Topological and magnetic properties of the QCD vacuum probed by overlap fermions The calculations were done at the GSI Batch Farm (Darmstadt) and on the BIRD farm at DESY

# Topological and magnetic properties of the QCD vacuum probed by overlap fermions 1

## Abstract

We study some of the local CP-odd and magnetic properties of the non-Abelian vacuum with use of overlap fermions within the quenched lattice gauge theory. Among these properties are the following: inhomogeneous spatial distribution of the topological charge density (chirality for massless fermions) in SU(2) gluodynamics (for uncooled gauge configurations the chirality is localized on low-dimensional defects with , while a sequence of cooling steps gives rise to four-dimensional instantons and hence a four-dimensional structure of the chirality distribution); finite local fluctuations of the chirality growing with the strength of an external magnetic field; magnetization and susceptibility of the QCD vacuum in SU(3) theory; magnetic catalysis of the chiral symmetry breaking, and the electric conductivity of the QCD vacuum in strong magnetic fields.

\ShortTitle

Topological and magnetic properties of the QCD vacuum probed by overlap fermions \FullConferenceXth Quark Confinement and the Hadron Spectrum
8–12 October 2012
TUM Campus Garching, Munich, Germany

## 1 Introduction

{floatingfigure}

[r] Isosurfaces of the chirality distribution, see [2] for details. Recently both aspects of our studies, properties of the QCD vacuum in strong magnetic fields and its non-trivial topological structure, attracted much attention in light of ongoing heavy-ion experiments. The magnitudes of the magnetic fields possibly created there are of order of , which may change the critical temperature of the chiral transition, several electromagnetic properties of QCD and also give rise to new phenomenological effects potentially measurable experimentally. For the origin and properties of the magnetic fields in heavy-ion collisions see the review [1].

A non-trivial gluonic background may create an imbalance between numbers of left- and right-handed quarks and lead to the local strong CP-violation [3, 4]. A typical spatial distribution of such imbalance is shown in Fig. 1, where the colors denote some fixed positive (red) and negative (blue) values of chirality. It is remarkable, that for uncooled gauge field configurations the distribution is irregular and, as we show later, does not follow the instanton pattern, but forms low-dimensional structures.

## 2 Magnetic-field-induced effects

### 2.1 Technical details

We use a quenched lattice gauge theory with the tadpole-improved Lüscher-Weisz action [5]. To generate statistically independent gauge field configurations we implement the Cabibbo-Marinari heat bath algorithm. The lattice size is , and lattice spacing . All observables we discuss later have a similar structure: for VEV of a single quantity or for dispersions or correlators. Here , , are some operators in spinor and color space. These expectation values can be expressed through the sum over low-lying2 but non-zero eigenvalues of the chirally invariant Dirac operator (Neuberger’s overlap Dirac operator [7]):

 ⟨¯ΨOΨ⟩=∑|k|

and

 Extra open brace or missing close brace (2)

where all spinor and color indices are contracted and we drop them for simplicity. The uniform magnetic field is introduced as described in [8]. To perform calculations in the chiral limit we calculated the expression (1) or (2) for a non-zero and averaged it over all configurations of the gauge fields. Then we repeated the procedure for other quark masses and extrapolated the VEV to limit. Simulation details for the Sections 3 and 4 are similar, but for two colors, and are described in the mentioned there papers.

### 2.2 Chiral condensate

In this section we present our results for the chiral condensate (),

 Σ≡−⟨0|¯ΨΨ|0⟩, (3)

as a function of the magnetic field . The result is shown in Fig. a. The general tendency for to grow with has been obtained in various models (see [9] for a review) and usually referred to the magnetic catalysis of the chiral symmetry breaking3. We perform the fit of the results by the following function

 Σfit(B)=Σ0[1+(eBΛ2B)ν], (4)

where . The obtained fitting parameters are

 Σ0=[(228±3)MeV]3,ΛB=(1.31±0.04)GeV,ν=1.57±0.23. (5)

The value of the condensate in absence of the magnetic field and the exponent have reasonable values, compare with e.g. [12, 13].

### 2.3 Magnetization and susceptibility

In this section we calculate the quantity

 ⟨¯ΨσαβΨ⟩=χ(F)⟨¯ΨΨ⟩qFαβ, (6)

where and is a coefficient of proportionality (susceptibility), which depends on the field strength.

This quantity was introduced in [14] and can be used to estimate the spin polarization of the quarks in external magnetic field. The magnetization can be described by the dimensionless quantity , so that

 ⟨¯Ψσ12Ψ⟩=μ⟨¯ΨΨ⟩. (7)

The expectation value (6) can be calculated on the lattice by (1) with . The result is shown in Fig. b. We can see, that the 12-component grows linearly with the field, which agrees with [14]. This allows us to find the chiral susceptibility from the slope of the curve. After a linear approximation , where4

 Ωfit≡−13χfit0Σ0, (8)

we obtain and

 χfit0=−4.24±0.18GeV−2. (9)

This value fits well into the range of present theoretical estimations (see [15] for a review and [16] for recent results). We also have to mention a well known analytic result obtained by the OPE combined with the idea of pion dominance [17], supported by two holographic ones [18, 19], but giving a too high value comparing to our results. This disagreement seems to be an important puzzle to solve, because it will lead to a better understanding of the pion dominance assumption and the large limit.

