Quantum phase transition without gap opening
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Quantum phase transition without gap opening
X. M. Yang , G. Zhang and Z. Song ∗ School of Physics, Nankai University, Tianjin 300071, China College of Physics and Materials Science, Tianjin Normal University, Tianjin 300387, China
Quantum phase transitions (QPTs), including symmetry breaking and topological types, alwaysassociated with gap closing and opening. We analyze the topological features of the quantum phaseboundary of the XY model in a transverse magnetic field. Based on the results from graphs in theauxiliary space, we find that gapless ground states at boundary have different topological characters.On the other hand, In the framework of Majorana representation, the Majorana lattice is shownto be two coupled SSH chains. The analysis of the quantum fidelity for the Majorana eigen vectorindicates the signature of QPT for the gapless state. Furthermore numerical computation showsthat the transition between two types of gapless phases associates with divergence of second-orderderivative of groundstate energy density, which obeys scaling behavior. It indicates that a continuesQPT can occur among gapless phases. The underlying mechanism of the gapless QPT is alsodiscussed. The gap closing and opening are not necessary for a QPT.
I. INTRODUCTION
Understanding phase transitions is one of challeng-ing tasks in condensed matter physics. No matterwhich types of a quantum phase transition (QPT),the groundstate wave function undergoes qualitativechanges . There are various signature phenomena mani-fest the critial points, such as symmetry breaking, switchof topological invariant, divergence of entanglement, etc..Among them, energy gap closing and opening seem neverabsent. It is important for a deeper understanding ofQPTs to find out the role of the energy gap takes dur-ing the transition. Usually, a continuous QPT is char-acterized by a divergence in the second derivative ofthe groundstate energy density, assuming that the firstderivative is discontinuous . A natural question iswhether the gap closing and opening are necessary fora QPT. The aim of this paper is to clarify the rela-tion between gap and a second-order QPT through aconcrete system. We take a 1D quantum XY modelwith a transverse field, which can be mapped onto thesystem of spinless fermions with p -wave superconductiv-ity. It plays an important role both in traditional andsymmetry-protected topological QPTs, received intensestudy in many aspects .The quantum phase boundary of the model is wellknown based on the exact solution. However, it mainlyarises from the condition of zero energy gap. There aremany other signatures to identify the critical point, suchas the quantities of the ground state from the point ofview of quantum information theory , the ground-statefidelity susceptibility . Then the ground states atdifferent locations of the boundary may belong to differ-ent quantum phases, although they are not protected bya finite energy gap. In this paper, we are interested in thepossible QPT along the boundary, at which the groundstate is always gapless state. Our approach consists ofthree steps: Bogoliubov energy band, Majorana fidelity,and finite-size scaling. Firstly, we use Bogoliubov energyband to construct a group of graphs that can capture the charaters of quantum phases in every regions, includ-ing all the boundaries, which indicate the distinctions ofboundaries. Secondly, we employ the fidelity of Majo-rana eigen vector to detect the QPT between two gap-less phases. Thirdly, we investigate the scaling behaviorof the critical region to show a gapless QPT has the sameperformance as a standard continuous QPT. The resultindicates that a continues QPT can occur among gaplessphases. The gap closing and opening are not necessaryfor a QPT.This paper is organized as follows. In Section II, wepresent the model Hamiltonian and the quantum phasediagram. In Section III, we investigate the phase diagrambased on the geometric properties. Section IV gives theconnection between the model to a simpler lattice modelby Majorana transformation. In Section V, we presentthe scaling behavior about the groundstate energy den-sity to demonstrate the characteristic of continuous QPTamong the gapless phases. Finally, we give a summaryand discussion in Section VI. II. MODEL AND PHASE DIAGRAM
