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Tachyonic Instability and Dynamics of Spontaneous Symmetry Breaking

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Gary Felder, Lev Kofman, Andrei Linde

High Energy Physics - Theory · Submitted 19 Jun 2001 · 37 references

Pith's verdict: The long version of the tachyonic preheating PRL: the quadratic-model headline is a restatement, but the n>2 potentials, the full 3D lattice spectra, and the public code make it the reference to cite.

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arXiv IDhep-th/0106179 (version 1)
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CitationAndrei Linde, Gary Felder, Lev Kofman. "Tachyonic Instability and Dynamics of Spontaneous Symmetry Breaking." arXiv:hep-th/0106179 (2001). https://pith.science/paper/CYJOJZI2
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The paper

The paper claims

A scalar field falling off the top of its potential breaks the symmetry in a single oscillation, because the tachyonic instability turns the vacuum energy into colliding waves and defects before any coherent rolling can set in.

The reading

The paper asks how a scalar field actually leaves the top of its effective potential when a symmetry breaks, and argues that the textbook picture — a homogeneous field oscillating many times about the minimum at $|phi| = v$ before giving up its energy — does not describe what happens. Because the curvature at $phi = 0$ is negative, every mode with wavelength longer than the inverse tachyonic mass grows exponentially, so the field fragments into domains instead of rolling coherently; the authors follow this through three-dimensional lattice simulations of the classical field equations seeded by quantum vacuum fluctuations. They find that most of the potential energy $V(0)$ is converted into gradient energy of colliding classical waves and topological defects within a single oscillation of the field distribution, so the symmetry is effectively broken almost immediately. The claim matters because it fixes the starting conditions for reheating and defect production in cosmological phase transitions, and because it marks a regime where the usual perturbative bookkeeping — Hartree or $1/N$ truncations built on a universe-averaged $langle delta phi^2 rangle$ — gives a demonstrably wrong answer. The authors also show that the mechanism is not universal in detail: for potentials behaving as $-lambda phi^n$ with $n > 2$ the curvature vanishes at the origin, and the instability must be seeded by tunneling, by rare Gaussian peaks, or by both.

What carries the argument

Two pieces carry the argument. The first is the tachyonic instability band itself: for $V = -lambda phi^n/2$ with the field rolling from rest, energy conservation gives $dot{phi} = sqrt{lambda (phi^n - phi_0^n)}$, and since long-wavelength fluctuations obey the same equation as $dot{phi}$, they grow as $phi_k propto phi^{n/2}$ — so $phi_k phi = text{const}$ for $n = 2$, $phi_k phi propto phi^{1/2}$ for $n = 3$, and $phi_k phi propto phi$ for $n = 4$. This single rule explains why the instability is so much faster than ordinary parametric resonance and why it becomes more violent as $n$ grows. The second is the lattice method: the quantum modes with large occupation numbers are treated as classical waves, evolved under $ddot{phi} - nabla^2 phi + V'(phi) = 0$ on a grid with a staggered leapfrog step, initialised in momentum space with random phases and Gaussian amplitudes $langle |phi_k|^2 rangle = 1/(2k)$. The paper also makes a conceptual point that carries real weight: because a large part of $langle delta phi^2 rangle$ comes from fluctuations longer than the domain scale, averaging over the whole universe is the wrong operation — one must average locally inside each domain, and no amount of accuracy within the Hartree or $1/N$ approximation repairs that. A renormalization step is introduced for the cubic model, where lattice modes induce a spurious linear term $lambda v langle delta phi^2 rangle phi$ that must be subtracted to restore the intended potential.

