Thermodynamics of Gauge Theory from Gradient Flow
A novel method to study the bulk thermodynamics in lattice gauge theory is proposed on the basis of the Yang–Mills gradient flow with a fictitious time . The energy density and the pressure of gauge theory at fixed temperature are calculated directly on lattices from the thermal average of the well-defined energy-momentum tensor obtained by the gradient flow. It is demonstrated that the continuum limit can be taken in a controlled manner from the -dependence of the flowed data.
pacs:05.70.Ce; 11.10.Wx; 11.15.Ha
The symmetric energy momentum tensor (EMT), , which is the generator of the Poincaré transformations, is a fundamental operator in quantum field theory Caracciolo:1990emt (). Since , , and correspond to the energy density, the momentum density, and the momentum-flux density, respectively, the EMT and its correlation functions provide useful information on the bulk and transport properties at finite temperature (). For example, the energy density and the pressure are given by and , respectively, with being the thermal average. Also, the shear viscosity can be extracted from the two-point correlation, . In quantum chromodynamics (QCD), these observables are particularly important in formulating the relativistic hydrodynamics for the quark-gluon plasma Romatschke:2009im (). Therefore, high precision and non-perturbative evaluation of the -point EMT correlations in lattice QCD is called for.
To calculate such correlations in numerical lattice simulations, we first need to define proper EMT on the lattice which is ultra-violet (UV) finite and is conserved in the continuum limit. Such a construction is not a trivial task due to the explicit breaking of the Poincaré invariance on the lattice. (See Refs. Giusti:2012yj (); Robaina:2013zmb (); Giusti:2013sqa (); Giusti:2013mxa () for recent developments.) This is the reason why and at finite have been mainly studied by an indirect “integral method” without the explicit use of the EMT Lombardo:2012ix ().
Recently, one of the present authors has shown that the proper EMT keeping all the nice features can be naturally constructed Suzuki:2013gza () on the basis of the Yang–Mills gradient flow Luscher:2010iy (); Luscher:2011bx (); Luscher:2013vga (). (See also related works, Refs. Borsanyi:2012zs (); Borsanyi:2012zr (); Fodor:2012td (); Fritzsch:2013je (); Luscher:2013cpa ().) In this Letter, we demonstrate, for the first time, that the thermal gauge theory can be studied by the direct lattice measurement of the proper EMT by considering and as examples. The key idea is to represent the EMT in the continuum limit by UV-finite and local operators obtained from the gradient flow. Then, by taking the limit of small flow time and small lattice spacing in an appropriate way, as discussed later, accurate thermodynamic observables are obtained with modest statistics.
Let us first recapitulate the basic idea of Ref. Suzuki:2013gza () in the continuum space-time. The Yang–Mills gradient flow is a deformation of the gauge configuration along a fictitious Euclidean time ; with , where and are the covariant derivative and the field strength of the flowed gauge field , respectively. The color indices are suppressed for simplicity. A salient feature of the gradient flow is its UV finiteness: Any correlation functions of , , … for are UV finite without the wave function renormalization if they are written in terms of the renormalized coupling Luscher:2011bx (). This is owing to the fact that the diffusion in naturally introduces a proper-time regulator of the form , where denotes a typical loop momentum. In particular, the correlation functions are free from UV divergences even at the equal-point, for positive . For example, the following gauge-invariant local products of dimension are UV finite for : and .
For , local products of flowed fields can be expanded in terms of four-dimensional renormalized local operators with increasing dimensions Luscher:2011bx (): The expansion coefficients are governed by the renormalization group equation and their small behavior can be calculated by perturbation theory thanks to the asymptotic freedom. For the operators mentioned above, we have Suzuki:2013gza (); DelDebbio:2013zaa ()
where is vacuum expectation value and is the correctly-normalized conserved EMT with its vacuum expectation value subtracted. Abbreviated are the contributions from the operators of dimension or higher, which are suppressed for small .
where the perturbative coefficients are found to be Suzuki:2013gza ()
Here denotes the running gauge coupling in the scheme with the choice, , and , , with , , and . Note that a non-perturbative determination of is also proposed recently DelDebbio:2013zaa ().
The formula Eq. (3) indicates that can be obtained by the small limit of the gauge-invariant local operators defined through the gradient flow. There are two important observations: (i) The right-hand side of Eq. (3) is independent of the regularization because of its UV finiteness, so that one can take, e.g. the lattice regularization scheme; (ii) since flowed fields at depend on the fundamental fields at in the space-time region of radius , the statistical noise in calculating the right hand side of Eq. (3) is suppressed for finite .