We consider a 1D spin-1 / XY model in a transversemagnetic field λ on N -site lattice. The Hamiltonian hasthe form H = N X j =1 (cid:18) γ σ xj σ xj +1 + 1 − γ σ yj σ yj +1 + λσ zj (cid:19) , (1)where σ αj ( α = ± , z ) are the Pauli operators on site j ,and satisfy the periodic boundary condition σ αj ≡ σ αj + N .Now we consider the solution of the Hamiltonianof Eq. (1). We start by taking the Jordan-Wignertransformation σ xj = − j − Y i =1 (cid:16) − c † i c i (cid:17) (cid:16) c † j + c j (cid:17) σ yj = − i j − Y i =1 (cid:16) − c † i c i (cid:17) (cid:16) c † j − c j (cid:17) σ zj = 1 − c † j c j (2)to replace the Pauli operators by the fermionic operators c j . The parity of the number of fermionsΠ = N Y l =1 ( σ zl ) = ( − N p (3)is a conservative quantity, i.e., [ H, Π] = 0, where N p = P Nj =1 c † j c j . Then the Hamiltonian (1) can be rewrittenas H = X η =+ , − P η H η P η , (4)where P η = 12 (1 + η Π) (5)is the projector on the subspaces with even ( η = +) andodd ( η = − ) N p . The Hamiltonian in each invariantsubspaces has the form H η = N − X j =1 ( c † j c j +1 + γc † j c † j +1 ) − η ( c † N c + γc † N c † )+H . c . − λ N X j =1 c † j c j + N λ. (6)Taking the Fourier transformation c j = 1 √ N X k ± e ik ± j c k ± , (7)for the Hamiltonians H ± , we have H η = − X k η [2 ( λ − cos k η ) c † k η c k η + iγ sin k η ( c − k η c k η + c †− k η c † k η ) − λ ] , (8)where the momenta k + = 2 ( m + 1 / π/N , k − =2 mπ/N , m = 0 , , , ..., N −
1. Employing the Bogoli-ubov transformation γ k η = cos θ k η c † k η + i sin θ k η c − k η , l o g N (cid:1) N (cid:1) N (cid:1) (cid:1) o (I) (II) (III) abc gihedf N (cid:1) (III) FIG. 1. (Color online) Phase diagram of the XY spin chainon the parameter λ − γ plane. The blue lines indicate theboundary, which separate the phases with winding number 0(yellow) and non-trivial topological phase with winding num-ber 1 (green). The green lines separate the phases with wind-ing number 0 (yellow) and non-trivial topological phase withwinding number − − a ) − ( i ) at typical positions are indicated. The correspondingloops about the vector B are given in Fig. 2. where tan (cid:0) θ k η (cid:1) = γ sin k η λ − cos k η , (9)one can recast Hamiltonian H η to the diagonal form H η = X k η ǫ k η ( γ † k η γ k η −
12 ) , (10)with spectrum being ǫ k η = 2 q ( λ − cos k η ) + γ sin k η . (11)The lowest energy in η subspace is − P k η ǫ k η for λ < − P k η ǫ k η + − η ǫ for λ >
1. The groundstateenergy for finite N is the foundation for the analysis ofscaling behavior in the Sec. V. In the thermodynami-cal limit, the difference between two subspaces can beneglected and the groundstate energy density can be ex-pressed as ε g = − π Z π − π ǫ k dk, (12)by taking k = k η . The quantum phase boundary can beobtained from ǫ k = 0 as λ = ± , and γ = 0 for | λ | < . (13)The phase diagram is presented in Fig. 1. There arefour regions separated by five lines as boundaries of quan-tum phases. In the next section, quantum phases andboundaries will be examined from the geometrical pointof view. III. TOPOLOGICAL INVARIANTS
In this section, we will investigate the topological char-acterization for the phase diagram. We demonstrate thispoint by rewriting the Hamiltonian in the form H = X k> (cid:0) c † k c − k (cid:1) h k (cid:18) c k c †− k (cid:19) , (14)where h k = 2 (cid:18) cos k − λ iγ sin k − iγ sin k λ − cos k (cid:19) . (15)The core matrix can be expressed as h k = B ( k ) · σ k , (16)where the components of the auxiliary field B ( k ) =( B x , B y , B z ) are B x = − γ sin kB y = 2 cos k − λB z = 0 . (17)The Pauli matrices σ k are taken as the form σ x = (cid:18) − ii (cid:19) , σ y = (cid:18) − (cid:19) , σ z = (cid:18) (cid:19) . (18)The winding number of a closed curve in the auxiliary B x B y -plane around the origin is defined as N = 12 π I C ( ˆ B y d ˆ B x − ˆ B x d ˆ B y ) (19)where the unit vector ˆB ( k ) = B ( k ) / | B ( k ) | . N is aninteger representing the total number of times that