Lattice Field Theory Study of Magnetic Catalysis in Graphene

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4-24-2017

Lattice field theory study of magnetic catalysis in graphene

Carleton DeTar

Christopher Winterowd

Savvas Zafeiropoulos College of William and Mary, [email protected]

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Recommended Citation DeTar, Carleton; Winterowd, Christopher; and Zafeiropoulos, Savvas, Lattice field theory study of magnetic catalysis in graphene (2017). PHYSICAL REVIEW B, 95(16). 10.1103/PhysRevB.95.165442

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Lattice field theory study of magnetic catalysis in graphene Carleton DeTar, Christopher Winterowd, and Savvas Zafeiropoulos Phys. Rev. B 95, 165442 — Published 24 April 2017 DOI: 10.1103/PhysRevB.95.165442 Lattice Field Theory Study of Magnetic Catalysis in Graphene

Carleton DeTar,1 Christopher Winterowd,1 and Savvas Zafeiropoulos2, 3, 4 1Department of Physics and Astronomy University of Utah, Salt Lake City, Utah 84112, USA 2Institut fürTheoretische Physik - Johann Wolfgang Goethe-Universität Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany 3Jefferson Laboratory 12000 Jefferson Avenue, Newport News, Virginia 23606, USA 4Department of Physics College of William and Mary, Williamsburg, Virgina 23187-8795, USA (Dated: March 16, 2017) We discuss the simulation of the low-energy effective field theory (EFT) for graphene in the presence of an external magnetic field. Our fully nonperturbative calculation uses methods of lattice gauge theory to study the theory using a hybrid Monte Carlo approach. We investigate the phenomenon of magnetic catalysis in the context of graphene by studying the chiral condensate which is the order parameter characterizing the spontaneous breaking of chiral symmetry. In the EFT, the symmetry breaking pattern is given by U(4) → U(2) × U(2). We also comment on the difficulty, in this lattice formalism, of studying the time-reversal-odd condensate characterizing the ground state in the presence of a magnetic field. Finally, we study the mass spectrum of the theory, in particular the Nambu-Goldstone (NG) mode as well as the Dirac quasiparticle, which is predicted to obtain a dynamical mass.

PACS numbers: 11.15.Ha, 71.10.Pm, 73.22.Pr, 73.43.Cd

I. INTRODUCTION hΨΨ¯ i, as well as a dynamical mass for the fermion. This transition can also be seen by studying the change in the conductivity after the gap is formed9 The discovery of graphene counts as one of the most important developments in recent years1. The study of Including short-range interactions in addition to the graphene has seen tremendous growth, in part due to its long-range Coulomb interaction brings forth a rich novel many-body and electronic properties2. In partic- phase diagram with the existence of an antiferromag- ular, there have been many studies of graphene in the netic (AFM) phase10, spontaneous Kekulédistortion11, presence of an external magnetic field3. and a topological insulating phase12 in addition to the Graphene, a two-dimensional hexagonal lattice of Car- semimetal and CDW phase. Introducing an on-site Hub- bon atoms, has an unusual band structure consisting of bard interaction and realistic nearest and next-nearest- 13 a conduction and a valence band that come together at neighbor interactions to the model allows these phases two inequivalent corners of the Brillouin zone. At these to be accessed by varying the relevant couplings. In par- valleys, or so-called “Dirac points”, the two bands take ticular, graphene undergoes a transition to the AFM the shape of cones. The consequence of this band struc- phase when the on-site interaction exceeds its critical ture is that the low energy excitations are described by value U > Ucr. Consistent with the results for the EFT massless Dirac fermions, which have linear dispersion. mentioned above, for small U, the system undergoes a These Dirac fermions have a Fermi velocity that satis- transition from the semimetal phase to the CDW phase when α > αcr14,15. By working directly with the hexag- fies vF /c ≈ 1/300, and thus one does not have Lorentz g g invariance. Furthermore, this small Fermi velocity ren- onal lattice Hamiltonian of graphene, one can construct ders the interaction between the Dirac quasiparticles to a coherent state path-integral representation which in- 16–18 be essentially Coulombic. cludes general two-body interactions . The effective coupling of the theory is particularly In this study, we perform calculations in the graphene 2 large, αg ≡ 2e /(( + 1)vF 4π) > 1, and thus this is EFT at half-filling with U = 0, where one can con- a strong-coupling problem. Here, the strength of the struct a positive-definite, real action. The short-distance interaction between the quasiparticles is controlled by physics due to the underlying lattice can, however, be the dielectric constant, , of the substrate on which the taken into account by including various four-fermi oper- graphene layer sits. For suspended graphene = 1, while ators in the EFT10. One can incorporate these terms into for graphene on a silicon oxide substrate Si02 ≈ 3.9. It is a lattice gauge theory calculation by introducing auxil- known that for sufficiently large αg, the EFT undergoes a iary scalar fields which couple linearly to the appropri- phase transition from a semimetal to a state with charge ate fermion bilinears19. The issue of simulating away density wave (CDW) order, as has been shown in sev- from half-filling is much more involved. By directly ap- eral previous studies4–8. This transition is characterized plying traditional Monte Carlo methods, one encounters by the appearance of a nonzero value of the condensate, the “sign-problem” associated with a theory whose action 2 is not positive-definite. Considerable progress has been trons. The sublattice degree of freedom appears due to made to systems suffering from the sign problem using the fact that the hexagonal lattice is bipartite, consisting a variety of different approaches20–25. In the case of lat- of two inequivalent triangular sublattices. The interac- tice gauge theory, one either tries to circumvent the sign tion is introduced via a scalar potential which lives in problem by working with imaginary chemical potential, (3 + 1) dimensions. The continuum Euclidean action is performing reweighting, or via Taylor expansions around given by zero chemical potential. Z 2 X In the presence of an external magnetic field, many SE = dtd x Ψ¯ σD6 [A0]Ψσ field theories, such as quantum electrodynamics (QED) σ=1,2 and the Nambu-Jona-Lasinio (NJL) model, are thought ( + 1) Z to undergo spontaneous symmetry breaking. This phe- 3 2 + 2 dtd x (∂iA0) . (1) nomenon is known as magnetic catalysis and is hypoth- 4e esized to be universal. In particular, this means that The Dirac operator is given by the microscopic details of the interaction are irrelevant and the mechanism should work in any case where there X D6 [A0] = γ0 (∂0 + iA0) + vF γi∂i. (2) is attraction between fermions and antifermions. In the i=1,2 context of the graphene EFT, the magnetic catalysis scenario leads to the appearance of a dynamically generated The basis used for the four-component Dirac spinors is Dirac mass and is proposed to account for several of the as follows quantum-Hall plateaus that appear at large values of the 27–29 > magnetic field . Ψσ = ψK+Aσ, ψK+Bσ, ψK−Bσ, ψK−Aσ , (3) Magnetic catalysis has also been the focus of recent where K ,K refer to the Dirac points, σ refers to the interest in lattice quantum chromodynamics (LQCD). + − electron’s spin, and A, B refer to the sublattices. Here we Several studies have shown that the chiral condensate use a reducible representation of the gamma matrices in increases with magnetic field at zero temperature (pion (2+1) dimensions, constructed from the two inequivalent mass m = 200−400 MeV, eB up to 1 GeV2)30–32. This π irreducible representations, which is given by confirms the idea that the background magnetic field aids chiral symmetry breaking in the vacuum. However, it was σµ 0 also discovered that with T ≥ 140 MeV and quark masses γµ = , µ = 0, 1, 2, (4) 0 −σµ tuned such that the pion mass was the value observed in nature, the opposite occurs and the chiral condensate de- where σ0 ≡ σ3. In (3 + 1) dimensions, a similarity trans- 33,34 † creases with the external magnetic field . formation, S γµS = −γµ, relates the two representa- The remainder of this article is organized as follows. In tions, with S = γ5. In (2 + 1) dimensions, γ5 does not section II we discuss the continuum EFT, its symmetries, 2 Y as well as our lattice setup. In section III we discuss exist as σµ ∝ 1. One can also define additional ma- the physical mechanism behind magnetic catalysis and µ=0 review the results that apply to graphene. In section IV trices which generate symmetry transformations in the we review the methods used to sample the Feynman path graphene EFT integral numerically. Section V introduces the various observables that we use to study the graphene EFT in 0 1 0 1 γ˜4 = , γ˜5 = , (5) the presence of an external magnetic field. Section VI 1 0 −1 0 contains our results for the various condensates, and the 1 0 mass spectrum of the EFT. Finally, in Section VII, we γ˜4,5 ≡ −γ˜4γ˜5 = , (6) conclude and discuss the interpretation of these results. 0 −1 Some preliminary results of this article first appeared in where {γ˜ , γ } = {γ˜ , γ } = {γ˜ , γ˜ } = 0. Ref.35 and Ref.36. 4 µ 5 µ 4 5 The fermionic part of the action in (1) has a global U(4) symmetry whose generators are given by II. GRAPHENE EFT 1 ⊗ 1, 1 ⊗ σµ, γ˜4,5 ⊗ 1, γ˜4,5 ⊗ σµ (7)