Extended reading notes

Core claim

The central claim is that spontaneous symmetry breaking driven by a tachyonic mass term completes within a single oscillation of the field distribution, not over a long stage of homogeneous oscillations. For $V = -m^2 phi^2/2 + lambda phi^4/4$ the growth of modes with $k < m$ proceeds until $sqrt{langle delta phi^2 rangle} sim v$, at which point the universe is already divided into domains of size slightly greater than $m^{-1}$ with $phi sim pm v$; the gradient energy of the resulting domain walls is of the same order as the initial potential energy $V(0) = m^4/(4 lambda)$, so the field cannot coherently return to $phi = 0$. Lattice simulations confirm this: the probability distribution of the field never comes back near the origin and settles at two peaks around $pm v$ within one oscillation, with occupation numbers reaching $n_k sim 10^2 lambda^{-1}$ for $k ll m$, far above the thermal value $n_k sim lambda^{-1/4}$ that full thermalization would eventually require. The authors then show that the dynamics depend qualitatively on the shape of the potential near its maximum. In the Coleman-Weinberg and $-lambda phi^4$ cases the curvature vanishes at the origin, so no mode grows from an infinitesimal seed; decay proceeds by barrier-free tunneling described by the $O(4)$-symmetric instanton family $phi(r) = 2sqrt{2/lambda}, rho/(r^2 + rho^2)$, whose action $S_E = 8 pi^2/(3 lambda)$ is independent of $rho$, followed by bubble growth, oscillation of the field inside the bubble, and bubble wall collisions. In the cubic case $-lambda v phi^3/3 + lambda phi^4/4$ there are no instantons at all, yet the state $phi = 0$ is still unstable: rare long-wavelength Gaussian peaks supply a local negative curvature, and the resulting fragmentation looks like bubble formation even though no Euclidean solution describes it.

Load-bearing premise

The simulations switch the negative mass squared on instantly at $t = 0$ and seed the field with the massless vacuum spectrum $langle |phi_k|^2 rangle = 1/(2k)$; a gradual quench, a thermal initial spectrum, or an initial displacement comparable to the fluctuation level would change the seed amplitudes, and hence the growth time and the domain size that the single-oscillation conclusion rests on. The authors check independence from the momentum cutoff but not from the quench rate, and in the quartic case they acknowledge starting from an analytic instanton rather than waiting for a bubble to nucleate.

Editorial extensions

If this is right

  • Reheating in models with concave potentials is dominated at its first stage by tachyonic preheating rather than by parametric resonance of a homogeneous oscillating inflaton, which changes the initial spectrum of produced particles.
  • Topological defects — domain walls for a real field, strings for a complex field, and their higher-dimensional analogues — are produced as an integral part of symmetry breaking rather than as a small nonperturbative correction, and they absorb a comparable share of the vacuum energy.
  • Full thermalization lags far behind symmetry breaking: occupation numbers immediately after the transition are many orders of magnitude above the thermal distribution, so a long intermediate stage of quasi-equilibrium with an effective chemical potential must intervene.
  • In hybrid-inflation-type models with a cubic tachyonic term, the onset of instability is a stochastic process of rare Gaussian peaks growing into bubble-like configurations, so the timing of the transition is set by tail statistics rather than by a homogeneous roll.
  • In the quartic and Coleman-Weinberg cases the transition proceeds by tunneling without a barrier, and the interior of each bubble behaves like an open universe whose oscillations send out a train of waves of nearly undiminished amplitude rather than a single thin shell.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The single-oscillation result is a statement about the shape of the potential at its maximum, so one could recast it as a classification: any potential with $V''(0) < 0$ breaks fast, and the interesting discriminator is not the coupling strength but whether $V''(0)$ is finite, zero, or reached only through fluctuations.
  • If domain size at formation is set by $m^{-1} ln^{1/2}(C/lambda)$ and grows only by domain wall migration afterwards, the resulting defect network should carry a memory of the coupling constant in its correlation length, which is in principle a distinguishable signature from networks produced by a thermal quench.
  • The argument against universe-averaged $langle delta phi^2 rangle$ applies wherever an infrared contribution dominates the variance, so it should also constrain how backreaction of long-wavelength inflationary fluctuations on the expansion rate is computed.
  • The cubic case suggests a broader class of potentials with vanishing curvature at the origin where the decay rate is controlled by the tail of a Gaussian random field rather than by an instanton, which makes the transition rate sensitive to the initial spectrum in a way the quadratic case is not.
Original abstract