curvetravels counterclockwise around the origin. Actually, thewinding number is simply related the loop described byequation ( B x ) γ + ( B y + 2 λ ) , (20)which presents a normal ellipse in the B x B y -plane.The shape and rotating direction of the ellipse dic-tated by the parameter equation (17) have the follow-ing symmetries. First of all, taking γ → − γ , wehave [ B x ( k ) , B y ( k )] → [ B x ( − k ) , B y ( − k )], representing (a) (b) (c)(i)(g)(d) (e) (f)(h) N (cid:1) N (cid:1) N (cid:1) N (cid:1) N (cid:1) (cid:1) FIG. 2. (color online) Schematic illustration of nine typesof phases by the geometry of graphs in the auxiliary B x B y -plane. Red filled circle indicates the origin. (a), (b) and (c)present graphs of trivial topological phase with zero windingnumber. (g) and (i) present graphs of non-trivial topologicalphases with winding number ±
1, which are separated by (h).(d), (e) and (f) are graphs of boundary. They correspond toellipses with various shapes, but passing through the origin.We see that from (d) to (f), the graph becomes a segment.Although (d) and (f) have the same shape, they have oppositedirections, indicating two different types of gapless phases. the same ellipse but with opposite rotating direction.Secondly, taking λ → − λ , we have [ B x ( k ) , B y ( k )] → [ B x ( k ) , B y ( k ) + 4 λ ], representing the same ellipse butwith a 4 λ shift in B y , while the 4 λ shift cannot affectthe relation between the graph and the origin. We plotseveral graphs at typical positions in Fig. 2 to demon-strate the features of different phases from the geometricpoint of view. We are interested in the loops for the pa-rameters at the boundaries in Eq. (13). (i) For λ = ± γ = 0, the ellipse always passes through the origin(0 ,
0) one time. Along the boundary, only the length ofthe semiaxis of the ellipse changes. (ii) For γ = 0 and | λ |
1, the loop reduces to a segment, passing throughthe origin twice times. Along the boundary, only thelength of the segment varies. According to the connectionbetween loops and QPTs , when a loop passes throughthe origin, the first derivative of groundstate energy den-sity experiences a non-analytical point. In general, thisprocess is associated with a gap closing. However, wenote that when parameters vary along λ = ± l- g+ g- ia b ia N ia - N ia b N b - N b FIG. 3. (Color online) Lattice geometries for the MajoranaHamiltonian described in Eq. (26), Solid (empty) circle indi-cates (anti) Majorana modes. It represents two-coupled SSHchains. through γ = 0, there is always one point in the curve atthe origin, while the ellipse becomes a segment. In theAppendix, we show that such a system also experiences anon-analytical point but without associated gap closingand opening. IV. MAJORANA LADDER
In this section, we investigate the phase diagram fromalternative way, which gives a clear physical picture byconnecting the obtained results to the previous work. Incontrast to last section, where a graph is extracted fromthe Bogoliubov energy band, we will mark the phase dia-gram from the behavior of wave functions. For quantumspin model, Majorana representation always make thingssimpler since it can map a Kitaev model (like the formin Eq. (8)) to a lattice model in a real space with thetwice number of site of the spin system . The bulk-edgecorrespondence has demonstrated this advantage .We introduce Majorana fermion operators a j = c † j + c j , b j = − i (cid:16) c † j − c j (cid:17) , (21)which satisfy the relations { a j , a j ′ } = 2 δ j,j ′ , { b j , b j ′ } = 2 δ j,j ′ , { a j , b j ′ } = 0 , a j = b j = 1 . (22)The inverse transformation is c † j = 12 ( a j + ib j ) , c j = 12 ( a j − ib j ) . (23)Then the Majorana representation of the Hamiltonian H η is H η = i N − X j =1 [(1 + γ ) b j a j +1 + (1 − γ ) b j +1 a j ] − iη γ ) b N a + (1 − γ ) b a N ]+ iλ N X j =1 a j b j + H . c .. (24) -0.5 0 0.5 γ F i de li t y ∆ γ =0.005 0.010.015 0.020.025(a)-0.5 0 0.5 γ F i de li t y ∆ γ =0.005 0.020.0150.025(b) FIG. 4. (Color online) The fidelity of the Majorana latticewith λ = − N = 20, (b) N = 100and various ∆ γ . To make the structure of Majorana lattice clear, we writedown the Hamiltonian in the basis ϕ T = ( ia , b , ia , b ,ia , b , ... ) and see that H η = ϕ T h η ϕ, (25)where h η represents a 2 N × N matrix. Here matrix h M is explicitly written as h η = 14 N − X l =1 [(1 + γ ) | l, i h l + 1 , | + (1 − γ ) | l + 1 , i h l, | ] − η