γ˜4 ⊗ 1, γ˜4 ⊗ σµ, iγ˜5 ⊗ 1, iγ˜5 ⊗ σµ. (8) A. Continuum These generators have the four-dimensional sublattice- The continuum EFT, describing the low-energy prop- valley subspace tensored with the two-dimensional spin erties of monolayer graphene, contains two species of subspace. The appearance of a Dirac mass term, of the X four-component Dirac spinors in (2 + 1) dimensions in- form m Ψ¯ σΨσ breaks the U(4) symmetry down to 37 teracting via a Coulomb interaction . The counting of σ=1,2 the fermionic degrees of freedom is as follows: 2 sublat- U(2)×U(2), whose generators are given by (7). Thus, the tices × 2 Dirac points × 2 spin projections of the elec- formation of a nonzero value of the condensate hΨ¯ σΨσi, 3 would signal spontaneous symmetry breaking and the ap- bandwidth, vF /a, where a is the spacing of the hexago- pearance of eight Nambu-Goldstone (NG) bosons. In Ap- nal lattice39. Taking these renormalized couplings into pendix B, we provide the details of the fermion bilinears account can have a decisive effect on the selection of relevant to this study. the ground state in the full theory. In our lattice gauge As we have mentioned, the action in (1) is diagonal in model, we consider only a long-range Coulomb interac- spin space and has a U(4) symmetry described by the tion mediated by a scalar potential and ignore the Zee- generators in (7) and (8). Although our lattice simu- man interaction as well as other lattice-scale interactions. lations do not take this into account, in real graphene this U(4) symmetry is explicitly broken in the presence of an external magnetic field by the Zeeman term in the B. Lattice Hamiltonian Z Taking the continuum EFT (1), which is valid up to 2 † HZ = −µBB dtd x Ψ σ3Ψ, (9) some cutoff Λ, we now discretize it on a cubic lattice. It is important to emphasize that this lattice is not directly related to the original honeycomb lattice of graphene. where µB is the Bohr magneton and σ3 acts in spin space. Although we are dealing with Dirac fermions, this Lattice methods in gauge theories have shown to be use- term is needed as the four-dimensional spinor structure ful in elucidating the nonperturbative aspects of strongly interacting theories such as quantum chromodynamics is constructed from the sublattice and valley degrees of 40 freedom. In the graphene EFT, the orbital and pseu- (QCD) . Thus, we chose to apply these methods to the dospin quantum numbers determine the Landau-level in- study of the graphene EFT. dex, while the electron’s spin degree of freedom couples We represent the gauge potential, A0(n), via the vari- to the magnetic field in a nonrelativistic way. In con- able U0(n) ≡ exp (iatA0(n)), which lives on the temporal trast, for relativistic Dirac fermions coupled to an ex- links of the lattice. We take the fermion fields to live on the sites of the lattice. To avoid complications arising ternal magnetic field, the spin degree of freedom in the 41 direction of the field combines with the orbital quantum from an unwanted bulk phase transition , we use the number to determine the Landau-level index. The Zee- so-called noncompact U(1) gauge action 2 man term explicitly breaks the U(4) symmetry down to (NC) ( + 1) X S = a a A (n) − A (n + î) , (14) U(2)↑ × U(2)↓ with the generators of U(2)σ given by G s t 4e2 0 0 n,i

1 ⊗ Pσ, γ˜4 ⊗ Pσ, iγ˜5 ⊗ Pσ, γ˜4,5 ⊗ Pσ, (10) where as and at are the lattice spacings in the spa- where we have introduced the spin projection operator tial and temporal directions respectively. Here n = (n0, n1, n2, n3) labels the lattice site and ˆi is the unit 1 vector in the ith direction. Using the dimensionless ﬁeld Pσ ≡ (1 ± σ3). (11) 2 Ac0(n) = atA0(n), we write the above action as

Although the Zeeman term lifts the spin degeneracy of 3 2 (NC) β X X ˆ S = Ac0(n) − Ac0(n + i) , (15) each Landau level, the size of this perturbation is very G 2 small even in the presence of large magnetic fields. This n i=1 can be seen by considering 2 where β = ξ( + 1)/2e and ξ ≡ as/at = 3vF is the q anisotropy parameter, controlling the ratio of the spa- 2 p LL ≡ ~vF |eB|/c = 26 B[T] meV, (12) tial lattice spacing to the temporal lattice spacing. This choice has no significance as a trivial redefinition of the ≡ µ B = 5.8 × 10−2B[T] meV, (13) Z B mass and coupling can produce the same theory studied5. The discretization of the fermions has well-known where is the energy separating the lowest Landau LL complications encoded in the Nielsen-Ninomiya no-go level (n = 0) from its neighbor (n = 1), and is the Zee- Z theorem42. Namely, one wants to remove unphysical man energy. For even the largest magnetic fields available “doubler” modes while still preserving some part of the in the laboratory (B ∼ 50 T), is only a small fraction Z continuum U(4) symmetry. The method we use in of . LL this study employs the staggered-fermion formulation43, Apart from the long-range Coulomb interaction which eliminates some of the doublers while preserving a present in (1), in the complete lattice theory there are nu- remnant of the U(4) symmetry. The staggered-fermion merous short-range electron-electron interactions. These action reads lattice-scale interaction terms are allowed by the point- group symmetry of the underlying hexagonal lattice, 2 X 1 † ˆ 38 SF = asat χ¯n U0(n)χn+0ˆ − U0 (n − 0)χn−0ˆ C6v . As a result, these terms break the much larger 2at U(4) symmetry present in our continuum EFT. The cou- n 1 X plings associated with these terms can vary in sign and + v η (n)¯χ χ − χ 2a F i n n+î n−î are strongly renormalized at energies on the order of the s i=1,2 4

+ mχ¯ χ , (16) in staggered language, corresponds to the taste degree of n n freedom. Due to taste-breaking terms in the staggered action at O(a) (see Appendix A), this becomes impossi- where the fermions live in (2+1) dimensions and η1(n) = ble as these terms mix the various tastes, or in graphene n0 n0+n1 (−1) , η2(n) = (−1) are the Kawamoto-Smit language, spin projections. phases which appear due to spin diagonalization of the In this study we use the improved staggered fermion naive fermion action. The fields χ andχ ¯ are one- action, asqtad45. We outline the various improvements component Grassmann variables. We have introduced in the following paragraphs. a mass term which explicitly breaks the remnant chiral Although staggered fermions have some advantages, symmetry but is needed in our simulations as an infrared being cost effective in numerical simulations and retain- regulator and to investigate SSB. We will later take the ing a remnant chiral symmetry, it is known that at fi- infinite-volume limit followed by the chiral limit, m → 0. nite lattice spacing the violations of taste symmetry can We can also write the fermion action in terms of the di- be significant. In the case of QCD, for example, simula- ˆ mensionless lattice quantities χ¯n = asχ¯n,χ ˆn = asχn, tions with unimproved staggered fermions on fine lattices mˆ = atm (a = 0.05 fm) obtain splittings within the pion taste- X 1 † multiplet (O(100 MeV)) which are of the order of the S = χˆ¯ U (n)ˆχ − U (n − 0)ˆˆ χ 46 F 2 n 0 n+0ˆ 0 n−0ˆ pion mass in nature . This is a completely unaccept- n able situation if one is attempting to accurately study vF X + η (n)χˆ¯ χˆ − χˆ low-energy properties of QCD. It was found that taste 2ξ i n n+î n−î i=1,2 violations are a result of the exchange of high-momentum gluons47. These couplings are eliminated by a process of +m ˆ χˆ¯nχˆn . (17) link “fattening”, which replaces the links Uµ(n) with a combination of paths that connect the site n with n +µ ˆ. In (2 + 1) dimensions, each species of staggered fermions For clarity, we describe the process of constructing one represents two identical four-component Dirac spinors. of the terms contained in the fat-link. Consider making In lattice QCD simulations, this degree of freedom, com- the following replacement for a link theµ ˆ direction monly referred to as “taste”, is unwanted, and consid- 2 X l erable effort goes into eliminating it. In our case, the Uµ(x) → Uµ(x) + λa ∆ν Uµ(x), (20) taste degree of freedom is desirable, as we are attempt- ν6=µ ing to simulate two identical species of Dirac fermions l representing the low energy excitations of graphene. The where λ is an undetermined coefficient and ∆ν is a taste degree of freedom becomes more apparent once one “Laplacian” operator acting on a link variable. This is performs a change of basis, the details of which are rel- seen in its definition egated to Appendix A. At zero mass, the lattice action l 1 † in (17) retains a U(1) × U(1) remnant of the contin- ∆ Uµ(x) ≡ Uν (x)Uµ(x +ν ˆ)U (x +µ ˆ) ν a2 ν uum U(4) symmetry. The U(1) symmetry corresponds † to fermion number conservation and is given by + Uν (x − νˆ)Uµ(x − νˆ)Uν (x − νˆ +µ ˆ) − 2U (x) . (21) χ(x) → exp (iα) χ(x), (18) µ χ¯(x) → χ¯(x) exp (−iα) . Expanding the links to linear order and transforming to momentum space, the relation in (20) yields The U(1), or “even-odd” symmetry, is given by χ(x) → exp (iβ(x)) χ(x), (19) X h Aµ(p) → Aµ(p) + λ 2Aµ(p) (cos(ˆpν ) − 1) χ¯(x) → χ¯(x) exp (iβ(x)) , ν6=µ i x +x +x where (x) ≡ (−1) 0 1 2 . In the continuum limit, one + 4 sin(ˆpµ/2) sin(ˆpν /2)Aν (p) . (22) expects to recover the full symmetry of the action in (1). The U(1) symmetry, which is a subgroup of the full con- One can verify that by choosing the coefficient λ = 1/4, tinuum chiral group, is spontaneously broken with the the gauge field has no support at momentum p where appearance of a nonzero value of the condensate, hχχ¯ i one transverse component is equal to π/a. A similar and is a reliable indicator of the spontaneous breaking construction is used to eliminate support at momentum of the continuum symmetry44. Thus, on the lattice, one with varying numbers of transverse components equal to expects there to be a single NG boson as a result of spon- π/a. This leads to a five-link staple, a seven-link staple, taneous symmetry breaking. For the study of the time- and a final term which corrects for discretization errors reversal-odd condensate, the calculation is a bit more introduced as a result of the fattening. intricate with staggered fermions. Namely, in order to Another improvement which has proven useful in compare with continuum results, one is interested in dis- lattice QCD simulations is the so called tadpole tinguishing the spin projections of the condensate, which improvement48. This, too, is used in the asqtad action. 5