Our procedure to calculate the EMT on the lattice has the
following four steps:
Step 1: Generate gauge configurations at on a space-time lattice with the lattice spacing and the lattice size .
Step 2: Solve the gradient flow for each configuration to obtain the flowed link variables in the fiducial window, . Here, is an infrared cutoff scale such as or . The first (second) inequality is necessary to suppress finite corrections (non-perturbative corrections and finite volume corrections).
Step 3: Construct and in Eqs. (1) and (2) in terms of the flowed link variables and average over the gauge configurations at each .
Step 4: Carry out an extrapolation toward , first and then under the condition in Step 2.
The thermodynamic quantities are obtained from the diagonal elements of the EMT: A combination of and called the interaction measure is related to the trace of the EMT (the trace anomaly):
Also, the entropy density at zero chemical potential reads
To demonstrate that the above four Steps can be indeed pursued, we consider the gauge theory defined on a four-dimensional Euclidean lattice, whose thermodynamics has been extensively studied by the integral method Boyd:1996bx (); Okamoto:1999hi (); Umeda:2008bd (); Borsanyi:2012ve (). For simplicity, we consider the Wilson plaquette gauge action under the periodic boundary condition on lattices with several different ( being the bare coupling constant). Gauge configurations are generated by the pseudo-heatbath algorithm with the over-relaxation, mixed in the ratio of . We call one pseudo-heatbath update sweep plus five over-relaxation sweeps as a “Sweep”. To eliminate the autocorrelation, we take – Sweeps between measurements. The number of gauge configurations for the measurements at finite is . Statistical errors are estimated by the jackknife method.
To relate and corresponding for each , we first use the relation between ( is the Sommer scale) and given by the ALPHA Collaboration Guagnelli:1998ud (). The resultant values of are then converted to by using the result at in Ref. Boyd:1996bx (). Nine combinations of and corresponding obtained by this procedure are shown in Table 1.
The gradient flow in the -direction is obtained by solving the ordinary first-order differential equation. We utilize the modified second-order Runge–Kutta method in which the error per step () is . We take , and confirm that the accumulation errors are sufficiently smaller than the statistical errors.
To extract the EMT from Eq. (3), we measure written in terms of the clover leaf representation on the lattice. To subtract out the contribution, , we carry out simulations on a lattice for each in Table 1. Note that this vacuum subtraction is required for the trace anomaly , but not for the entropy density . For in and in Eqs. (4) and (5), we use the four-loop running coupling with the scale parameter determined by the ALPHA Collaboration, Capitani:1998mq (). We confirmed the previous finding Luscher:2010iy () that the lattice data of in the fiducial window matches quite well with its perturbative estimate in the continuum, with the four-loop running coupling and the above .
Shown in Fig. 1 is our results for the dimensionless interaction measure () and the dimensionless entropy density () at as a function of the dimensionless flow parameter . The bold bars denote the statistical errors, while the thin (light color) bars show the statistical and systematic errors including the uncertainty of . In the small region, the statistical error is dominant for both and , while in the large region the systematic error from becomes significant for . For instance, the statistical (systematic) errors of the data for are () for and () for at .
The fiducial window discussed in Step 2 is indicated by the dashed lines in Fig. 1. The lower limit, beyond which the lattice discretization error grows, is set to be , where we consider the size of our clover leaf operator. The upper limit, beyond which the smearing by the gradient flow exceeds the temporal lattice size, is set to be .
The data in Fig. 1 show, within the error bars, that (i) the plateau appears inside the preset fiducial window () for each , and (ii) the plateau extends to the smaller region as increases or equivalently as decreases. Similar plateaus as in Fig. 1 also appear inside the fiducial window for other temperatures, and , with comparable error bars. These features imply that the double extrapolation in Step 4 is indeed doable.
Our lattice results at fixed with three different lattice spacings allow us to take the continuum limit. First, we pick up a flow time which is in the middle of the fiducial window. Then we extract and for each set of and . We have checked that different choices of do not change the final results within the error bar as long as it is in the plateau region. In Fig. 2, resultant values taking into account the statistical errors (bold error bars) and the statistical plus systematic errors (thin error bars) are shown. The lattice data for with the same lattice setup at and in Ref. Boyd:1996bx () are also shown by the cross (green) symbols in the top panel; the statistical error of our result on lattice for () is about () times smaller than the one in Ref. Boyd:1996bx () obtained on the same lattice. In this way, our results with gauge configurations have substantially smaller error bars at these points.