14 [(1 + γ ) | N, i h , | + (1 − γ ) | , i h N, | ]+ λ N X l =1 | l, i h l, | + H . c .. (26)where basis {| l, σ i , l ∈ [1 , N ] , σ = 1 , } is an or-thonormal complete set, h l, σ | l ′ , σ ′ i = δ ll ′ δ σσ ′ . The FIG. 5. (Color online) The Laplacian of the groundstate energy density ε g as a function of the field for the case N = 8 ,
40 and68. The maxima (ridge) mark the pseudo-boundary quantum phases. We see that there are peaks on the ridge near the jointsfor the case of N = 68 (see also Fig. 6(a)) . basis array is ( | , i , | , i , | , i , | , i , ... | N − , i , | N − , i , | N, i , | N, i ), which accords with ϕ T .Schematic illustrations for structures of h η are describedin Fig. 3. The structure is clearly two coupled SSHchains. The situation with η = + corresponds to thecase with half quanta flux through the double SSH rings.In the previous work , the gapless states in two coupledSSH chains with η = − has been studied. It is shownthat the quantum phase boundary γ = 0 for | λ | < λ = ± γ, λ ) = (0 , ±
1) are boundaries separated the gap-less phases with different topological characterizations.We refer these points as transition point between twogapless phases.Majorana matrix in Eq. (26) with 2 N dimension con-tains all the information of the original H in Eq. (1) with2 N dimension. Although the eigen vectors of h η have di-rect relation to the ground state of H , it is expectedthat the signature of QPT between gapless phases canbe manifested from them. We will investigate the changeof the eigen vector of h + along λ = −
1. By the similarprocedure, the eigen problem of the equation h + | k + , ρ i = ǫ k + ,σ | k + , ρ i (27)with ρ = ± , can be solved as | k + , ρ i = 1 √ N X l ( e ik + l | l, i + ρe − iφ e ik + l | l, i ) , (28)with the eigen value ǫ k + ,ρ = ρ q (cos k + + 1) + ( γ sin k + ) . (29)Here the parameter φ is defined astan φ = − γ tan k + . (30)We focus on the vectors {| k + , −i} with negative eigenvalues and taking | k + , −i = | k + i . We employ the quantum fidelity to detect the suddenchange of the eigen vectors, which is defined as F ( γ, ∆ γ ) = Y k + O k + = Y k + |h k + ( γ − ∆ γ ) | k + ( γ + ∆ γ ) i| , (31)i.e., the modulus of the overlap between two neighborvectors with γ ± ∆ γ . Direct derivation shows that O k + ≈ − (cid:12)(cid:12)(cid:12)(cid:12) ∂ | k + ( γ ) i ∂γ (cid:12)(cid:12)(cid:12)(cid:12) (∆ γ ) = 1 − tan k + (cid:16) γ tan k + (cid:17) (∆ γ ) . (32)We note that the minimum of O k + always locates at γ = 0 for any values of k + , leading to the minimumof F ( γ, ∆ γ ). It indicates that the fidelity approach canbe employed for Majorana eigen vector to witness thesudden change of ground state.Fig. 4 shows the fidelity of h + for a given finite systemas a function of γ with various parameter difference ∆ γ .As expected, the point γ = 0 is clearly marked by asudden drop of the value of fidelity. The behavior canbe ascribed to a dramatic change in the structure of theMajorana vector, indicating the QPT at zero γ . V. SCALING BEHAVIOR
In this section, we investigate what happens for thegroundstate energy when a gapless QPT occurs. Thenon-analytical point of energy density is more fundamen-tal to judge the onset of a QPT. We start from the linewith λ = 1, on which the density of groundstate energyin thermodynamic limit has the form ε g = − π Z π q (1 − cos k ) + γ sin k d k, (33) -1.5 0 1.5 γ ∇ ǫ g N=8N=40N=68 1.5 3 4.5 ln N l n ( ∇ ǫ g ) m numerical datalinear fitting1.5 3 4.5 ln N -4-3-2-1 l n γ m numerical datalinear fitting(a) (b) (c) FIG. 6. (color online). The characteristics of second-order QPT for the present system. (a) Plots of of ▽ ε g as a functionof γ for different values of N at λ = 1. (b) The scaling law of pseudo critical point γ m as a function of N . (c) The scalingbehavior for ( ▽ ε g ) m , which is plotted as a function of N . The plots are fitted by solid lines ln γ m = − .
897 ln N + 0 .
456 andln (cid:0) ▽ ε g (cid:1) m = 0 .