This improvement is motivated by the observation that 1 , nx 6= Ns − 1 U (n) = 2 . (26) x −ia eBNxny new vertices, with no continuum analog, appear in lat- e s , nx = Ns − 1 tice perturbation theory. These vertices are suppressed by powers of the lattice spacing but can lead to UV- where Ns = Nx = Ny, and nx, ny = 0, 1,...,Ns − 1. divergent diagrams which cancel the lattice spacing de- Furthermore, the toroidal geometry demands that the pendence of the vertex. In the graphene EFT, the tad- magnetic flux through the lattice be quantized as follows pole factor, u0, is defined in the following way eB NB 1/2 Φ ≡ = , (27) (p) B 2 u0 = hU i , (23) 2π Ls where hU (p)i is the volume-averaged expectation value of where Ls = Nsas, is the lattice extent in the spatial the space-time-oriented plaquette. In practice, the tad- direction and NB is an integer in the range pole factor is determined self-consistently for each lattice N 2 ensemble. Tadpole improvement consists of dividing each 0 ≤ N ≤ s . (28) B 4 link in the temporal direction by u0. Thus, each fat link lt receives a factor 1/u0 , where lt is the number of steps We note that the spatial links, which describe the exter- the path takes in the temporal direction. nal magnetic field, are static and are not updated during One can further improve the staggered-fermion action the sampling of the path integral. by introducing a third-nearest-neighbor term (third be- cause the unit cell is 23). This is known as the Naik 49 term . With an appropriate weighting of the nearest III. MAGNETIC CATALYSIS neighbor and the third-nearest neighbor terms, one can obtain an improved free dispersion relation. This is seen by considering the following replacement The phenomenon of magnetic catalysis is a fascinating example of dynamical symmetry breaking53–56. In this Naik ∇µχ(x) → c1∇µ + c3∇µ χ(x), scenario, an external magnetic field acts as a catalyst for c fermion-antifermion pairing, even if they are weakly cou- = 1 (χ(x +µ ˆ) − χ(x − µˆ)) 2a pled. First studied in the context of the Nambu-Jona- c + 3 (χ(x + 3ˆµ) − χ(x − 3ˆµ)) . (24) Lasinio model and quantum electrodynamics in both 8a (2 + 1) and (3 + 1) dimensions, magnetic catalysis has The coefficients of the one and three-hop terms are fixed been predicted to occur also for planar condensed-matter 57–60 by the condition that the dispersion relation reproduces systems, including graphene . Although various per- its continuum version with O(a4) discretization errors. turbative approaches have been used to study magnetic For the interacting theory, one introduces the appropriate catalysis in various settings, it is useful to apply lattice number of links into the terms in the above expression methods, as one is able to take a fully nonperturbative in order to maintain gauge invariance. Thus, using all approach. of the above-mentioned ingredients, one has a tadpole- One can begin to understand the mechanism responsi- improved action which is free of discretization errors up ble for magnetic catalysis by first considering free Dirac to O(a2). This is what is known as the tadpole-improved fermions in the presence of an external magnetic field. asqtad action45. This mechanism involves a dimensional reduction, D → 61 Although previous lattice studies have used various D − 2, due to the magnetic field . The Dirac equation levels of improvement for staggered fermions50,51, fur- in (2 + 1) dimensions is given by ther reducing taste violations by using the asqtad action µ should come closer to realizing the continuum theory. In (iγ Dµ − m)Ψ = 0, (29) the presence of an external magnetic field, the contin- 2 where Dµ = ∂µ − ieAµ. Solving (29), one finds that the uum limit is realized by taking the limit as|eB| → 0. In this limit, one could examine various dimensionless energy levels are given by ratios such as hΨΨ¯ i/|eB| which should have improved p 2 values with our improved staggered action. En = ± 2|eB|n + m , (30) The introduction of an external magnetic field on the lattice proceeds in the following manner52. In the con- where n = 0, 1, 2, ··· is the Landau-level index and is tinuum, a homogenous magnetic field perpendicular to given by a combination of orbital and spin contributions, the sheet of graphene can be described by the Landau- n ≡ k + sz + 1/2. The Landau levels are highly de- gauge vector potential, A = δ Bx . The spatial links generate, with a degeneracy per unit area of |eB|/2π for µ µ,2 1 n = 0, and |eB|/π for n > 0. When the Dirac mass is are similarly modified, with a special prescription at the p boundary of the lattice due to the periodic boundary con- much smaller than the magnetic scale, m |eB|, the ditions satisfied by the gauge links low energy sector is completely dominated by the lowest Landau level. Furthermore, the levels at E = ±m do 2 0 ia eBnx Uy(n) = e s , (25) not disperse which confirms the kinematic aspect of the 6 reduction, (2 + 1) → (0 + 1), due to the presence of the and SE is our Euclidean lattice action. The equations of magnetic field. motion are then An interesting consequence of an external field on ˙ dAb0(n) Dirac fermions in (2 +1) dimensions is the appearance of Ab0(n) ≡ = π(n), (35) a nonzero value for the condensate, hΨΨ¯ i. This supports dτ dπ(n) ∂S the existence of spontaneous symmetry breaking. One π˙ (n) ≡ = − E . (36) can calculate the free fermion propagator in the presence dτ ∂Ab0(n) of an external field62, and use the result to arrive at63,64 Integrating these equations over a time interval of length 1 √ 1 m2 T , one generates a new configuration which one either hΨΨ¯ i = − m 2eBζ , 1 + 2π 2 2eB accepts or rejects based on a Monte-Carlo Metropolis decision. In this study we have used the second-order + eB − 2m2 , (31) leapfrog method in integrating (35) and (36). The main question one has when selecting an algorithm to simulate dynamical fermions is how best to deal with where ζ(s, a) is the Hurwitz zeta function. Taking the the fermion determinant. In our case, we are interested limit m → 0 of (31), one obtains in simulating two continuum species, and thus, for stag- eB gered fermions, one can take advantage of the even-odd lim hΨΨ¯ i(B, m) = − . (32) symmetry of the staggered Dirac operator. Namely, one m→0+ 2π first introduces a complex pseudofermion field φ which Of course, by introducing interactions among the lives at each site of the lattice. Based on the identity fermions, this result will be modified. Calculations using Z −1 † † −φ†(M †M) φ Schwinger-Dyson equations predict the dynamical mass det(M M) = Dφ Dφe , (37) of the fermion to satisfy61 √ one can use the pseudofermions to replace the fermion mdyn ∝ αg eB. (33) action by making the identification M ≡D 6 st + m, where the unimproved staggered Dirac operator is given by In addition to the condensate that one normally con- 1 X siders when discussing spontaneous chiral symmetry (D6 ) = η (x)(U (x)δ st x,y 2 µ µ y,x+ˆµ breaking, there are other condensates which can de- µ scribe the ground state of the graphene EFT in the pres- − U †(x − µˆ)δ ). (38) ence of a magnetic field. In our study we also con- µ y,x−µˆ ¯ sider a time-reversal-odd condensate, ∆H ≡ hΨ˜γ4,5Ψi. Noting that M †M decouples even and odd sites, the de- While the former condensate leads to a Dirac mass in terminant on the LHS of (37) overcounts the number of the graphene EFT, the latter condensate gives rise to degrees of freedom by a factor of two. This can be cor- 65 a Haldane mass . The ground state at filling factor rected by restricting the pseudofermions to the even sites, ν = 0, corresponding to a half-filled lowest Landau level, for example. is posited to support a nonzero value for both conden- The Φ algorithm starts by generating a random com- sates in the absence of Zeeman splitting. In this work, plex field distributed according to (37) as well as a we study the flux dependence of both condensates in the Gaussian-distributed momentum. One can then inte- chiral and zero-temperature limits. grate the equations of motion over a trajectory. Specif- ically, the difficult part in doing this is computing the “force” given by (36). More specifically, we can write IV. SIMULATION DETAILS this expression as