Spontaneous symmetry breaking usually occurs due to the tachyonic (spinodal) instability of a scalar field near the top of its effective potential at $\phi = 0$. Naively, one might expect the field $\phi$ to fall from the top of the effective potential and then experience a long stage of oscillations with amplitude O(v) near the minimum of the effective potential at $\phi = v$ until it gives its energy to particles produced during these oscillations. However, it was recently found that the tachyonic instability rapidly converts most of the potential energy V(0) into the energy of colliding classical waves of the scalar field. This conversion, which was called "tachyonic preheating," is so efficient that symmetry breaking typically completes within a single oscillation of the field distribution as it rolls towards the minimum of its effective potential. In this paper we give a detailed description of tachyonic preheating and show that the dynamics of this process crucially depend on the shape of the effective potential near its maximum. In the simplest models where $V(\phi) \sim -m^2\phi^2$ near the maximum, the process occurs solely due to the tachyonic instability, whereas in the theories $-\lambda\phi^n$ with n > 2 one encounters a combination of the effects of tunneling, tachyonic instability and bubble wall collisions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,256 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.79 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Spontaneous symmetry breaking usually occurs due to the tachyonic (spinodal) instability of a scalar field near the top of its effective potential at $\phi = 0$. Naively, one might expect the field $\phi$ to fall from the top of the effective potential and then experience a long stage of oscillations with amplitude O(v) near the minimum of the effective potential at $\phi = v$ until it gives its energy to particles produced during these oscillations. However, it was recently found that the tachyonic instability rapidly converts most of the potential energy V(0) into the energy of colliding classical waves of the scalar field. This conversion, which was called "tachyonic preheating," is so efficient that symmetry breaking typically completes within a single oscillation of the field distribution as it rolls towards the minimum of its effective potential. In this paper we give a detailed description of tachyonic preheating and show that the dynamics of this process crucially depend on the shape of the effective potential near its maximum. In the simplest models where $V(\phi) \sim -m^2\phi^2$ near the maximum, the process occurs solely due to the tachyonic instability, whereas in the theories $-\lambda\phi^n$ with n > 2 one encounters a combination of the effects of tunneling, tachyonic instability and bubble wall collisions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Spontaneous symmetry breaking usually occurs due to the tachyonic (spinodal) instability of a scalar field near the top of its effective potential at $\phi = 0$. background 1.00 0.05 0.04 asserted 0.98
s1 Naively, one might expect the field $\phi$ to fall from the top of the effective potential and then experience a long stage of oscillations with amplitude O(v) near the minimum of the effective potential at $\phi = v$ until it gives its energy to particles produced during these oscillations. background 0.99 0.04 0.02 speculative 0.95
s2 However, it was recently found that the tachyonic instability rapidly converts most of the potential energy V(0) into the energy of colliding classical waves of the scalar field. background 0.62 0.06 0.14 asserted 1.00
s3 This conversion, which was called "tachyonic preheating," is so efficient that symmetry breaking typically completes within a single oscillation of the field distribution as it rolls towards the minimum of its effective potential. result 0.62 0.16 0.18 asserted 0.98
s4 In this paper we give a detailed description of tachyonic preheating and show that the dynamics of this process crucially depend on the shape of the effective potential near its maximum. contribution 1.00 0.90 0.29 asserted 1.00
s5 In the simplest models where $V(\phi) \sim -m^2\phi^2$ near the maximum, the process occurs solely due to the tachyonic instability, whereas in the theories $-\lambda\phi^n$ with n > 2 one encounters a combination of the effects of tunneling, tachyonic instability and bubble wall collisions. result 0.84 0.30 0.19 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.86 · evidence from proof or derivation 0.42.

