An anomaly in quantum phases induced by borders

The stationary behavior of a quantum system is determined by its Hamiltonian and its boundary conditions. All quantum phase transitions (QPT) reported previously were induced by changing the Hamiltonian. In a circular spin model with Heisenberg XY interactions and no magnetic field, we observe an anomaly in quantum phases caused by a qualitative change of the boundary condition. The unexpected anomaly features an infinite number of single-particle levels, in the same pattern as the single-photon-triggered quantum phase transition in the Rabi model.

www.nature.com/scientificreports www.nature.com/scientificreports/ from the hard-core Boson Hubbard model having N sites with a single hard-core boson, i.e., the average particle occupation for each site is n 0 = 1/N. Thus in the thermodynamical limit N → ∞, the system approaches a Mott insulator with n 0 = 0. A complete analysis may be found in Method.
The Hamiltonian of the total system is written as where g is the exchange coupling strength between nearest neighbor spins, and X and Y indicate the Pauli matrices σ x and σ y , respectively. The index s is an arbitrary site number ranging from 1 to N. Units will be so chosen that the coupling strength g is equal to one in the following discussion. The full exciton number is conserved because n n tot , where σ ≡ Z n n z . The quantum phase transition of interest can be observed in the subspace with one exciton. Note that under such conditions the constituents at sites of the system can be replaced by harmonic oscillators (modelling coupled single-mode cavities) or other bosonic modes. Hence the interaction term for two sites can be expressed as + a a a a n n n n 1 1 , where a (a † ) is the annihilation (creation) operator. Our Hamiltonian (1) is reminiscent of many models (for example, ref. 26 ) that are used to study impurity or disorder effects in the one-dimensional XY model or Heisenberg model. In contrast to these works, the existence of an impurity in the bulk will destroy the U(1) and Z 2 symmetries in our Hamiltonian and the critical behavior observed here (the dramatic distinction between the ground states in the single-exciton subspace with J < 0 and J ≥ 0) has not been reported previously.

Results
Symmetry analysis of the system. In the language of network topology the model of Eq. (1) describes a hybrid system consisting of a circle or ring topology and a line segment or bus topology. Figure 1(a) illustrates that the connecting strengths between the nearest-neighbor sites are homogenous except that between the pair s and s + 1, which is equivalent to the addition of a small line segment described by H(J). In the following analysis, we choose s = N (then s + 1 = 1) with no loss of generality, i.e., we concentrate on the boundary condition between the neighbor sites 1 and N in periodic condition. When J = −1 (in units where g = 1), the whole system is an open-ended line segment. When J = 0, the system becomes a circle that is translational-invariant along the sites. The line and the circle are not homeomorphic because the line can be disconnected by removing one site but the circle can not. The circle and the line however are locally homeomorphic only when they have an infinite number of sites. When J > 0 and −1 < J < 0 the system has no translational invariance. Thus our anomaly in quantum phases is induced by strengthening the boundary sites in the close-ended condition. In contrast, in the single-photon-triggered Rabi model 20,21 , the phase transition occurs in the limit that the energy splitting of the qubit is much larger than that of the bosonic mode. Thus in the strong-coupling regime the population of the www.nature.com/scientificreports www.nature.com/scientificreports/ bosonic mode in the ground state can be enhanced suddenly from zero when the coupling strength reaches the critical value.
For the physical realization of our model, as J is increased from −1 to positive values the system can be regarded as a spin-line segment with increased bending [see Fig. 1(b)]. This effectively changes the boundary condition of the system because of the increasing coupling between the spins at the two ends. This behavior may be expected to occur for any physical system where the site-site interaction is proportional to certain powers of the site separation distance. For example, a Coulomb-like interaction is inversely proportional to the square of the distance between the subsystems and a 10% variation in the site-site distance yields a greater than 25% variation in the mutual coupling strength. Then when the one-dimensional system is folded as in Fig. 1(b) until its two ends approach each other, their mutual interaction depends sensitively on their separation distance. Thus the model in Eq. (1) can also be used to investigate the effect of boundary conditions. In a large system with N sites the boundary terms are conventionally expected to be of order 1/N, and thus to have negligible influence on macroscopic physical quantities. However, we find in this model that microscopic boundary conditions can have a dramatic effect because they can lead to an anomaly in quantum phases.
Henceforth we confine attention to the effect of the control parameter J in Eq. (1). Roughly the passage from J = −1 to J = 0 is expected to be connected closely to the transition from one well-defined topology to another well-defined topology, while the passage from J = 0 to J > 0 corresponds to a transition from a well-defined topology to a hybrid one. As will now be demonstrated, we observe an anomaly in quantum phases around J = 0 in this system that is reflected in the singularity properties of the ground and the first-excited states.

Ground state properties.
To understand the singularity in the ground state, we first consider the dependence of the ground state energy on system size, which is a functional of the boundary condition parameterized by J, or the extra strength between two selected neighbor sites 1 and N. We display the two lowest derivatives of the ground state energy E 0 with respect to J ∈ [−1,1] in Fig. 2(a-d) for systems with N = 50, 100, 1000, and 10,000 sites, respectively. The variation of the energies for the ground and the first excited states as functions of J indicates an anomalous transition from a gapless phase to a gapped phase. Neither derivative changes significantly until J approaches zero. The first derivative ∂ E J 0 exhibits a rapid decrease with increasing J after passing through J = 0, while the second derivative ∂ E J 2 0 2 exhibits a cusp at J = 0, with both behaviors becoming more sharply defined as the site number is increased. Thus, the variation of the energy derivatives indicates that the model experiences an 1 , which exhibits the finite-size scaling shown in the lower-left inset of Fig. 3. The critical exponent v is found numerically to be around 0.63. Even more interesting, after J increases through the critical point J = 0 the ground state wavefunction jumps immediately to another state ψ | 〉 > J 0 that is fully orthogonal to ψ | 〉 −1 (their overlap vanishes) and nearly orthogonal to ψ | 〉 = J 0 (their overlap is also very close to zero). Thus Fig. 3 indicates unequivocally an anomalous transition at J = 0 involving a level crossing in which the ground state is replaced by a new ground state essentially orthogonal to the original state. Combining these results with those for the derivatives of the ground state energy in Fig. 2, this new kind of anomaly can be categorized as a single-exciton quantum phase transition that is of second order.