∂SG ∂SF We generated our gauge configurations using a U(1) π˙ n = − − , (39) 66 variant of the Φ algorithm . This algorithm falls un- ∂Ab0(n) ∂Ab0(n) der a broad class of algorithms collectively known as hy- 67 where the first term represents the force coming from the brid Monte Carlo (HMC) . This procedure introduces gauge action, and the second term represents the force a fictitious time variable, τ, and a real-valued canonical coming from the fermion action. Expressing the fermion momentum, πn, which is conjugate to the gauge poten- action in terms of φ, φ† one has tial A0(n). One is then able to construct a molecular- dynamics Hamiltonian (f) ∂ † † −1 Fn = − φ M M φ , (40) ∂A (n) 1 X b0 H(MD) = π2(n) + S , (34) 2 E where the calculation of the derivative with respect to n Ab0(n) involves a significant number of terms for the asq- where the momentum term is a Gaussian weight which tad action. The calculation of the fermion force also in- factors out during the calculation of expectation values, volves solving the sparse linear system M †M X = φ, 7 which is performed with an iterative solver. The gauge where in our lattice partition function we integrate over force has a concise expression which can be obtained by the noncompact gauge potential A0. In order to de- differentiating (14) with respect to Ab0(n), which leads to termine if magnetic catalysis is present in the graphene EFT, we will be interested in taking the zero-temperature (g) X ˆ ˆ as well as massless limit for the chiral condensate. Fn = −β 2Ab0(n) − Ab0(n + i) − Ab0(n − i) . (41) i Another condensate which characterizes magnetic catalysis in the graphene EFT is the Haldane conden- After each trajectory, the pseudofermion field φ and the sate. This condensate is time-reversal odd and takes the gauge momenta are refreshed. This strategy has been form known to improve the amount of phase space explored by the algorithm and goes under the name of refreshed ∆H ≡ hΨ¯ (˜γ4,5 ⊗ 1)Ψi HMC. Z Z 1 −Seff[A ] = DA Ψ¯ (˜γ ⊗ 1)Ψe E 0 , (46) In our simulations we have taken the trajectory length VZ 0 4,5 to be 1, and varied the step size for a given ensemble in V order to obtain a Metropolis acceptance rate of 70%. We R where V denotes an integral over the space-time volume. have discarded the first 200 − 250 trajectories in order to In the one-component basis, this takes the form account for the equilibration of each ensemble. For ensembles where we were interested in the mass spectrum of 1 1 Z eff X −S [A0] ∆ = DA χ¯ (y)χ 0 (y)e E , (47) the theory, we have generated roughly 800 independent H V Z 0 η η y;η 6=η0 configurations. For those where we were interested only µ µ in the condensates, we generated roughly 100 independent configurations. To achieve statistical independence, where the details of the transformation connecting (46) we have saved configurations after every tenth trajectory. and (47) are given in Appendix A. As opposed to (45), the An analysis of the autocorrelation function for various fields in (47) are located at diagonally opposite corners of 3 observables confirms that these configurations are suffi- the 2 cube. Although the appearance of a nonzero value ciently statistically independent. for this condensate does not signal spontaneous symmetry breaking of the U(1) symmetry, its value and dependence on the external magnetic field characterize the V. OBSERVABLES ground state of the graphene EFT. To compute condensates on the lattice, one must use The primary observable of this study is the chiral con- stochastic methods even for modest lattice volumes. For densate. In the massless limit, this observable serves as example, the trace in (45) involves computing the prop- an order parameter for the transition between the insula- agator from a given site back to the same lattice site for tor phase, where it is nonzero, and the semimetal phase, each site on the lattice. To estimate the trace, we gener- where it vanishes. In the continuum graphene EFT, the ate an ensemble of Gaussian-distributed complex vectors chiral condensate is defined as with support on each lattice site that satisfy

1 ∂ log Z Nv hΨΨ¯ i = ∗ 1 X (k)† (k) hΦi Φji = δij ≈ Φ Φ , (48) V ∂m Nv k=1 1 1 Z eff −1 −S [A0] = DA0 Tr (D6 + m) e E , (42) V Z where the expectation value is over the Gaussian distribu-

eﬀ tion, i and j are lattice-site labels, and Nv is the number where SE [A0] = SG [A0] − Tr log (D6 + m), and the par- of stochastic vectors chosen from the distribution. We tition function is given by then can obtain

Z ¯ ¯ −SE [A0,Ψ,Ψ] Nv Z = DA0DΨDΨe . (43) 1 X Tr M −1 ≈ Φ(k)†M −1Φ(k). (49) Nv k=1 The fermions have been integrated out of the partition function using the identity Typically, to obtain a good estimate of the chiral conden- Z sate, we need approximately 100 stochastic vectors per det (D6 + m) = DΨ¯ DΨe−Ψ(¯ D6 +m)Ψ. (44) configuration. To estimate the condensate in (47), one must alter the procedure done for the chiral condensate as the fields For staggered fermions in the one-component basis, the reside at opposite corners of the cube. We first gener- scalar density takes the formχχ ¯ , and thus the chiral ate an ensemble of Gaussian-distributed complex vectors condensate becomes satisfying (48) with support only on sites with a given Z 1 1 −1 −Seff[A ] η = (ηt, ηx, ηy), where ηµ labels a particular site within hχχ¯ i = DA Tr (D6 + m) e E 0 , (45) V Z 0 st the cube. We then shift the source to the opposite corner 8 of the cube using parallel transport In the one-component basis, this operator becomes 1 X X Φ˜ (k) = Sˆ Sˆ Sˆ Φ(k), (50) O (y) = (x)¯χ (y)χ (y), (57) 6 Pµ Pν Pλ PS η η P η where the sum is over the six permutations taking us to η0+η1+η2 where xµ = 2yµ + ηµ, (x) = (−1) , and the sum the corner opposite η. Here Pµ,Pν ,Pλ are permutations runs over the sites of the cube labeled by y. The details of of ±t, ±x, ±y, and we have introduced the shift operator the transformations connecting (56) and (57) are given in † Appendix A. The spatial pseudoscalar correlator is also ˆ Uµ(x − µˆ)Φi−µˆ , − S±µΦ = . (51) of interest and is given by i Uµ(x)Φi+ˆµ , +

(s) X i(ωlτ+py y) Using the source in (50), we can calculate the Haldane GPS(x; ωl, py) = e condensate as follows x,y

Nv × hOPS(x, y, τ)OPS(0, 0, 0)i, (58) X 1 X (k)† −1 (k) χ¯ (y)χ 0 (y) ≈ Φ M Φ˜ . (52) η η N 0 v where ωl = 2lπT, l = 0, 1, 2,..., are the bosonic Matsub- y;ηµ6=ηµ k=1 ara frequencies. where the sum over η goes over all eight sites of the cube. Stochastic estimation relies on cancellations of the noise to find the signal, which decays exponentially with the VI. RESULTS separation between source and sink. Therefore, for operators such as the Haldane condenstate, which are nonlo- A. Chiral Condensate cal, it can be difficult to obtain a good signal-to-noise ratio. In order to overcome this, we have taken N ≈ 1000 v In this section we will discuss the results of our calcula- in (52). tions regarding magnetic catalysis in the graphene EFT. The spontaneous breaking of the U(1) symmetry also Using the lattice methods described above, we will at- has observable consequences for the fermion quasiparti- tempt to characterize the symmetry breaking using var- cle, which receives a dynamical mass, m , which remains F ious observables. Apart from the condensates defined nonzero even as the bare mass vanishes. In order to study above, we will also investigate the quasiparticle and pseu- the dynamical fermion mass, we calculate the quasipar- doscalar mass. Thus, various methods which have been ticle propagator in the temporal direction used to study chiral symmetry breaking in the context of (τ) X G (τ; ~p) = ei~p·~rhχ(x, y, τ)¯χ(0, 0, 0)i, (53) QCD can be applied to the graphene EFT. F In the absence of an external magnetic field, the ~r graphene EFT is known to exhibit two phases: a where ~p ≡ (px, py) and ~r ≡ (x, y). In order to investigate semimetal phase at weak coupling (large dielectric con- effects of the finite spatial size of the box, we are also stant ) and an insulating phase at strong coupling (small interested in the quasiparticle propagator in the spatial ). In the language of chiral-symmetry breaking, the direction semimetal phase corresponds to the chirally symmetric (s) X phase while the insulating phase corresponds to the bro- G (x; ω , p ) = ei(ωlτ+py y)hχ(x, y, τ)¯χ(0, 0, 0)i,(54) F l y ken phase. Various aspects of this transition were studied τ,y previously using lattice methods5–7,50. The transition is where ωl = (2l + 1)πT, l = 0, 1, 2,..., are the fermionic believed to be of second order as indicated by the results Matsubara frequencies. in6. As a result of the spontaneous breaking of the U(1) The situation changes as one turns on the external 2 symmetry on the lattice, there should arise a pseu- magnetic field. Namely, the critical βcr = ξ(cr + 1)/2e doscalar NG boson. This state is analogous to the pion in which determines the boundary between the two phases QCD. As in QCD, one expects this state to become mass- shifts to larger values. This has been shown previously 68 less as the bare mass vanishes. Its mass, mπ, is studied in , where at fixed temperature, the authors obtained through the following two-point correlator in Euclidean a phase diagram in the (B, β)-plane. One would ex- time pect that the phase boundary has a temperature dependence, where at T = 0, the authors of53–56,58–60 predict (τ) X i~p·~r GPS(τ; ~p) = e hOPS(x, y, τ)OPS(0, 0, 0)i, (55) that an infinitesimal attraction between fermions and an- x,y tifermions will lead to pairing, and thus magnetic catalysis. Another early analysis of a graphene-like theory where OPS is a staggered bilinear operator with pseu- showed that at extremely weak coupling, a nonzero con- doscalar quantum numbers. In the spin-taste basis, the 69 operator we have used is densate was obtained in the chiral limit We began our calculations by identifying the semimetal OPS(y) = Ψ(¯ y) (˜γ4 ⊗ 1) Ψ(y). (56) phase through a scan in β, coupled with an investigation 9