What was examined, and what was not

sources read
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not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1. Evolution of the occupation numbers for the fluctuati
Figure 2. FIG. 2. Same as in Fig 1 for
Figure 3. FIG. 3. The process of symmetry breaking in the model (1) for
Figure 4. If one continues the calculation for a much longer time, one can see much more clearly the formation and growth of domains with φ = ±v. In the beginning these domains are small, but then they “eat” each other and grow. To illustrate this process we performed simulations in a 2D box of size 1024 x 1024. This allowed us to perform the calculations out to a much greater time and see domain formation on a much greater scale; see
Figure 5. Quadratic model, Two dimensional simulation with N = 10242 , L = 1600, dt = .1, kcut = 0, λ = 10−2 . The slices shown are plots of 1282 points where each point represents an average over an 8 × 8 box of gridpoints. This averaging was done simply to make plotting easier and reduce the resulting image sizes. Figures 7 and 8 Complex model, N = 2563 , L = 600, dt = .2, λ = 10−4 . The momentum cutoff was kcut = 2 and kcut = 0 for Figs. 7 and 8 respectively. The PDF’s in
Figure 6. FIG. 6. The process of symmetry breaking in the model (1) for
Figure 7. FIG. 7. The process of symmetry breaking in the model (1) for a
Figure 8. shows the occupation numbers nk of produced particles. During the first oscillation these numbers grow up to 107 – 108 for k < m (k < 1 in the figure). Then the occupation numbers at k < m slightly decrease, whereas the occupation numbers at k > m begin to grow. Complete thermalization takes a very long time. FIG. 8. Occupation numbers nk of particles produced during tachyonic preheating in the model of a complex scalar field φ with effective potential V = −m2φ ∗φ + λ(φ ∗φ) 2 with λ = 10−4 . In the beginning (lower curves), nk grows for k ∼< m (k ∼< 1 in this figure), but then…
Figure 9. Complex model, N = 1283 , L = 40, dt = .01, kcut = √ 2, λ = 10−4 . A string was defined as the collection of points on the lattice for which |φ| 2 < .02. Figures 10 and 11 Quartic model, N = 2563 , L = 500, dt = .1, kcut = 0, λ = 5 × 10−5 . Both runs used as initial conditions the instanton solution (26) at t = 0 with φ0 = .02v. The run shown in
Figure 10. FIG. 10. Field values on a partial 2D slice through the lattic
Figure 11. FIG. 11. The process of symmetry breaking in the model
Figure 12. FIG. 12. Field values on a 2D slice through the lattice for
Figure 13. FIG. 13. Histograms describing the process of symmetry brea
Figure 14. FIG. 14. Occupation numbers of particles produced during ta

Figures reproduced from arXiv:hep-th/0106179 by the authors.

References
How later papers used it

93 later papers cite it

Evidence: substantial. 88 of the citing papers share no author with it, and 10 of those carry a confident label from Pith's citation classifier.

Pith read 17 of the 145 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its data or values 1 paper · 1 by the same authors

By the same authors

Compares with it 1 of 5 papers with this label
How the literature uses it

84 arXiv papers cite this in 135 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 72 sentences without a confident label

Uses its method or tool 3 papers
Uses its data or values 1 paper
  • From the numerial results presented in [ref] , we estimate the typical size of the inhomogeneous structures of $σ$ when they fragment as R_* ∼5/√(λ) v √(ln(100/λ)) The resulting frequency ([ref]) and amplitude ([ref]) associated with the peak of the gravity wave spectrum are then given by f_* ∼λ^1/4 10^10 Hz/√(ln(100/λ)) and h^2 Ω_gw^* ∼2 ×10^-6 ln(100/λ) (v/)^2 Note that $f_*$ depends on the energy scale only through the coupling constant $λ$, and that this dependence is very weak, essentially varying as $λ^1/4$.

    Theory and Numerics of Gravitational Waves from Preheating after Inflation · Expectations for Preheating after Hybrid Inflation · 0.99

Compares with it 5 papers
Cites it as support 2 papers
  • This crucial observation implies that any description in terms of a homogeneous spectator field rolling down a potential is inaccurate to start with [ref] .

    Quintessential inflation: A tale of emergent and broken symmetries · Spectator field dynamics · 0.98

  • Analytical investigation of this issue accompanied by lattice simulations shows that the field distribution typically experiences just a single oscillation before it relaxes near the minimum of the potential [ref] .

    Update of D3/D7-Brane Inflation on K3 x T^2/Z_2 · On reheating after inflation and entropy fluctuations · 0.98

Cites it as background 40 papers
  • At the end of hybrid inflation, the inflaton decays via a spinodal instability of the inhomogeneous modes accompanying the symmetry breaking, in the process of tachyonic preheating [ref] .

    Theory and Numerics of Gravitational Waves from Preheating after Inflation · Expectations for Preheating after Hybrid Inflation · 0.87

  • Reheating in hybrid inflation was first studied in [ref] , and then in greater depth in [ref] where, in particular, the tachyonic instability channel was investigated [See also [ref] for an application of tachyonic decay to the cosmological moduli problem.].

    A Study of Structure Formation and Reheating in the D3/D7 Brane Inflation Model · Reheating in the D3/D7 Brane Inflation Model · 0.96

  • Given the shape of the potential ([ref]), very different (p)reheating mechanisms could in principle take place, from a tachyonic production in the region between the end of inflation and the inflection point [ref] , an instant preheating mechanism [ref] and a parametric resonance effect around the minimum of the potential [ref] .

    Preheating in the Standard Model with the Higgs-Inflaton coupled to gravity · Reheating in the Standard Model of particle physics · 0.80

Discussion

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