Discussion
The preceding results are remarkable in that the anomaly in quantum phases may be attributed entirely to a microscopic boundary condition of the quantum system, without invoking any internal or applied external fields. According to the Hellmann -Feynman theorem 27 , where ψ | 〉 J denotes the ground wavefunction of the system with special J. Thus the expectation value of the boundary interaction term X 1 X N + Y 1 Y N may be taken as the order parameter, in analogy with the single-photon-triggered quantum phase transition in the Rabi model. The dimensionless interaction strength J between a single pair of sites along the circle, which is a quantity that could be modified easily in experiments, previously was thought to be unimportant in the thermodynamical limit N → ∞.
Therefore one can see that the transition of the system from a chain to a circle as J varies from J = 1 (a mixture of line-segment and circle topologies) to J = 0 (a perfect circle) causes no phase transition due to the invariance of the ground state. However, the transition of the boundary condition from a state with translation symmetry (J = 0) However, this result will fail as J → 0 + from J ≤ 0, as indicated by Fig. 2. In the limit N → ∞, the ground state becomes the ground state of . The population distributions over the bases of these two ground states with respect to J are obviously different, as shown in Fig. 4. For large N one finds that 〈 + 〉 = − ψ Z Z 2 N 1 J for −1 < J ≤ 0, which means that population of the spin-flip on the boundary (the two ends) is negligible in this conventional range. In contrast, 〈 + 〉 ψ Z Z N 1 J increases suddenly at the critical point J = 0 + , and this effect on the boundary becomes sharper as the size of system increases. Thus the system exhibits a clear transition from a delocalized phase to a localized one at the critical point, analogous to that found in the Rabi model 28 .
Our work also shows that the anomaly in quantum phase can be viewed as a transition between a gapless phase and a gapped one. Figure (5) provides a detailed plot of the eigen-energy structure for our model with N = 10 3 . One can clearly see that the energies of the states labelled E 1 to E 4 are quasi-degenerate for all J, while the ground-state energy E 0 (see the dashed red line) drops suddenly just when J moves over 0 + along the direction from the negative to the positive axis.  www.nature.com/scientificreports www.nature.com/scientificreports/ The microscopic details of the individual building blocks of a system generally are believed to not be important to quantum phase transitions. Rather, the collective behavior is thought to be controlled by general properties of the interaction between the building blocks (spins or harmonic oscillators), and the quantum phase transition is induced by tuning an "external" parameter (field strength, pressure, doping concentration, …). We have exhibited here a new class of anomalies in quantum phases induced by a microscopic boundary condition as well as translational symmetry in a finite-size, one-dimensional spin model with no external field applied that defies this common understanding. The anomaly in quantum phases exhibited here does not require a noncommuting condition between different constituents of the Hamiltonian. At the critical point, we demonstrate a nonanalytic behavior of the second-order derivative of the ground eigen-energy and a sudden change of the ground state wavefunction. This special anomaly in quantum phases might also be understood using concepts developed for symmetry-protected topological phases except that the system analyzed here has a local order parameter. With respect to the analytical result, we apply the Jordan-Wigner transformation to get the ground state (3) when the additional boundary coupling strength J ≤ 0, while have to resort to the numerical simulation when J moves over the critical point 0 + to the other phase regime. It is then interesting to explore the effects of boundary condition in other strongly correlated systems in the future with the numerical methods, such as the Monte Carlo simulation 29 and the dynamical mean-field theory 30 .

Method
Optical lattices of ultracold atoms offer a realization of the boson Hubbard model, and exhibit the superfluid -insulator transition 31 . Here we shall present a hidden anomaly in quantum phases in a model that could be realized by a relevant physical system described by a one-dimensional boson Hubbard model; it is of interest for quantum computation, quantum information, and ultra-cold atoms 4,32 . The Hamiltonian of the boson Hubbard model neglecting off-site and longer-range repulsions consists of three terms, n n n n n n n n n n 1 Here the first term allows site-hopping of the bosons through the creation and and annihilation operators † b n and b n , with g the hopping matrix element, μ in the second term represents the chemical potential of bosons, and the on-site repulsion U in the third term sets the energy scale for the problem. By relying on specific physical conditions, a given system can be constrained to have a fixed total number of bosons. We study in this work a hard-core boson Hubbard model having N sites, with a single hard-core boson that can also be represented as a spin-1 2 particle. In the thermodynamical limit N → ∞ the system approaches a Mott insulator 33 . For an open-ended chain or a closed homogeneous ring, the Heisenberg XY model can be solved analytically using a Jordan -Wigner transformation [34][35][36] , which converts a spin-1 2 system into a free spinless fermion chain with nearest-neighbor hopping.
For our model, in the limit of hard-core bosons on every site of the lattice in a Hubbard model, only the two Mott insulator states with n 0 = 0 or n 0 = 1 are permitted. This model can be mapped to a magnet of S = 1/2 spins with nearest-neighbor exchange interactions. The Hamiltonian is written as Under the periodic boundary condition, the Hamiltonian (5) for the one-dimensional case can be rewritten as  mod N if n ↓ is even. Up to a constant number, we have