0.8 2 β Φ 0.5 2 β Φ 20x10x60,=0.80, B=0 30x10x60,=0.80, B=0.125 2 β Φ 2 β Φ 0.7 20x10x60,=0.80, B=0.125 20x10x60,=0.80, B=0.125 2 β Φ 8 x10x60,=0.80, B=0.125 0.4 0.6

0.5 0.3

σ 0.4 σ

0.3 0.2

0.2 0.1 0.1

0 0 0 0.05 0.1 0.15 0.2 0.25 0 0.01 0.02 0.03 0.04 0.05 m m

¯ 2 FIG. 1: σ ≡ hΨΨi, the chiral condensate (in units of as), as FIG. 2: The chiral condensate as a function of the fermion 2 a function of the bare fermion mass (in units of at) at zero mass for different spatial volumes Ns at magnetic flux ΦB = 2 external magnetic field (black points) and at magnetic flux 0.125. The volumes are listed in the form Ns × Nz × Nτ , ΦB = 0.125 (pink points). Here, the magnetic flux is mea- where the fermions live in the xy-plane and the gauge field is 2 sured in units of as. The volumes are reported in the form present throughout the entire volume. 2 Ns × Nz × Nτ . We note that σ vanishes with m at nonzero field as well as at zero external magnetic field. Thermal effects 0.4 are argued to be the reason behind the vanishing of the con- m=0.05 densate in the presence of the magnetic field. The statistical 0.35 m=0.02 errors on each point are not visible on this scale. m=0.01 m=0.00167 0.3 of the condensate’s behavior as a function of the bare 0.25 mass in the chiral limit. At large values of β, which correspond to values of the graphene fine-structure con- σ 0.2 stant which are less than the critical value of αcr = g 0.15 2 5 2e /(4πvF (cr + 1)) ≈ 1.1 , we expect to be inside the semimetal phase. In this symmetric phase, one expects 0.1 σ ≡ hΨΨ¯ i to be linear in the bare mass for small values of m. One can compare this expectation with the behavior 0.05 of hΨΨ¯ i as a function of the bare mass when the external magnetic field is turned on. This is illustrated in Fig. 1, 0 where one sees an approximately linear behavior at zero 0 1 2 3 4 5 field and a strongly nonlinear behavior of the condensate T/m as a function of m for the magnetic flux ΦB = 0.125. In our calculations we have fixed β = 0.80, which corre- FIG. 3: The chiral condensate plotted as a function of the sponds to αg ≈ 0.3. This further assures us that we are ratio T/m for the ensembles with ΦB = 0.125 and Ns = 8. deep within the semimetal phase. One sees that at small values of T/m, the condensate increases Notice, however, that in both cases, as the explicit and plateaus towards a nonzero value. symmetry breaking parameter is taken to zero, the condensate also goes to zero. Even if SSB occurs, this still may occur due to the finite spatial volume. Referring to and the pseudoscalar. The screening masses are obtained Fig. 2, one can see that this is not the case. For a broad by calculating the correlation functions in (54) and (58) range of spatial extents Ns, the condensate shows little and projecting to py = 0 and the lowest Matsubara fre- variation. This can be explained by observing that the quency (ω = πT and ω = 0, respectively). We find p 0 0 magnetic length, lB ≡ ~c/eB, which characterizes the that the masses characterizing the decay of the above fermion’s cyclotron orbit, satisfies 1 < lB < Ns, in units correlation functions in space satisfy MsLs 1, where of as. Ls = Nsas is the lattice extent in the spatial direction. We have also confirmed the independence of the results For the pseudoscalar, we found that mπ,sLs ≈ 18 − 20, on the finite spatial size of the box by indepedently calcu- for the ensembles with volume 202 × 10 × 60 and flux lating the screening masses for the fermion quasiparticle ΦB = 0.125. For the fermion propagator, we obtained 10

0.36 rection to the infinite-volume result is less than 2%. 0.35 Thermal effects are also known to play a role in sym- 0.34 metry restoration. The temperature of the system is related to the extent of the lattice in Euclidean time, 0.33 T = 1/Nτ at. We have investigated the effect of fi- 0.32 nite temperature on the condensate and illustrated it in Fig. 3, where σ(T/m) is plotted. When the temperature σ 0.31 is large compared with the fermion mass, one can see 0.3 that the condensate tends towards zero. On the other hand, when the temperature is small compared with the 0.29 fermion mass, one can see that the condensate increases 0.28 and tends asymptotically towards a nonzero value. By 0.27 first taking the limit T → 0, we are able to study magnetic catalysis in the ground state. Using this strat- 0.26 egy, we have first extrapolated to zero temperature at 0 0.01 0.02 0.03 0.04 0.05 0.06 fixed bare fermion mass. We have extrapolated using m two different methods. First, we have taken points at small T which form a plateau and fit them to a constant. FIG. 4: The chiral limit of the chiral condensate using the These plateaus can be seen in Fig. 3, for magnetic flux points extrapolated to T = 0. The error bars on each point ΦB = 0.125. We have then included one other point at were determined from both systematics i.e. choice of model larger T and extrapolated using a quadratic fit (except for the zero-temperature extrapolation, as well as statistics. for bare mass m = 0.01 in Fig. 3, where the third point is The value of the intercept for the linear fit is 0.2721(7) 2 too far away from T = 0). We use the difference between (χ /d.o.f. ≈ 0.6). the two results for σ at T = 0 to estimate a systematic 0.3 uncertainty associated with this extrapolation, with the central value taken as the average. A similar procedure has been also carried out for the other three magnetic 0.25 fluxes. We then use these points for our chiral extrapolation, which we performed using a linear fit. This is 0.2 illustrated in Fig. 4, where we plot the mass dependence of the T = 0 extrapolated points.

σ 0.15 Proceeding in an analogous manner for three other fluxes, we were able to obtain the behavior of the zero- 0.1 temperature, chirally extrapolated condensate as a function of the magnetic flux, ΦB. These results are shown in Fig. 5. The relationship between the condensate and the 0.05 2 magnetic flux is fit to the form σT =m=0 = eBc1+(eB) c2. The errors on the points are those calculated in the chiral 0 extrapolation. These results overwhelmingly support the 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 scenario of magnetic catalysis in the graphene EFT. Φ B

B. Haldane Condensate FIG. 5: The zero-temperature, chirally-extrapolated values of the condensate, plotted as a function of the external magnetic Long before the realization of graphene in the lab, flux, ΦB = eB/2π. The points at ΦB = 0.083 and ΦB = 0.056 have a spatial size of Ns = 12, while those at ΦB = 0.125 and F.D.M. Haldane considered a graphene-like lattice model 65 ΦB = 0.0625 have a spatial size of Ns = 8. The error bars on which he used to study the quantum Hall effect (QHE) . the points come from the chiral extrapolations at T = 0. The In this model, he considered a magnetic field which was data have been fit to a quadratic constrained to pass through periodic in space such that the flux through the unit cell the origin (χ2/d.o.f. ≈ 3.6/2). was zero. This is in stark contrast to normal realiza- tions of the QHE where the electrons are subjected to a uniform magnetic field, which implies the formation of mF,sLs ≈ 11 − 14, for the same ensembles mentioned Landau levels. In Haldane’s model, the electrons retain above. This is further proof that the corrections arising their Bloch character, and thus it is referred to as the from the finite spatial extent of the box are well under QHE without Landau levels. As in graphene, the points control. The dependence of the condensate on Nz, the in his model where the valence and conductance bands spatial direction perpendicular to the plane of graphene, touch inside the Brillouin zone are symmetry-protected was also checked. We found that for Nz ≥ 10, the cor- (inversion and time-reversal). If time-reversal invariance 11