The horizontal axis of Fig. 2, , is a variable suited for making continuum extrapolation of the thermodynamic quantities Boyd:1996bx (). We consider two extrapolation: A linear fit with the data at , , and (the solid lines in Fig. 2), and a constant fit with the data at and (the dashed lines in Fig. 2). In both fits, the correlation between the errors due to the common systematic error from is taken into account. The former fit is used to determine the central value in the continuum limit whose error is within even at our lowest temperature. The latter is used to estimate the systematic error from the scaling violation whose typical size is at high temperature and at low temperature.
We have analyzed various systematic errors; the perturbative expansion of , the running coupling , the scale parameter, and the continuum extrapolation. We found that the dominant errors in the present lattice setup are those from and the continuum extrapolation, which are included in Fig. 2. To reduce these systematic errors, finer lattices are quite helpful: They make the plateau in wider by reducing the lower limit of the fiducial window, so that the continuum extrapolation becomes easier. We also note that our continuum extrapolation with fixed would receive the finite volume effect especially for lower Borsanyi:2012zs (). Larger aspect ratio would be helpful to guarantee the thermodynamic limit. Moreover, the scale setting procedure could be improved to have better accuracy: Instead of the Sommer scale adopted in this Letter, more precise scale determination, e.g. by or in the gradient flow approach Luscher:2010iy (); Fodor:2012td (), will be useful.
Finally, we plot, in Fig. 3, the continuum limit of and obtained by the linear fit of the , , and data (the solid lines) in Fig. 2 for , , and . For comparison, the results of Ref. Boyd:1996bx (); Okamoto:1999hi (); Borsanyi:2012ve () obtained by the integral method are shown by the magenta, green, and blue data in Fig. 3. The results of the two different approaches are consistent with each other within the statistical error.
In this Letter, we have proposed and demonstrated a novel way to study thermal gauge theory on the lattice. The key ingredient is the conserved and UV-finite energy-momentum tensor defined from the the UV-finite operators ( and ) obtained from the Yang–Mills gradient flow with the matching coefficients () Suzuki:2013gza (). From the simulations on lattices with modest statistics ( gauge configurations), we found that the dimensionless interaction measure and entropy density, and , show plateau structure inside the fiducial window () with small statistical errors, so that the double extrapolation can be taken appropriately for given .
Major advantages of the gradient flow applied to the lattice thermodynamics are as follows: (i) One can simulate and independently at any fixed through the direct measurement of the well-defined EMT. There is no need of integration by or , which requires a boundary condition and the numerical interpolation. (ii) There is no need of constant subtraction in entropy density . The interaction measure needs one subtraction of its value, which is obtained by the accurate measurement of or by its perturbative evaluation at small . (iii) The statistical noise is substantially reduced at finite flow time due to the effective smearing of the operators with the radius , so that the extrapolation of the results back to is well under control.
Although we studied only the thermal average of EMT in this Letter, there is no conceptual difficulties in applying our method to -point EMT correlations Suzuki:2013gza (). This opens the door to investigate transport coefficients (such as shear and bulk viscosities), fluctuation observables in the hot plasma, glueballs at zero and finite temperatures. Here we note that there is no difficulty in measuring thermodynamic quantities even at extremely high temperature in this method since no temperature integration is necessary. It is also an interesting direction to study the dilation mode or the -function of (nearly) conformal theory Appelquist:2010gy (); Latorre:1997ea () using the present method. Furthermore, including fermions in the present framework extends the scope even further Makino-Suzuki (). Some of these issues as well as the simulations with finer lattice with larger volume are already started and will be reported elsewhere.
We would like to thank S. Aoki, F. Karsch, M. Lüscher, and H. Nagatani for useful discussions and comments. We are also grateful to H. Matsufuru for his help of the code development. Numerical simulation was carried out on NEC SX-8R and SX-9 at RCNP, Osaka University, and Hitachi SR16000 at KEK under its Large-Scale Simulation Program (Nos. T12-04 and 13/14-20). M. A., M. K., and H. S. are supported in part by a Grant-in-Aid for Scientific Researches 23540307, 25800148, and 23540330, respectively. E. I. is supported in part by Strategic Programs for Innovative Research (SPIRE) Field 5. T. H. is supported by RIKEN iTHES Project.
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