Finite density O(3) non-linear sigma model and low energy physics
Falk Bruckmann, Christof Gattringer, Thomas Kloiber, Tin Sulejmanpasic
FFinite density O ( ) non-linear sigma model and lowenergy physics Falk Bruckmann ∗ Universität Regensburg, Institut für Physik, Universitätstraße 31, 93053 Regensburg, Germany [email protected]
Christof Gattringer, Thomas Kloiber †‡ Universität Graz, Institut für Physik, Universitätsplatz 5, 8010 Graz, Austria [email protected] , [email protected] Tin Sulejmanpasic
North Carolina State University, Department of Physics, Raleigh, NC 27695-8202, USA [email protected]
We present lattice results for simulations of the O ( ) non-linear sigma model at finite chemicalpotential. The complex action problem is overcome by a dual variable representation of the model.We discuss two aspects of the theory at finite density: 1) The relation of the finite density datato scattering phases and wave-functions. 2) The phase-structure of the theory as a function ofchemical potential and the possibility of a Kosterlitz-Thouless phase transition. The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan ∗ Supported by the DFG (BR 2872/6-1). † Speaker. ‡ Supported by the Austrian Science Fund, FWF, DK
Hadrons in Vacuum, Nuclei, and Stars (FWF DK W1203). c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] D ec inite density O ( ) non-linear sigma model Thomas Kloiber
1. Introduction
Due to asymptotic freedom, dynamical mass gap generation and non-trivial topology, the O ( ) non-linear sigma model in 2D is not only an excellent toy model for Yang-Mills theory and QCDin 4D, but also an important condensed matter system. Physically the system describes a one-dimensional chain of O ( ) rotors. Although many analytic results with varying level of assump-tions exist [1], the system is far from well understood. The related O ( N ) and CP ( N − ) systemsare susceptible to large N calculations, and we mention in passing that there was an interesting de-velopment in the diagrammatic Monte-Carlo study of the O ( N ) system [2] where a bosonic systemwith non-trivial IR physics was successfully numerically analysed using a diagrammatic expansion.The model has a global O ( ) symmetry, and if this symmetry is unbroken, the particles (orlow energy excitations) are classified by the charge they carry with respect to internal symmetries.Studying the system at non-zero charges is of interest if one wants to understand the low energydynamics of the theory. One can couple chemical potentials which can be used to control thecharges and therefore the particle content of the system. In particular a chemical potential caneasily be coupled to one of the O ( ) subgroups of O ( ) , corresponding to one of the generators t a , a = , ,
3. Singling out one of the generators we can add a chemical potential to a conservedcharge, which in the present case corresponds to the total angular momentum component of the onedimensional chain of O ( ) rotors . The choice of a generator represents a choice of the axis withrespect to which we wish to induce angular momentum.Recently, based on low energy symmetries and large N computations, a possible phase diagramof the O(3) model at finite density was conjectured in [3]. Since the system is gapped, the increaseof chemical potential enforces condensation of charges at some critical chemical potential µ = µ c . This value represents the dynamically generated mass of the charged excitations. Once thishappens the O ( ) symmetry is broken down to O ( ) by the presence of charges for one of thethree generators (i.e., it is broken explicitly by singling out one direction of angular momentumwhich is populated). The low energy effective theory is therefore a two-dimensional O ( ) theory,which can have vortex defects. In [3] it was argued that the effective coupling of the effective O ( ) model for µ (cid:38) µ c is strong enough so that the vortices would percolate until some µ = µ KT > µ c (see Section 3, however), which would be signaled by a Kosterlitz-Thouless-Berezinskii transition,forcing vortices to combine into pairs. The vortex pairs are instantons of the O(3) model, whosesize modulus is related to the distance between the vortex and the anti-vortex they are made of. Thevacuum picture is that of small instantons.Incidentally this is very similar to what is believed to happen in QCD at asymptotically