0.0003 2 β Φ is broken, he found that a ν = ±1 integer QHE state is 8 x10x480,=0.80, B=0.125 formed, where ν is referred to as the filling factor. Recent studies of symmetry breaking in graphene have 0.0002 brought up the possbility of the formation of the time- 59,60,70 reversal-odd condensate, ∆H . This condensate gives rise to the so-called Haldane mass in the low-energy 0.0001 theory. The reason for the interest in these symme- H try breaking scenarios is as follows. The appearance ∆ of anomalous QHE states in graphene at previously un- Re 0 known filling factors, ν = 0, ±1, ±2, is due to the splitting of the degeneracy of the lowest Landau level (LLL) at large external magnetic fields (B ∼ 45T). Although -0.0001 there are several scenarios to explain these states, (see71 for a nice discussion), the authors of59,60 advocate the magnetic catalysis scenario whereby a Dirac mass as well -0.0002 as possibly a Haldane mass are generated due to the 0 0.01 0.02 0.03 0.04 0.05 0.06 strong, weakly-screened electron-electron interactions in m graphene. In the Schwinger-Dyson approach employed in59, FIG. 6: The real part of the Haldane mass as a function of the 2 where a simplified contact interaction was used, the fermion bare mass m for 8 × 10 × 480, β = 0.80, ΦB = 0.125. ground state of the EFT in the presence of an external magnetic field was found to be described by 0.004 ∆˜ ↑ = ∆˜ ↓ = 0, ∆H,↑ = −∆H,↓ = M, (59) ˜ where ∆σ corresponds to the Dirac mass for a given spin 0.002 projection, and ∆H,σ corresponds to the Haldane mass for a given spin projection. In this expression, M is the dynamically generated mass scale which is a function of H ∆ 0 the cutoff Λ and the dimensionful coupling constant characterizing the contact interaction, G. We note that the Re spin index in (59) directly corresponds to the taste index for staggered fermions. In60, a more realistic long-range -0.002 Coulomb interaction in the instantaneous approximation was used. The results were similar to that of59, with the qualitative difference that the gap parameters depend on -0.004 the Landau-level index. Although we ignore it in our lattice simulations, the 0 200 400 600 800 1000 Zeeman term plays a significant role in the selection of trajectory the ground state. According to59,60, the solution FIG. 7: Monte Carlo time history of the real part of the Haldane mass for 82 × 10 × 480, β = 0.80, m = 0.01, Φ = ∆˜ ↑ = ∆˜ ↓ = M, ∆H,↑ = ∆H,↓ = 0, (60) B 0.125. The errors on the points are the standard deviation of is degenerate with (59) in the absence of Zeeman split- the mean of the stochastic estimation of the Haldane mass. ting. This makes the comparison with continuum results more difficult and is an avenue for future work. In our simulations, we have used operators which are removing the explicit mass term after taking the infinite all taste singlets and thus have not distinguished the vari- volume limit. Using lattices generated with the action ous “spin” components. Taste-nonsinglet operators have (17), we can look at the ensemble average of the oper- zero expectation value as the vacuum does not have a ator as well as look for the appearance of a first-order preferred direction in taste space unless one introduces a phase transition in the time history of the taste-singlet taste-breaking term into the action. Thus, one is unable operator characterized by a tunneling between negative to project out the various spin components of the order and positive values. parameters, which makes a comparison with the contin- We have computed the taste-singlet time-reversal-odd uum results much more difficult. In light of this fact, we condensate on the lattice ensembles with the largest flux have looked at the taste-singlet Haldane mass character- ΦB = 0.125 for atT = [0.002, 0.016]. These values of ized by the operator Ψ¯ (˜γ4,5 ⊗ 1) Ψ. In contrast with the atT are smaller than the dynamical fermion mass, as one chiral condensate, we have not added a term like this to can see from Fig. 10. Thus, one expects that if the the action and thus cannot perform the same analysis of ground state did support a Haldane condensate, these 12

2 2 temperatures would be sufficiently small and the mag- 8 x10, T=0,β=0.80, N=0, O(e) 0.08 2 B netic flux sufficiently large to observe this. Instead, we 8 x10, T=0,B N=0 find that our data do not support a nonzero taste-singlet 0.07 free fermion pole Haldane condensate. In Fig. 6, we have plotted the real part of the Haldane mass, computed with 1000 stochas- 0.06 tic sources on 100 gauge configurations, versus m for 0.05

Nτ = 480, ΦB = 0.125. Our results are consistent F

with zero. We have also checked, at the same value m 0.04 of the flux and bare mass, that as the temperature increases, the Haldane mass remains consistent with zero 0.03 as to be expected from the solutions of the gap equa- 0.02 tions. In Fig. 7, we plot the Monte Carlo time history of the taste-singlet Haldane mass. The values are regu- 0.01 larly distributed around the mean and do not show any indication of tunneling, which would characterize a first- 0 order phase transition. Thus we do not see any evidence 0 0.01 0.02 0.03 0.04 0.05 for the formation of a taste-singlet Haldane condensate m in our calculations. Keeping in mind the limitations of our approach, one would need to perform further simu- FIG. 8: The fermion dynamical mass as a function of the bare lations with an explicit taste-singlet mass in the action fermion mass for zero magnetic flux. We have included the in order to test for spontaneous breaking of the discrete result for the pole of the fermion propagator at O(e2) (upper √ symmetry. curve) along with the free fermion pole, log m + m2 + 1 (lower curve). One can see that the perturbative result is close to our nonperturbative result. C. Dirac Quasiparticle explicitly checked that the propagator in the forward di- We also studied the fermion quasiparticle propagator. rection calculated with this setup gives the same results Due to the appearance of a nonzero value for the conden- ¯ for the quasiparticle mass as if one had ignored this back- sate hΨΨi, one expects that the dynamical mass remains ground field as long as the source and sink positions sat- nonzero in the limit that the bare mass vanishes. Al- isfy 0 < tsource < tsink < Nτ − 1 and tsink − tsource Nτ . though the propagator itself is gauge variant, the pole of This can be explained by the fact that the lattices used the propagator in Coulomb gauge is a familiar quantity. have a large temporal extent and thus thermal effects In order to extract information from the quasiparticle involving higher-order contributions are suppressed. propagator, we apply a transformation on the temporal To determine the dynamical fermion mass, mF , we fit links. Our gauge fixing procedure sets the average value the propagator in (53) to the following form of the potential in all but one time slice to zero, −mF τ τ −mF (Nτ −τ) X GF (τ, ~p = 0) = A e + (−) e . (64) A0(r, τ) = 0, (61) r The fits that we performed for the quasiparticle prop- where τ = 0, 1,...,Nτ − 2 and r ≡ (x, y, z). To do this, agator necessitated proper care in order to accurately we make the following substitution determine the ground state mass. To do this one must select a fit range, [τ , τ ], where the correlator is free 1 min max X of excited-state contributions. We have found that the A0(r, τ) → A0(r, τ) − 2 A0(r, τ). (62) N Nz s r selection of τmin primarily depends on the value of the external magnetic flux, ΦB. This is due to the fact that At the boundary of the lattice in the Euclidean time di- the excited state with the same oscillating behavior as rection, due to the periodic boundary conditions on the the ground state consists of the fermion promoted to the gauge potential, we must set next highest LL. One can also have an excited state consisting of a fermion along with a pseudoscalar particle A0(r, τ = Nτ − 1) → A0(r, τ) N −2 having nonzero orbital angular momentum. Our results 1 X Xτ for particle masses suggest that this contribution dies out − A (r, τ). (63) N 2N 0 quite quickly in Euclidean time, τ. s z r τ=0 At zero magnetic field, we expect the dynamical Thus, although one cannot enforce the condition hA0i = fermion mass to vanish in the chiral limit. In Fig. 8, one 0, one can effectively restrict this background field to lie can see our nonperturbative determination of the dynam- on the last time-slice of the lattice. Several earlier studies ical fermion mass. However, it appears that making an have noted this and dealt with the external field explic- extrapolation to zero bare mass, based on the nonper- itly in their extraction of the fermion mass72,73. We have turbative results, leaves us with a nonzero value for the 13

0.002 2 2 0.14 2 8 x10x120,β=0.80, N=0, O(e) 8 x10, T=0,Φ =0.125 B 2 B free fermion pole 12x10, T=0,Φ =0.083 0.12 2 B 8 x10, T=0,Φ =0.0625 2 B 12x10, T=0,Φ =0.055 0.0015 B 2 Φ 0.1 8 x10, T=0,B =0.0

0.08 F 0.001 F m m 0.06

0.04 0.0005

0.02

0 0 0 2e-05 4e-05 6e-05 8e-05 0.0001 0 0.01 0.02 0.03 0.04 0.05 0.06 m m

FIG. 9: The pole of the fermion propagator at O(e2) (upper FIG. 10: The dynamical fermion mass plotted as a function curve) along with the free fermion pole (lower curve) as one of the bare mass for zero external magnetic ﬂux and all four approaches the chiral limit, m → 0. One can see that the nonzero external magnetic ﬂuxes. To keep T/m small, we 2 pole at O(e ) vanishes in this limit, as expected. However, have chosen Nτ = 120, 240, 240, 240, 480 for the bare fermion the curvature that causes this behavior can be observed only masses m = 0.05, 0.02, 0.015, 0.01, 0.005, respectively. as one moves to extremely small fermion bare masses.

