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Free energy of the self-interacting relativistic lattice Bose gas at finite density
High Energy Physics - Lattice · Submitted 24 Oct 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Using fits inspired by the approximate validity of mean-field theory, which is confirmed by our simulations, we extract the critical chemical potential as the non-analyticity point in the overlap free energy, obtaining a value that is in agreement with other determinations.
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The abstract, sentence by sentence
The density of state approach has recently been proposed as a potential route to circumvent the sign problem in systems at finite density. In this study, using the Linear Logarithmic Relaxation (LLR) algorithm, we extract the generalised density of states, which is defined in terms of the imaginary part of the action, for the self-interacting relativistic lattice Bose gas at finite density. After discussing the implementation and testing the reliability of our approach, we focus on the determination of the free energy difference between the full system and its phase-quenched counterpart. Using a set of lattices ranging from $4^4$ to $16^4$ , we show that in the low density phase, this overlap free energy can be reliably extrapolated to the thermodynamic limit. The numerical precision we obtain with the LLR method allows us to determine with sufficient accuracy the expectation value of the phase factor, which is used in the calculation of the overlap free energy, down to values of ${\cal O}(10^{-480})$. When phase factor measurements are extended to the dense phase, a change of behaviour of the overlap free energy is clearly visible as the chemical potential crosses a critical value. Using fits inspired by the approximate validity of mean-field theory, which is confirmed by our simulations, we extract the critical chemical potential as the non-analyticity point in the overlap free energy, obtaining a value that is in agreement with other determinations. Implications of our findings and potential improvements of our methodology are also discussed.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The density of state approach has recently been proposed as a potential route to circumvent the sign problem in systems at finite density. | background 1.00 | 0.04 | 0.02 | hedged possibility 0.93 |
| s1 | In this study, using the Linear Logarithmic Relaxation (LLR) algorithm, we extract the generalised density of states, which is defined in terms of the imaginary part of the action, for the self-interacting relativistic lattice Bose gas at finite density. | contribution 0.88 | 0.97 | 0.05 | asserted 1.00 |
| s2 | After discussing the implementation and testing the reliability of our approach, we focus on the determination of the free energy difference between the full system and its phase-quenched counterpart. | contribution 0.91 | 0.86 | 0.06 | asserted 0.92 |
| s3 | Using a set of lattices ranging from $4^4$ to $16^4$ , we show that in the low density phase, this overlap free energy can be reliably extrapolated to the thermodynamic limit. | result 0.97 | 0.96 | 0.18 | asserted 1.00 |
| s4 | The numerical precision we obtain with the LLR method allows us to determine with sufficient accuracy the expectation value of the phase factor, which is used in the calculation of the overlap free energy, down to values of ${\cal O}(10^{-480})$. | result 0.96 | 0.97 | 0.10 | asserted 1.00 |
| s5 | When phase factor measurements are extended to the dense phase, a change of behaviour of the overlap free energy is clearly visible as the chemical potential crosses a critical value. | result 1.00 | 0.94 | 0.17 | asserted 1.00 |
| s6 | Using fits inspired by the approximate validity of mean-field theory, which is confirmed by our simulations, we extract the critical chemical potential as the non-analyticity point in the overlap free energy, obtaining a value that is in agreement with other determinations. | result 0.94 | 0.96 | 0.34 | asserted 0.99 |
| s7 | Implications of our findings and potential improvements of our methodology are also discussed. | future work 0.89 | 0.46 | 0.03 | unclear 0.72 |
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The paper
Original abstract
The density of state approach has recently been proposed as a potential route to circumvent the sign problem in systems at finite density. In this study, using the Linear Logarithmic Relaxation (LLR) algorithm, we extract the generalised density of states, which is defined in terms of the imaginary part of the action, for the self-interacting relativistic lattice Bose gas at finite density. After discussing the implementation and testing the reliability of our approach, we focus on the determination of the free energy difference between the full system and its phase-quenched counterpart. Using a set of lattices ranging from $4^4$ to $16^4$ , we show that in the low density phase, this overlap free energy can be reliably extrapolated to the thermodynamic limit. The numerical precision we obtain with the LLR method allows us to determine with sufficient accuracy the expectation value of the phase factor, which is used in the calculation of the overlap free energy, down to values of ${\cal O}(10^{-480})$. When phase factor measurements are extended to the dense phase, a change of behaviour of the overlap free energy is clearly visible as the chemical potential crosses a critical value. Using fits inspired by the approximate validity of mean-field theory, which is confirmed by our simulations, we extract the critical chemical potential as the non-analyticity point in the overlap free energy, obtaining a value that is in agreement with other determinations. Implications of our findings and potential improvements of our methodology are also discussed.
Human review
How the literature uses it
9 arXiv papers cite this in 14 sentences
Papers citing it · 8 sentences without a confident label
Uses its method or tool 3 papers
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The LLR approach [ref] – [ref] computes the slopes $k_n^(J)$ by finding the zeros of the lhs. of Eq. ([ref]) using an iterative process, while the FFA method [ref] – [ref] combines simulations at different values of $λ$ and determines the $k_n^(J)$ from one-parameter fits of the data with the rhs. of ([ref]).
