93/a is less than 6 times m ∞ . We expect that with p c /W 3 (4) = π/0.93 ∼ 3.4 cutoff effects are small. A similarly good ratio holds for our 2-d study.

CONDENSATION THRESHOLDS AND MULTI-PARTICLE ENERGIES
Using the worldline representation summarized in the previous section we now analyze the particle number N as a function of the chemical potential µ. We have already discussed that in the worldline representation the particle number N is represented by the temporal winding number ω[k] of the worldlines, and thus we study the expectation value N ≡ ω[k] , where the vacuum expectation value on the rhs. of this equation is evaluated in the worldline representation. In Fig. 1 we show the expectation value of the particle number N as a function of the chemical potential µ for the 4-d case with N s = 6. We split the figure into 3 plots, choosing the ranges of µ such that we see the condensation thresholds for 1-, 2-and 3-particle condensation, i.e., the values µ crit n , n = 1, 2, 3 where N quickly climbs from N = n − 1 to N = n. One observes that N has shoulders at the integers 0, 1, 2, 3 and shows a rapid increase in between. In the zero temperature limit, i.e., for N t → ∞ this turns into steps, which here at finite N t are rounded due to temperature effects. Near the transition from N = n−1 to N = n we fit the data with a logistic function shifted by a constant, N = [1 + exp(−a n [µ−µ crit n ])] −1 +n−1. The plots show that this 2-parameter (a n and µ crit n ) fit function describes the data very well and allows us to determine the critical values µ crit n , n = 1, 2, 3 of the chemical potential. As a cross-check we determined the critical values µ crit n also from the peaks of the particle number susceptibility and found excellent agreement of the two determinations.
In [2] it was pointed out, that the critical values µ crit n , n = 1, 2, 3 at low temperature are related to the The physical mass m and the 2-and 3-particle energies W2 and W3 for the 4-d case as a function of the spatial extent Ns. We compare the results from the n-particle energy thresholds (squares) to the results from a direct determination with correlators (diamonds). The fits to m and W2 are shown as solid red curve, and the prediction for W3 based on these fits is shown as a dashed purple curve.
1-, 2-and 3-particle energies at finite volume, i.e., where m is the renormalized physical mass, W 2 the 2particle energy, and W 3 the 3-particle energy. When changing the spatial extent N s of the lattice one observes a shift of the condensation steps µ crit n and thus of m, W 2 and W 3 . In Fig. 2 we show the N s -dependence of these three quantities (squares), which in the next section we will use to determine scattering data.
Before we come to the analysis of the finite volume effects and their connection to scattering data we present an important cross-check of our worldline results. The mass m, as well as W 2 and W 3 can also be computed from the exponential decay of Euclidean 2n-point functions in the conventional representation. More specifically we consider the spatially Fourier transformed fields at zero momentum, φ t = (N s ) −3 x φ x,t , and compute the connected 2n-point functions for n = 1, 2, 3: where E 1 = m and E n = W n for n = 2, 3. From a fit to the correlators we determined the values for m, W 2 and W 3 , which in Fig. 2 are shown as diamonds. As a crosscheck we determined m, W 2 and W 3 also using a full correlation matrix and got values agreeing very well with those from the 2n-point functions (5), which indicates that contributions of excited states are negligible. Fig. 2 shows that the values for m, W 2 and W 3 as determined from the condensation steps agree very well with the values from the 2n-point functions. This establishes that the condensation steps observed for the particle number are indeed determined by the corresponding n-particle energies via the relations (4). The same comparison was done also for the 2-d case and again we found excellent agreement of the energies determined from the critical µ crit n and those from the 2n-point functions.