highdensities: the color Cooper pairs form, and Higgs the gauge fields. This in turn, forces instan-tons to be small. As one reduces the chemical potential the instantons start to grow, and it wasconjectured that at some critical density instantons fractionate into smaller objects [4]. Althoughthere is no complete understanding of what these instanton constituents are in four-dimensions,it is possible that they are related to instanton-monopoles which have been a topic of increasinginterest in recent years, both in theoretically controllable regimes as well as in lattice field theoryand phenomenology (see e.g. [5] and references therein). Any choice of a generator, or linear combination of them is equivalent, as it only amouns to the redefinition of thegenerators. inite density O ( ) non-linear sigma model Thomas Kloiber
Although QCD and non-linear sigma models are undoubtedly different, these similarities arestriking. It is therefore of interest to study the phase diagram of the O(3) model from first principles,i.e., from lattice simulations. There is a technical difficulty here, however, generically present insystems with chemical potential: The complex action problem. Although plenty of work andattempts have been made to overcome this problem, no satisfactory solution exists to date for QCDat finite baryon density. However, an extremely elegant solution for several bosonic and somefermionic systems exists, which employs an exact rewriting of the partition function in terms of dual variables living on links which are generically integer valued for compact symmetries. Whatis very important is that the dual variable fugacity is real and positive, which permits importancesampling and allows Monte Carlo simulations to be performed. The results we present here arebased on such simulations.
2. Lattice action, dual variables and chemical potential
Before we present the lattice results let us discuss some generalities of the system. The latticeaction of the O ( ) model is given by S = − J ∑ x , ν nnn ( x ) · nnn ( x + ˆ ν ) , (2.1)where nnn ( x ) = ( n ( x ) , n ( x ) , n ( x )) is a 3-component field with unit length and where J = / g is theinverse coupling. It is clear that the transformation nnn ( x ) → Onnn ( x ) , O ∈ O ( ) is a global symmetryof the above actions, and due to Noether’s theorem, there is a conserved charge associated withit. We use spherical coordinates such that nnn ( x ) = ( cos φ ( x ) sin θ ( x ) , sin φ ( x ) sin θ ( x ) , cos θ ( x )) . Wenow add a chemical potential for the charge associated with the O ( ) ∈ O ( ) symmetry, whichchanges φ ( x ) → φ ( x ) + const. The simplest way to do this is to introduce a background gauge fieldfor the symmetry and to set A ν ( x ) → i µ δ ν , , where µ is the chemical potential. In this way weobtain the action S = − J ∑ x , ν (cid:104) cos θ ( x ) cos θ ( x + ˆ ν ) + sin θ ( x ) sin θ ( x + ˆ ν ) cos ( φ ( x + ˆ ν ) − φ ( x ) − i δ ν , µ ) (cid:105) . (2.2)The action has a non-vanishing imaginary part, i.e., it suffers from the complex action problem.However, as already mentioned, there exists an exact rewriting of the partition function, whichuses flux variables living on links [6]: Z = ∑ { k , m , ¯ m } (cid:32) ∏ x , ν J k x , ν k x , ν ! ( J / ) | m x , ν | + m x , ν ( | m x , ν | + ¯ m x , ν ) ! ¯ m x , ν ! (cid:33) e − µ ∑ x m x , ∏ x I (cid:18) ∑ ν ( k x , ν + k x − ˆ ν , ν ) , + ∑ ν (cid:2) m x , ν + m x − ˆ ν , ν + ( ¯ m x , ν + ¯ m x − ˆ ν , ν ) (cid:3)(cid:19) ∏ x δ (cid:18) ∑ ν ( m x , ν − m x − ˆ ν , ν ) (cid:19) E (cid:18) ∑ ν ( k x , ν + k x − ˆ ν , ν ) (cid:19) , (2.3)where m x , µ ∈ Z and ¯ m x , µ , k x , µ ∈ N are integers living on links (the dual variables). Here δ ( n ) denotes the Kronecker delta, and E ( n ) is the evenness function which is 1 for even n and vanishesfor odd n , while I ( a , b ) = Γ (( a + ) / ) Γ (( b + ) / ) / Γ (( a + b + ) / ) .3 inite density O ( ) non-linear sigma model Thomas Kloiber -0.0050.0000.0050.0100.0150.0200.0250.0300.0350.040 0.85 0.9 0.95 1 1.05 1.1 〈 n 〉 µ / m N t = 200 400 800 1000 -5-4-3-2-1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 l o g ( m ) = l o g ( µ c ) J dual datac J exp(- c J) with c = 1167 ± 501 c = 6.61 ± 0.28log( 3 / J ) Figure 1:
Left: The charge density v.s. chemical potential µ , for four temporal lattice extents N t at fixedspatial volume N s =
100 and coupling J = .