Density of states approach for lattice gauge theory with a $\theta$-term · Determination of the densities with modern DoS techniques · 0.92
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Here and throughout this proceedings, we reconstruct the probability density $P_β(E)$ using both a naive trapezium-rule integration and a more robust polynomial fit technique [ref] .
Advances in using density of states for large-N Yang--Mills · SU(4) · 0.98
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We have compared two ways to reconstruct the normalizable probability density $P_β(u_P) = ρ(u_P) e^-6Vβu_P$ from our LLR results for $a(u_P)$: Simple trapezium-rule numerical integration and a polynomial fit technique first used in [ref] .
First-order bulk transitions in large-$N$ lattice Yang--Mills theories using the density of states · Bulk transition results · 0.99
Cites it as background 3 papers
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In the past decade, the Density of States (DoS) approach has been advocated as an efficient way to make the sign problem milder [ref] .
Real Time Simulations of Quantum Spin Chains: Density-of-States and Reweighting approaches · Introduction · 0.93
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DoS techniques were initially introduced to lattice field theory in [ref] , but saw a major revival based on a paper by Langfeld, Lucini and Rago [ref] , where a modern formulation with considerably improved accuracy was presented (see also [ref] – [ref] ).
Density of states techniques for fermion worldlines · Introductory comments · 0.90
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Inspired by the Wang-Landau [ref] approach an interesting new development was presented by Langfeld, Lucini and Rago [ref] .
New DoS approaches to finite density lattice QCD · Introduction · 0.85
What it cites · 15 sentences without a confident label
Uses the method or tool of 4 papers
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The density of states will be determined using the Linear Logarithmic Relaxation (LLR) algorithm [ref] , which has been shown to provide exponential error reduction in a range of Lattice Gauge Theories applications involving e.g., tunnelling suppression at a first order phase transition [ref] , the determination of the free energy for a system with shifted boundary conditions [ref] and the de-correlation of the topological charge near the continuum limit [ref] .
The density of states in gauge theories · Introduction · 0.90
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The density of states will be determined using the Linear Logarithmic Relaxation (LLR) algorithm [ref] , which has been shown to provide exponential error reduction in a range of Lattice Gauge Theories applications involving e.g., tunnelling suppression at a first order phase transition [ref] , the determination of the free energy for a system with shifted boundary conditions [ref] and the de-correlation of the topological charge near the continuum limit [ref] .
An efficient algorithm for numerical computations of continuous densities of states · Introduction · 0.90
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The density of states will be determined using the Linear Logarithmic Relaxation (LLR) algorithm [ref] , which has been shown to provide exponential error reduction in a range of Lattice Gauge Theories applications involving e.g., tunnelling suppression at a first order phase transition [ref] , the determination of the free energy for a system with shifted boundary conditions [ref] and the de-correlation of the topological charge near the continuum limit [ref] .
Ergodicity of the LLR method for the Density of States · Introduction · 0.90
Compares with 2 papers
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However, our calculation shows that our results are compatible with the mean-field calculations [ref] ($μ_c ≃1.15$) as well as with the value obtained in [ref] ($μ_c = 1.146 ±0.001$) with a dual formulation of the same theory in a work more focused to the study of the phase transition than the present one.
Complex Langevin dynamics at finite chemical potential: mean field analysis in the relativistic Bose gas · Phase structure away from criticality · 1.00
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However, our calculation shows that our results are compatible with the mean-field calculations [ref] ($μ_c ≃1.15$) as well as with the value obtained in [ref] ($μ_c = 1.146 ±0.001$) with a dual formulation of the same theory in a work more focused to the study of the phase transition than the present one.
Lattice study of the Silver Blaze phenomenon for a charged scalar phi-4 field · Phase structure away from criticality · 1.00
Cites as background 18 papers
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This model has been extensively studied in the context of the sign problem of lattice field theories via Complex Langevin dynamics and mean-field approximation [ref] as well as complete dualization [ref] .
Complex Langevin dynamics at finite chemical potential: mean field analysis in the relativistic Bose gas · Model: Relativistic Bose Gas · 0.99
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This model has been extensively studied in the context of the sign problem of lattice field theories via Complex Langevin dynamics and mean-field approximation [ref] as well as complete dualization [ref] .
Lattice study of the Silver Blaze phenomenon for a charged scalar phi-4 field · Model: Relativistic Bose Gas · 0.99
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A recent review of this approach can be found in [ref] .] Recent examples include the Complex Langevin approach [ref] , thimble regularisation [ref] and the density of states route, which, following the original proposal of [ref] and further refinements [ref] , has been recently revisited in [ref] .
Can stochastic quantization evade the sign problem? -- the relativistic Bose gas at finite chemical potential · Introduction · 0.95
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