874 ln N − . where we neglect the difference between k + and k − . Thefirst derivative of groundstate energy density with therespect to γ reads ∂ε g ∂γ = Z π ̥ ( k ) d k (34)where the integrand is defined as ̥ ( k ) = − γ sin kπ q (1 − cos k ) + γ sin k . (35)We are interested in the divergent behavior of ∂ ε g ∂γ when γ ∼
0. We note that the main contribution to the integralof ∂ ̥ ∂γ comes from the region k ∈ [0 , δ ] with δ ≪ π . Thecontribution to ∂ε g ∂γ from this region is approximately Z δ ̥ ( k ) d k ≈ | γ | γπ [1 − s (cid:18) δ γ (cid:19) ] , (36)which predicts that the first derivative of groundstateenergy density along λ = 1 has a non-analytical point at γ = 0. It is the standard characterization of second orderQPT. It is crucial to stress that such phase separationdoes not arise from the gap closing and opening.To characterize the behavior of groundstate energy in2D parameter space, we calculate the Laplacian of ε g ▽ ε g = ∂ ε g ∂λ + ∂ ε g ∂γ , (37)which will reduce to second derivative of the ground-state energy density of the standard transverse-field Isingmodel with respect to the transverse field λ when wetake γ = 0. In Fig. 5 we plot the Laplacian of ε g for the finite sized systems. We observe that as N increases, the regions of criticality are clearly markedby a sudden increase of the value of ▽ ε g . Remark-ably, there are higher order sudden increases around thepoints ( γ, λ ) = (0 , ± λ = ± γ = 1, we ascribe this type of behavior toa dramatic change in the structure of the gapless groundstate. The system undergoes a QPT along the boundary.In order to quantify the change of the ground statewhen the system crosses the critical point, we look at thevalue of ▽ ε g as a function of ( γ, λ ) for finite size system.The results for systems of different size are presented inFig. 6. We find the similar scaling behavior for suchkinds of QPT, which reveals the fact that the signatureof a second order QPT must not require the gap closing. VI. SUMMARY AND DISCUSSION
In summary, we have studied the the necessity of theenergy gap closing and opening for the presence of aQPT. We focused on the joint of three types of gap-less phases. The analysis based on the geometry ofthe ground state and Majorana representation indicatethe distinction of the gapless phases. Numerical com-putation for finite size system shows that the transitionsamong the gapless phases exhibit scaling behavior, whichhas been regarded as the fingerprint of continuous QPT.This provides an example to demonstrate that energy gapclosing and opening is not a necessary condition for theQPT. Mathematically, the existence of a gap dependson the function of dispersion relation, specifically, theupper bound of the negative band. The divergence ofthe second-order derivative of groundstate energy densityarises from the non-analytical point of the groundstateenergy, which is the summation of all negative energy k e g (a) k „ k = (b) k e g FIG. 7. (Color online) Energy spectra for the Hamiltonian P h k = γ ( k + k ) σ x as a function of γ , with (a) k = 0, and(b) k = 0, respectively. We see that the groundstate energiesof both cases have a non-analytical points at γ = 0. However,the gap closes only at γ = 0 in the case (a). levels. On the other hand, a typical negative energy levelwith non-analytical point is a simple level crossing at zeroenergy, for example, levels from matrix h k = γ ( k + k ) σ x for k, k ∈ [0 , π ]. The energy gap ∆ = 2 γk , which iszero only at γ = 0, for nonzero k . However, the energygap always vanishes for zero k . We plot the spectrumfor k = 0 and k = 0, in Fig. 7 to illustrate this point. VII. APPENDIX
In this appendix, we demonstrate the existence ofgapless QPT through a toy model. We consider aHamiltonian with a specific dispersion relation ε k = p x ( k ) + y ( k ) for Bogoliubov band, where ( x ( k ) , y ( k ))describes a rectangle with width 2 | γ | and length x in xy -plane (see Fig. 8). Here we use a rectangle to replacea ellipse in Eq. (20) in order to simplify the derivation.In the case of x ≫ | γ | , the main contribution to the en-ergy is the ε k at the long sides. The parameter equationsof the two long sides is (cid:26) x = − | γ | tan k, ( θ k π/ y = ± γ, (38)where θ is determined by tan θ = | γ | /x . The energy o x y ( , ) x γ− θ (0, ) γ FIG. 8. (Color online) Schematics of a graph in auxiliary xy -plane for the Bogoliubov band of a toy model, which is arectangle. As γ varies, the origin is always at one side of therectangle, keeping the ground state fixed at gapless state. density can be expressed as ε = − π Z π/ θ | γ | sin k d k = | γ | π ln tan( θ /
2) = | γ | π ln | γ | x . (39)We note that the derivative of ε∂ε∂γ = γπ | γ | (ln | γ | x + 1) , (40)has a jump at zero γ , indicating the critical point of QPT. ACKNOWLEDGMENTS
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