0.03 dynamical fermion mass. According to the arguments 0.025 and results previously discussed, one expects the dynamical mass to vanish in this limit as we are ﬁrmly in the 0.02 semimetal phase in the absence of the external magnetic ﬁeld. To investigate this result further, we calculated the 0.015 pole position of the lattice fermion propagator at O(e2) F

74 m using lattice perturbation theory . In this calculation 0.01 we employed a one-link staggered action with tadpole im- 2 provement determined at O(e ). As one can see in Fig. 8, 0.005 the perturbative result exhibits a significant renormaliza- tion of the mass compared with the free theory. 0 Furthermore, referring to Fig. 9, one can see that as one approaches the origin, the perturbative result shows -0.005 large curvature and eventually vanishes at the origin. Us- 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 ing this result as a heuristic explanation of our results at √eB zero magnetic field, one might expect the same to happen FIG. 11: The T = 0,√ chirally-extrapolated dynamical fermion in the chiral limit for a nonperturbative calculation. mass as a function eB. We have depicted a linear fit which gives an intercept which is consistent with zero (χ2/d.o.f. ≈ Fig. 10 shows results for the dynamical fermion mass 3.7/2). This confirms previous perturbative predictions. with nonzero and zero magnetic flux. One sees that at a given bare mass, the dynamical mass increases with the magnetic flux. Furthermore, the plot suggests that the values for all four nonzero magnetic fluxes extrapolate to renormalizaiton of the Fermi velocity v . nonzero values in the chiral limit. However, in light of F the behavior observed in the zero field case, one might As previously mentioned, perturbative approaches to want to be cautious in predicting the behavior at bare magnetic catalysis predict that for (2 + 1)-dimensional masses smaller than those plotted. For all of the ensem- field theories,√ the dynamical fermion mass scales linearly 61 bles depicted in Fig. 10, we have chosen an Nτ which with eB . In order to check this, one first needs to corresponds to a temperature where the chiral conden- extrapolate the dynamical mass to the chiral limit. Our sate plateaus. Thus, we expect that we are effectively results, depicted in Fig. 11, show that the chirally ex- at T = 0 when determining the dynamical mass. Fur- trapolated dynamical mass behaves as expected. This ther investigations of the fermion propagator are planned provides further evidence in favor of magnetic catalysis at nonzero spatial momentum where one can study the in the graphene EFT. 14

0.16 2 8 x10, T=0,Φ =0.125 the diﬃculties in studying the Haldane mass with stag- 2 B 12x10, T=0,Φ =0.083 0.14 2 B gered fermions. Although we found no evidence for the 8 x10, T=0,Φ =0.0625 2 B taste-singlet Haldane condensate, one must take into ac- 12x10, T=0,Φ =0.055 0.12 B count the limitations of our lattice setup in this regard. The evidence for a dynamically generated Dirac mass for 0.1 the quasiparticle, the NG boson, and the nonzero value

π for the chirally extrapolated, T = 0 chiral condensate all 0.08

m show that indeed, magnetic catalysis is occurring in the graphene EFT. Further studies of magnetic catalysis in 0.06 the graphene EFT could address other questions, such 68 0.04 as how the phase diagram presented in changes as one varies the temperature. This would necessarily involve 0.02 a scan in β which differs from our current study which was performed at fixed β = 0.80. Another direction one 0 could pursue would be to study the valley-sublattice and 0 0.01 0.02 0.03 0.04 0.05 0.06 spin densities which are order parameters for quantum m Hall ferromagnetism (QHF). In a study of the Schwinger- Dyson gap equations, it was found that the appearance of FIG. 12: The mass of the pseudoscalar bound state as a func- the condensates associated with magnetic catalysis and tion of the bare fermion mass for all four magnetic fluxes. those associated with QHF share the same dynamical origin59.

D. Pseudoscalar Acknowledgments The appearance of a pseudoscalar Nambu-Goldstone boson is expected as a consequence of the spontaneous chiral symmetry breaking. To determine this state’s mass This work was in part based on a variant of the MILC we fit the correlator in (55) to the following form collaboration’s public lattice gauge theory code. See http://physics.utah.edu/∼detar/milc qcd.html. (τ) Calculations were performed at the Center for High Per- G (τ; ~p = 0) = A e−mπ τ + e−mπ (Nτ −τ) PS formance Computing at the University of Utah, Fermi 0 −m0 τ −m0 (N −τ) National Accelerator Laboratory, and the LOEWE-CSC + A e π + e π τ , (65) high performance supercomputer of Johann Wolfgang Goethe-University Frankfurt am Main. S.Z. and C.W. where we have included two sets of exponentials char- would like to thank HPC-Hessen, funded by the State acterizing the ground state and the first excited state Ministry of Higher Education, Research and the Arts, respectively. In Fig. 12, we see the ground state mass for programming advice. Numerical computations have in the pseudoscalar channel, mπ, plotted as a function also used resources of CINES and GENCI-IDRIS as well of the fermion mass for the various magnetic fluxes. We as resources at the IN2P3 computing facility in Lyon. notice that the pseudoscalar mass shows little variation The authors would like to acknowledge discussions with with magnetic flux, which is not surprising since the pseu- Maksim Ulybyshev, Dima Pesin, Eugene Mishchenko, doscalar carries no charge. This is in contrast with QCD, and Kostas Orginos. SZ would like to acknowledge where electrically charged pions couple to the external 75 discussions with Wolfgang Unger. SZ would like to magnetic field . We also have determined that the pseu- acknowledge the support of the Alexander von Hum- doscalar mode is indeed a bound state. This can be seen boldt foundation and the National Science Foundation by referring to Fig. 10 where the dynamical fermion mass (USA) under grant PHY-1516509 and the Jefferson is plotted as a function of the bare mass. The fermion- Science Associates, LLC under U.S. DOE Contract antifermion scattering state has energy 2mF , which is #DE-AC05-06OR23177. CW and CD were supported higher than mπ for all simulated bare masses. by the US NSF grant PHY10-034278.

VII. CONCLUSION Appendix A: Spin-Taste Basis in (2 + 1) dimensions Through a thorough, fully nonperturbative study of the graphene EFT, we have shown the existence of spon- In this appendix we discuss the spin-taste basis in taneous symmetry breaking due to an external magnetic (2 + 1), dimensions which has some differences with the field. We have characterized the ground state of the sys- more familiar (3 + 1) dimensional case. The discussion tem by performing a zero-temperature extrapolation of follows that of76. Starting from the one-component fields our observables. Furthermore, we have commented on χ, χ¯ one performs a change of basis by introducing the 15

T following transformation where σµ refers to the transpose. The derivative operator now acts on a lattice of spacing 2a and is deﬁned as αa 1 X αa u (y) = √ Γη χη(y), (A1) 4 2 η 1 ∂µu(y) ≡ (u(y +µ ˆ) − u(y − µˆ)) . (A11) αa 1 X αa 2(2a) d (y) = √ B χη(y), (A2) 4 2 η η One can now deﬁne a four-component Dirac spinor as where one has introduced the matrices follows η0 η1 η2 Γη ≡ σ0 σ1 σ2 , (A3) u(y) η0 η1 η2 Ψ(y) = . (A12) Bη ≡ β0 β1 β2 , βµ ≡ −σµ. (A4) d(y)

We note that one labels a lattice site by nµ = 2yµ + ηµ, where y is an integer that labels the corner of the cube, Using the reducible set of gamma matrices in (4) and (5), µ one can write the action in the following compact form and ηµ = 0, 1 labels the sites within a cube. Using the † † identity Tr (ΓηΓη0 + BηBη0 ) = 4δηη0 , one can invert the 3 X relation in (A1) and (A2) to obtain Sst = (2a) Ψ(¯ y)(γµ ⊗ 1)∂µΨ(y) (A13) √ X ∗αa αa ∗αa αa y,µ χη(y) = 2 (Γη u (y) + Bη d (y)), (A5) α,a + aΨ(¯ y)(˜γ ⊗ σT )∂2Ψ(y) √ 5 µ µ X αa αa ¯αa αa χ¯η(y) = 2 (¯u (y)Γ + d (y)B ). η η 3 X ¯ α,a + (2a) m Ψ(y)(1 ⊗ 1)Ψ(y), y One can then rewrite the action in the spin-taste basis. For example, the mass term becomes Where the second derivative operator is defined as 3 X a m χ¯η(y)χη(y) = (A6) 2 1 y,η ∂ u(y) ≡ (u(y + 2ˆµ) + u(y − 2ˆµ) µ 4(2a)2 3 X ¯ (2a) u¯(y)(1 ⊗ 1)u(y) + d(y)(1 ⊗ 1)d(y) , − 2u(y)). (A14) y where we have used the following identities One sees from (A13) that the second derivative term, which is suppressed by a factor of the lattice spacing, is X αa ∗βb X αa ∗βb Γη Γη = Bη Bη = 4δαβδab, (A7) not invariant under a rotation in taste space. η η The residual symmetry of the staggered lattice action X αa ∗βb Γη Bη = 0. (A8) is U(1) × U(1). The form of these symmetries on the η one-component fields is as follows To rewrite the kinetic term, one first expresses the shifted χ(x) → exp(iα)χ(x), field as χ¯(x) → χ¯(x) exp(−iα), (A15) χ (y) = (A9) η+ˆµ χ(x) → exp(iβ(x))χ(x), √ † † χ¯(x) → χ¯(x) exp(iβ(x)), (A16) δηµ,0ηµ(η) 2 Tr Γηγµu(y) + Bηβµd(y)

√ where (x) ≡ (−1)x0+x1+x2 . In terms of the ﬁelds u and +δ η (η) 2 Tr Γ† γ u(y +µ ˆ) + B†β d(y +µ ˆ) . ηµ,1 µ η µ η µ d, these transformations become A similar expression exists for the backward shifted ﬁeld, u u χ (y), and thus one arrives at the following form for → exp(iα) , η−µˆ d d the staggered action u¯ d¯ → u¯ d¯ exp(−iα), (A17) Sst = u cos(β) i sin(β) u 3 X → , (2a) u¯(y)(σµ ⊗ 1)∂µu(y) d i sin(β) cos(β) d y,µ cos(β) i sin(β) ¯ T 2 u¯ d¯ → u¯ d¯ . (A18) + d(y)(βµ ⊗ 1)∂µd(y) + a[¯u(y)(1 ⊗ σµ )∂µd(y) i sin(β) cos(β) ¯ T 2 3 X + d(y)(1 ⊗ βµ )∂µu(y)] + (2a) m [¯u(y)(1 ⊗ 1)u(y) Thus, one can see that the formation of the condensate y hχχ¯ i spontaneously breaks the U(1) symmetry and leads + d¯(y)(1 ⊗ 1)d(y)], (A10) to the appearance of a single NG boson. 16