VOLUME DEPENDENCE AND SCATTERING
Having shown that the low temperature condensation steps are indeed governed by the n-particle energies we can now apply known finite volume relations to connect the condensation steps with scattering data.
We begin this analysis with the 2-d case following [10]. There the 2-particle energy W 2 is related to the relative momentum k of the particles via W 2 = 2 √ m 2 + k 2 , and we can invert this equation to determine the momentum k for each value of W 2 . On our finite lattice with spatial extent N s the momentum k is subject to the quantization condition e 2iδ(k) = e −ikNs , where δ(k) is the phase shift for that momentum. Combining this relation with the relation between W 2 and k, we can extract the scattering phase shift δ(k) from W 2 . Varying N s gives rise to different values of the relative momentum k such that δ(k) can be determined for a whole range of momenta.
In Fig. 3 we plot the results for δ(k) as a function of k and again compare the data determined from the nparticle energy thresholds (m = µ crit 1 , W 2 = µ crit 1 +µ crit 2 ) to the results obtained by determining m from 2-point functions and W 2 from 4-point functions. We find very good agreement of the two data sets and thus establish the relation of the condensation steps to the scattering phase shift for the 2-d case.
For the 4-d case we use the finite volume relations for m [11], the result [1,12] for the 2-particle energy W 2 (using the notation of [14]) and the results [13][14][15][16][17] for the 3-particle energy W 3 : Fit parameters are the infinite volume mass m ∞ , the amplitude A and the scattering length a. The numerical constants I and J are given by I = −8.914, J = 16.532. We identify L ≡ N s and fit the data for m(N s ) as determined from µ crit 1 in the range between N s = 4 and N s = 10 with the functional form (6). This fit gives the amplitude parameter A and the infinite volume mass m ∞ in lattice units. Subsequently we fit W 2 (N s ) determined from µ crit 1 + µ crit 2 with the functional form (7), again in the range between N s = 4 and N s = 10. We use m(N s ) from the previous step, such that this second fit is a 1parameter fit that gives the scattering length a in lattice units. From Fig. 2 it is obvious that the fits for m(N s ) and W 2 (N s ) describe the data very well and indeed the reduced χ 2 is close to 1 for both fits.
For W 3 (N s ) no additional free parameter is needed such that we simply can compare our data to the curve for W 3 that we obtain from (8) using the fit parameters of the previous fits as input. In Fig. 2 we observe good agreement of this "predicted" W 3 with the data from the condensation steps. Only for the smallest L (≡ N s ) we observe a deviation which shows that here higher order corrections in a/L start to play a role. However, it has to be remarked that cleanly determining the higher order power law corrections in W 2 and W 3 or exponential terms of the form e −L/R , where R is the range for an interaction with an exponential tail, is a non-trivial numerical challenge. Such terms contribute significantly only for very small extent L and for a serious determination one should resolve small L with several data points, which in turn implies working with very fine lattices and very high statistics.
For completeness we quote our results for the two physical parameters, i.e., the mass m ∞ and the scattering length a. In lattice units we obtain m ∞ = 0.168(1) and a = −0.078 (7) and a value of a m ∞ = −0.013(1) for their dimensionless product.

DISCUSSION AND CONCLUDING REMARKS
In this letter we determined the n-particle energy thresholds µ crit n , n = 1, 2, 3 in a worldline simulation of the charged φ 4 field at finite density. These thresholds correspond to the first three unit steps of the particle number expectation value N and emerge for small volumes and low temperatures. From the µ crit n we determined the mass m, as well as the 2-and 3-particle energies W 2 and W 3 using (4). We studied their dependence on the spatial extent N s and cross-checked the condensation results with those from 2n-point functions. We found very good agreement of the data for all values of N s we analyzed, thus demonstrating that the condensation thresholds are indeed governed by the lowest nparticle energies, which correspond to the mass and 2and 3-particle scattering states. This is the first time that this agreement is shown non-perturbatively in four dimensions, since the necessary worldline techniques for finite density lattice simulations became available only recently.
Subsequently we analyzed the volume dependence of W 2 and W 3 in order to make contact to scattering data. For the 2-d case such an analysis allows for a complete determination of the scattering phase δ(k) from W 2 . In four dimensions we fit the L-dependence of m and W 2 to determine the infinite volume mass m ∞ and the scattering length a. Inserting these parameters in the functional form for W 3 determines the 3-particle energy up to 1/L 6 corrections. We compared this "prediction" with our results for W 3 from the condensation data and found very convincing agreement. This demonstrates that the finite density condensation steps at low temperature are indeed governed by the scattering data of the theory.
We stress at this point that the scenario for the condensation which was exploited in our analysis requires a repulsive interaction. In case of an attractive interaction one expects condensation thresholds at the masses of the bound states, and a nice example of this scenario is documented in [18] for the model of QCD with the exceptional gauge group G 2 , which is free of sign problems.
It is well known that in four dimensions φ 4 theory is trivial. The renormalized coupling vanishes when approaching the continuum limit and one may expect that also the scattering length vanishes. Consequently our study in φ 4 theory deals with an effective theory which, however, has a large scaling region where it essentially behaves like a continuum theory at low energies, as has been demonstrated in [19]. In 2-d, on the other hand, one could try to run towards the Ising fixed point to construct a non-trivial continuum limit, where one expects a constant phase shift of δ(k) = −π/2 [20].
Having established the interesting connection between scattering data and thermodynamical properties at low temperature in a simple scalar theory it is of course in-teresting to ask if this can also been seen in more rich theories. The key issue is to be able to simulate the theory at finite density, which often is spoiled by the sign problem. However, two interesting cases can be addressed immediately, namely QCD with only two colors and QCD with an isospin chemical potential. In both cases the sign problem is absent, but the former case clearly is of more academic interest. However, using an isospin chemical potential one can condense pions in full QCD and study the relation of the second and third condensation thresholds to scattering data. This is a highly interesting connection that could be studied along the lines presented in this letter.