3. Right: The massgap as a function of coupling. The two solidcurves are strong coupling prediction (blue) and weak coupling fit. Note that the coefficient c is schemedependent, while the coefficient c is consistent with 2 π . Notice that the chemical potential couples directly to the temporal fluxes m x , , and that, bytaking a derivative with respect to the chemical potential, it is clear that the expectation value ofthese temporal fluxes are expectation values of the charge. Since the m -flux is conserved by theKronecker deltas in (2.3) we can identify the temporal winding number of the m -worldlines withthe charge of a configuration .
3. The results
We have simulated the O ( ) model with chemical potential in the dual representation (2.3).We begin with discussing the charge density (cid:104) n (cid:105) = Q / L , Q = ( ∂ / ∂ µ ln Z ) / β with L the spatialextent, as a function of chemical potential µ . We remind the reader that since the O ( ) modelhas dynamical mass gap generation, it is expected that if the temperature is low enough chargecondensation will set in at some sharply defined non-zero value of the chemical potential, µ = µ c = m , where m is the mass of the lightest charged excitation. In J (cid:29) J (cid:28) µ c isgiven by µ J (cid:29) c ∝ Λ = C a Je − π J , µ J (cid:28) c = log ( / J ) , (3.1)where Λ is the strong scale, a is the lattice spacing, and C is a scheme-dependent pure number.The former comes from the fact that the strong scale is the only scale in the continuum, and thelatter from an explicit calculation in strong coupling. As can be seen from Fig. 1, this is preciselywhat we observe (left panel): At sufficiently small temperature (cid:104) n (cid:105) = µ and chargecondensation sets in at a sharply defined value of µ which defines µ c . In the rhs. plot of Fig. 1 thevalues of µ c as a function of J are compared to the above two expressions for the continuum- andthe strong coupling limit. The data quite nicely match the continuum scaling for J (cid:38) .
4, and thestrong coupling expansion for J (cid:46) . inite density O ( ) non-linear sigma model Thomas Kloiber Q µ / m T/m = 0.045 = 0.023 = 0.011 = 0.0045
MASSGAP!! Charge 1 plateau Charge 2 plateau µ T I M E SPACE dual variable m-fluxes
Figure 2:
Left: The charge Q versus µ at N s = J = . Lm = .
4) for low temperatures correspondingto N t = , , , is decreased the steps become more pronounced. The physical reason for this is that for finitespatial extent there is always a finite energy gap between the charge Q and the charge Q + ( , , µ ) – the phases between the plateaus are called ’canted’ or ’unquantizedferromagnetic’ [7, Sec. 19.4].The small volume data are very useful for extracting information about the particle interac-tions. As was discussed in [8], knowing the two-particle interaction energy in 1+1 dimensionsallows one to extract the complete scattering data, while in higher dimensions it can be used toobtain the scattering length [9] . Since this interaction energy is exactly equal to the difference in µ between the onset of the charge 1 plateau and the charge 2 plateau of the lhs. panel of Fig. 2, thearguments of [9] can be used to obtain the scattering phase. We call this method for obtaining thescattering phases the charge condensation method , as it arises from data of the charge condensation[8]. The corresponding results for the phase shift are shown in the lhs. plot of Fig. 3. Note thatthis is a universal result for any system, independent of the dual variable representation, and canbe used whenever the sign problem can be overcome in a MC simulation, such as in two-flavor,two-color QCD or in QCD with isospin chemical potential.However, the dual flux representation has much more information about the nature of the lowenergy particle excitations. In fact the dual representation allows for the dual fluxes to be interpretedas particle worldlines (see Fig. 2 rhs.). This interpretation can then be used to construct the wave-function of the multi-particle ground states at finite volume, by analyzing the spatial distribution ofthe worldlines. This multi-particle ground state wave-function in principle has all the informationabout the scattering phases and interaction of the particles. In the lhs. plot of Fig. 3 we also presentthe results for the scattering phases extracted by this method which we call the Dual Wave-FunctionMethod (for more details see [8]). 5 inite density O ( ) non-linear sigma model Thomas Kloiber - π /2-3 π /8- π /40.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 δ ( k ) k / m exactJ = 1.3 CC1.4 CC1.5 CC1.3 DWF1.4 DWF1.5 DWF1.4 LW1.5 LW1.54 LW s l o p e o f 〈 w s 〉 v s . N t / N s µ / m Figure 3:
Left: Phase shifts obtained from the Charge Condensation Method (blue) and the Dual Wave-function method (red). The green symbols show data from Lüscher and Wolff [9] and the full curve theanalytic solution [1, first ref.]. Right: The stiffness J r as a function of chemical potential µ in mass units. Let us finally address the possibility of the conjectured Kosterlitz-Thouless-Berezinskii tran-sition. A good indicator of this transition is the so-called “stiffness”, which, for an effective O ( ) model is defined in the following way: Let φ ( x ) be the angular variable of the effective O ( ) model. In the continuum limit the kinetic Lagrangian is given by J ( ∂ µ φ ( x )) . Imposing the spatialboundary conditions to be such that φ ( x + L ) = φ + φ ( x ) , where L = aN s is the physical extent ofthe spatial direction. It can be easily checked that the classical free energy difference between thetwisted and untwisted system is then simply given by ∆ F class ( φ ) = J φ / L . Quantum mechanicallythe coupling J may be renormalized, and we can defineJ r ≡ L lim φ → φ ∆ F ( φ ) . (3.2)This quantity is usually called the stiffness . Note, however, that the twist φ / L plays the role of animaginary chemical potential in the spatial direction. It is simple to show that J r = L β (cid:10) w s (cid:11) , (3.3)where w s is the winding number of the dual flux variable m , the one coupling to µ , in the spatial direction. The stiffness indicates whether the system is in the gapped, vortex-percolating phase,(i.e., insensitive to the value of φ so that J r = J r >
0. The rhs. plotof Fig. 3 shows our data for the stiffness from measurements of (cid:10) w s (cid:11) as a function of µ . As can beseen, the data indicates that the stiffness starts to develop a non-zero value at µ = µ c , which seemsto indicate that both, charge condensation as well as the KT transitions, occur at the same point µ = µ c . However, one should keep in mind that the data shown is for small ratio β / L of the inversetemperature β and the volume L , and the system may have a quantum phase transition dependingon the order of limits L → ∞ and β → ∞ . A more detailed analysis of this scaling with temperatureand volume still needs to be performed before final conclusions can be made. We would like to thank Hans Gerd Evertz for pointing this out to us. inite density O ( ) non-linear sigma model Thomas Kloiber
4. Conclusions
We have presented several results from simulations of the O(3) non-linear sigma model atnon-zero charge density in the dual variable representation. We have discussed two aspects of thetheory: (1) The worldline interpretation of the dual fluxes and scattering phases [8]. (2) The phasestructure as a function of chemical potential and the possibility of a KT transition. Concerning (2)we have seen that preliminary results seem to indicate that the KT transition happens at the samecritical value of the chemical potential as the charge condensation transition. However, this resultmay be very sensitive to the order of infinite volume and zero temperature limits, and at the momentfinal conclusions must be deferred until a proper scaling analysis is performed. Concerning (1) wehave demonstrated [8] that the scattering phases can be reliably extracted from the distribution ofthe worldlines. This is not only an alternative approach to computing scattering data in latticesimulations, but also reveals a deep connection between different aspects of a quantum field theory,i.e., its phenomenology in the condensed phase and scattering properties.
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