Appendix B: Fermion Bilinears where the ± refers to electron and hole states respectively, and φ ≡ tan−1 ky describes the orientation of ~k kx In our study we are looking for spontaneous symmetry the vector ~k in the plane. Applying the transformation in breaking in the presence of an external magnetic field. (B7) to (B3), one can verify that this term is odd under Typically, this will involve the breaking of the SU(2) σ time-reversal. symmetry, where SU(2) is the largest non-abelian sub- σ On the lattice, we must calculate the relevant conden- group of U(2)σ, described in (10). In this appendix, we list the expressions for the various bilinear operators in sates in the language of staggered lattice fermions. In terms of the degrees of freedom on the hexagonal lat- (2+1) dimensions, a bilinear in the spin-taste basis takes tice as well as their representation in terms of staggered the form lattice fermions. Ψ(¯ y) (Γ ⊗ Γ ) Ψ(y), (B10) We first introduce the Dirac mass term S T † ∆˜ σΨ¯ PσΨ = ∆˜ σΨ (γ0 ⊗ Pσ)Ψ, (B1) where the four-component Dirac spinor Ψ is defined in (3). From the above discussion, one can see that which is a triplet with respect to spin and breaks SU(2) σ the four-dimensional spin space can be mapped to the down to U(1) with the generatorγ ˜ ⊗ P . The corre- σ 4,5 σ four-dimensional sublattice-valley subspace and the two- sponding order parameter for this term is hΨ¯ P Ψi, and, σ dimensional taste space can be mapped to the two- written in terms of Bloch components it can be expressed dimensional spin of the electron. as The Dirac mass is the usual staggered mass (ΓS = ∆˜ : ψ† ψ − ψ† ψ Γ = 1), and takes the form σ K+Aσ K+Aσ K+Bσ K+Bσ T † † + ψ ψK Aσ − ψ ψK Bσ. (B2) K−Aσ − K−Bσ − Ψ(¯ y)(1 ⊗ 1) Ψ(y) =u ¯(y)(1 ⊗ 1) u(y) One can interpret a nonzero value for this order parame- + d¯(y)(1 ⊗ 1) d(y) ter as an imbalance of charge between the two sublattices, 1 X = χ¯ (y)χ (y), (B11) A and B, corresponding to a charge density wave (CDW). 8 η η The Haldane mass term is given by η ¯ † ∆σΨ (˜γ4,5 ⊗ Pσ) Ψ = ∆σΨ (γ0γ˜4,5 ⊗ Pσ)Ψ, (B3) where we have used the following identity which is a singlet with respect to spin but is odd under † † time-reversal. The order parameter for the Haldane mass Tr (ΓηΓη0 + BηBη0 ) = 4δηη0 . (B12) is hΨ¯ (˜γ ⊗ P )Ψi, and its expression in terms of Bloch 4,5 σ We note that in our lattice calculations, we do not iso- components is given by late the individual spin contributions to the condensates † † ∆ : ψ ψ − ψ ψ (ΓT = 1 i.e. Pσ → 1). σ K+Aσ K+Aσ K−Aσ K−Aσ † † The Haldane mass in the spin-taste basis takes the − ψ ψK Bσ + ψ ψK Bσ. (B4) K+Bσ + K−Bσ − following form Thus, this order parameter can be seen to represent a Ψ(¯ y) (˜γ ⊗ 1) Ψ(y) =u ¯(y)(1 ⊗ 1) u(y) (B13) charge imbalance between the two valleys, K+ and K−. 4,5 To discuss the properties of (B3) under time-reversal, we − d¯(y)(1 ⊗ 1) d(y), introduce the Hamiltonian of the low-energy theory i X = χ¯ (y)χ 0 (y),(B14) 8 η η H(~k) = v τ ⊗ ~σ · ~k, (B5) 0 ~ F 0 ηµ6=ηµ where τ0 is a two-dimensional unit matrix in valley space which is tensored with the two-dimensional sublattice where we have employed the identity space. Time-reversal invariance imposes the following re- 0 † † 4i, ηµ 6= ηµ, ∀µ striction on the states as well as the valley Hamiltonians Tr ΓηΓη0 − BηBη0 = . (B15) themselves 0, otherwise T ψ = ψ∗ = ψ , (B6) K+(A,B) K+(A,B) K−(A,B) Thus the Haldane mass involves a bilinear with the fields −1 ∗ residing on diagonally opposite sites within the cube. T HK T = H . (B7) + K− This operator is left invariant by the staggered lattice The relation in (B6) can be shown by inspecting the ex- action’s U(1) symmetry. plicit form of the valley Hamiltonian eigenstates The subject of time-reversal for staggered fermions in (2 + 1) dimensions also deserves a discussion. In terms −iφ~k/2 K+ 1 e ψe.h. = √ iφ /2 , (B8) of the underlying graphene lattice, time-reversal inter- ±e ~k 2 changes Dirac points, reverses spin, and leaves the sub- iφ~k/2 K− 1 e lattice degree of freedom intact. Ignoring spin for the ψe.h. = √ −iφ /2 , (B9) 2 ±e ~k moment and using the basis defined in (3), the action 17 of time-reversal in the four-dimensional sublattice-valley fermions in the spin-taste basis is as follows subspace is as follows Ψ(~y, y ) → −iγ˜ γ γ Ψ(~y, −y ),  ~   ~  0 5 1 2 0 ψAσ(K+ + ~p) ψAσ(K− − ~p) ~ ~ 0 σ0 u(~y, −y0)  ψBσ(K+ + ~p)   ψBσ(K− − ~p)  = ,  ~  →  ~  (B16) −σ0 0 d(~y, −y0)  ψBσ(K− + ~p)   ψBσ(K+ − ~p)  ~ ~ σ0d(~y, −y0) ψAσ(K− + ~p) ψAσ(K+ − ~p) = . (B25) −σ0u(~y, −y0) = γ1γ˜5Ψσ(−~p), (B17) ~ To see the effect of time-reversal on the one-component where ψK±,A/B,σ(~p) ≡ ψA/B,σ(K± + ~p), and we have written the spinor in momentum space. Time-reversal basis, we note the following identity acts on the spin degree of freedom in the following way √ u(y) χ (y) = 2 Tr Γ† ,B† , (B26) ψ ψ η η η d(y) K±,A/B,↑ → K±,A/B,↓ , (B18) ψK±,A/B,↓ −ψK±,A/B,↑ where we regard the spinor as a matrix with the Dirac in- ψ dex representing the row and the taste index representing = iσ K±,A/B,↑ . (B19) 2 ψ the column. Thus to find the action of time-reversal on K±,A/B,↓ the one-component spinor we must evaluate the following Combining the action of time-reversal on the four- expression dimensional sublattice-valley space, which composes the four-dimensional spinor structure, and on the † † σ0d(~y, −y0) Tr Γη,Bη , (B27) two-dimensional spin space, which composes the two- −σ0u(~y, −y0) dimensional “flavor” space, we obtain the following result in momentum space which represents the projection of the spin-taste basis onto the one-component basis. Using the expressions for Ψ(~p) → (γ γ˜ ⊗ iσ ) Ψ(−~p). (B20) 1 5 2 the spinors u and d in (A1) and (A2), one must compute † † Tr Γησ0Bη0 − Bησ0Γη0 . The only nonzero contribution For the coordinate space spinor, time-reversal takes the 0 following form comes when η =η ˜ ≡ (η0 ± 1, η1, η2), where the “+” corresponds to η0 = 0 and the “−” corresponds to η0 = 1. The result is thus Ψ(~x,t) → (γ1γ˜5 ⊗ iσ2) Ψ(~x, −t). (B21) √ One can then show that the continuum action in † † σ0d(~y, −y0) 2 Tr Γη,Bη Minkowski space is left invariant by the following −σ0u(~y, −y0) 77 transformations η0+η1+η2 = (−) χη˜(~y, −y0). (B28)

Ψ(~x,t) → (γ1γ˜5 ⊗ iσ2) Ψ(~x, −t), (B22) This implies that time-reversal induces the following Ψ(¯ ~x,t) → −Ψ(¯ ~x, −t)(γ γ˜ ⊗ iσ ) (B23) 1 5 2 transformation on the one-component staggered ﬁeld A0(~x,t) → −A0(~x, −t), (B24)

η0+η1+η2 where, for a fermion bilinear of the form Ψ¯ AΨ, due to the χη(~y, y0) → (−) χη˜(~y, −y0), (B29) fact that time-reversal, T , is an anti-unitary operator, one has T AT −1 = A∗. In Euclidean space, the time- where an analogous expression holds forχ ¯. Using the reversal transformation takes a diﬀerent form. In this expressions for the time-reversal transformation in the case, time is not distinguished from the spatial coordi- spin-taste as well as the one-compent basis, one can show nates by a relative minus sign in the metric. In (2+1) di- that the operators in (B13) and (B14) change sign and mensions, the time-reversal transformation on staggered are thus, indeed, time-reversal-odd.

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