### 2.4 Evidences of the chiral magnetic effect

One example of a new effect mentioned in the Introduction is the chiral magnetic effect (CME), which generates an electric current along the magnetic field in the presence of a nontrivial gluonic background [3, 4]. This effect may naturally take place in heavy-ion collisions and is at the moment under active experimental search [20, 21, 22] (see also a review [23] on the interpretation of the experimental data). Lattice evidences of the effect can be found in [24, 26, 25, 27]. Here we implement the procedure from [24] for the case and study the local chirality

 ρ5(x)=¯Ψ(x)γ5Ψ(x)≡ρL(x)−ρR(x) (10)

and the electromagnetic current

 jμ(x)=¯Ψ(x)γμΨ(x). (11)

The expectation value of the first quantity can be computed by (1) with and with for the second quantity. Both VEV’s are zero, as expected, but the corresponding fluctuations obtained from (2) are finite and grow with the field strength (see Fig. b). Here we use the “IR” subscript to emphasize, that we subtract from the quantity its value at :

 ⟨Y⟩IR(B)=1V∫Vd4x⟨Y(x)⟩B−1V∫Vd4x⟨Y(x)⟩B=0 (12)

One can interpret the enhancement of the current fluctuations either as short-living quantum fluctuations or as a charge flow. We measured the conductivity of the vacuum to argue in support of the latter.

## 3 Conductivity of the QCD vacuum

Electric conductivity can be extracted from the correlator of two vector currents,

 Gij(τ)=∫d3→x⟨ji(→0,0)jj(→x,τ)⟩ (13)

Following [28], let us define the spectral function which corresponds to the correlator (13)

 Gij(τ)=+∞∫0dw2πK(w,τ)ρij(w),withK(w,τ)=w2Tcosh(w(τ−12T))sinh(w2T), (14)

where is the temperature. The Kubo formula for the electric conductivity then reads [29, 28]

 σij=limω→0ρij(ω)4T. (15)

In the limit of the weak time-independent electric field , one has . We performed the SU(2) quenched lattice computations (for details see [26]) and obtained results shown in Fig. 3. One can conclude from the results, that the QCD vacuum is an isotropic conductor in the deconfinement phase, and the conductivity does not depend on the strength of the magnetic field ( for ). In the confinement phase the conductivity in the transverse directions is zero within error range, while it is a growing function of B-field in the longitudinal direction. In other words, the strong magnetic field at small temperatures turns the QCD vacuum from an insulator to an anisotropic conductor.

Such a behavior may also lead to the enhancement in the soft photon and dilepton production rates in the direction transverse to the magnetic field [25].

## 4 Fractal dimension of the chirality distribution

We studied some properties of the spatial distribution of chirality (10), such that the volume occupied by the distribution and the fractal dimension of that volume, for various lattice spacings and cooling stages. To measure the former we used the inverse participation ratio (IPR),

 IPR50=N⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣∑x|ρ5(x)|2(∑x|ρ5(x)|)2⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦λ=0≡VlatticeVdistribution, (16)

where is the total number of sites of the lattice and the brackets denote an averaging over all zero modes and further averaging over all gauge field configurations. In general, the IPR is equal to the inverse fraction of sites occupied by the support of a distribution. The fractal (Hausdorff) dimension can be extracted from the IPRs at various lattice spacings ,

Lattice parameters and simulation details can be found in [2]. The results are shown in Fig. 4. Our conclusion is that the chirality distribution (and hence fermionic zero modes) has zero volume in the continuum limit (since IPR grows with decreasing ) and its fractal dimension equals for uncooled configurations (in agreement with [30, 31, 32, 33]), tending to after the cooling. This behavior supports the idea of [34], stating that the Yang-Mills vacuum structure is different for different resolutions of the measuring procedure. The low dimensional structure of the vacuum, if true beyond the probe quark limit, may lead to a new phenomenology relevant for the heavy-ion experiments [35, 36, 37, 38].

### Footnotes

1. thanks: The calculations were done at the GSI Batch Farm (Darmstadt) and on the BIRD farm at DESY
2. We believe that the IR quantities are insensitive to the UV cutoff realized by selecting some finite number of the eigenmodes [6]
3. A non-monotonic behavior in the vicinity of has been recently observed [10, 11] and named as inverse magnetic catalysis.
4. in our simulation we calculate the magnetization of the d-quark condensate, thus

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