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Confinement-Deconfinement Interplay in Quantum Phases of Doped Mott

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Confinement-Deconfinement Interplay in Quantum Phases of Doped Mott Confinement-deconfinement interplay in quantum phases of doped Mott insulators Peng Ye1, Chu-Shun Tian1;2, Xiao-Liang Qi3;4 and Zheng-Yu Weng1 1 Institute for Advanced Study, Tsinghua University, Beijing, 100084, P. R. China 2 Institut f¨urTheoretische Physik, Universit¨atzu K¨oln,D-50937 K¨oln,Germany 3 Microsoft Research, Station Q, Elings Hall, University of California, Santa Barbara, CA 93106, USA 4 Department of Physics, Stanford University, Stanford, CA 94305, USA It is generally accepted that doped Mott insulators can be well characterized by the t-J model. In the t-J model, the electron fractionalization is dictated by the phase string effect. We found that in the underdoped regime, the antiferromagnetic and superconducting phases are dual: in the former, holons are confined while spinons are deconfined, and vice versa in the latter. These two phases are separated by a novel phase, the so-called Bose-insulating phase, where both holons and spinons are deconfined. A pair of Wilson loops was found to constitute a complete set of order parameters determining this zero-temperature phase diagram. The quantum phase transitions between these phases are suggested to be of non-Landau-Ginzburg-Wilson type. PACS numbers: 74.40.Kb,74.72.-h Introduction.{The concept of fractionalization, find- eters? Substantial efforts [7{11] suggest that an exact ing its analog in quantum chromodynamics, is nowa- non-perturbative result, the so-called phase string effect, days a guiding principle of strongly correlated systems discovered for the t-J model [7] is at the core of these [1]. Specifically, a quasiparticle (like electron), at short issues. In this Letter, we present an analytical study of spacetime scales, is effectively fractionalized into a few these open issues based on this exact result. We stress degrees of freedom (like spin and charge) which, at large that our theory may be extended to other systems such as scales, are \glued" by emergent gauge degrees of free- the Hubbard model on the honeycomb lattice currently dom. As such, the gauge fields affect profoundly the undergoing intense investigations [12]. formation of various novel quantum phases [2]. A fun- damental issue is the nature of the quantum phase tran- sition between these unconventional phases. In the con- ventional Landau-Ginzburg-Wilson (LGW) symmetry- breaking paradigm, different phases are characterized by the presence or absence of certain local order parameters. δ On the contrary, the fluctuation of emergent gauge field antiferromagnet Bose insulator superconductor may contribute to low energy excitations of the system. In general, the transition between such phases cannot FIG. 1: (Color online) Zero temperature phase diagram of be described by the symmetry breaking paradigm, as is underdoped Mott insulators. The ball (in blue) and the arrow exemplified in Ref. [3]. (in red) stand for the holon and spinon, respectively. The vortex, in red (blue), surrounding a holon (spinon) arises from Practically, an important playground of fractionaliza- the spinon (holon) condensate. The wavy line stands for the tion is doped cuprates [1, 2]. Despite the debate about confinement. the nature of the cuprate phase diagram, it is widely ac- cepted that the essential physics of high Tc superconduc- Main results and qualitative discussions.{In essence, tivity is captured by a doped Mott insulator where the the phase string effect renders an electron \fractional- electron fractionalization into spinons and holons is seem- ized" into two topological objects, the spinon and the ingly inevitable [1, 2, 4, 5]. Indeed, there have been abun- holon, each of which is bosonic and carries a π-flux [13]. dant evidence indicating that the electron fractionaliza- A U(1) gauge field Ah (As ), radiated by the holons tion leads to new quantum phases [2] and their transitions µ µ arXiv:1007.2507v2 [cond-mat.str-el] 17 Mar 2011 (spinons), interacts with spinons (holons) through min- may not be understood in terms of symmetry breaking. imal coupling. (This is the so-called mutual statistical This opinion has been reiterated very recently in Ref. [6], interaction which was also found in different contexts where it was conjectured that the drastic change in the [12, 14]). The macroscopic electric current (density), jh, nature of quantum statistics{a direct manifestation of is fully carried by holons and driven by both the external electron fractionalization{is at the root of the pseudogap electric field E and the \electric field” Es resulting from phase found in the cuprates. s s Aµ induced by spinons. Interestingly, E finds its origin Many fundamental issues arise thereby. How does the analogous to that of Ohmic dissipation in type-II super- electron fractionalization turn an antiferromagnet into a conductors: each spinon mimics a \magnetic vortex" sus- superconductor upon doping? What does the phase dia- pending in holon fluids and, upon moving, generates an h s 2 h gram look like and what are the underlying order param- electric field antiparallel to j , i.e., E = −π σsj , with 2 σs the spinon conductivity characterizing the mobility of charge hopping process, respectively. An exact transfor- −1 h 2 h spinons. From the Ohm's law, i.e., σh j = −π σsj +E, mation [7] keeping track of the phase string effect trans- with σh the holon conductivity, we find that the (electric) forms the superexchange term into (α = 1; 2) resistivity, defined as σ−1jh = E, is X −iσAh (i) y y iσ0Ah (i) HJ / e α b b e α bi+^ασ0 bi−σ0 σ−1 = σ−1 + π2σ : (1) i−σ i+^ασ h s iασσ0 A microscopic derivation of this composition rule will be and the hopping term into given later. X s y h y For the doping δ sufficiently small, the antiferromag- iAα(i) −iσAα(i) Ht / e hi hi+^αe bi+^ασbiσ + h:c:; netic (AF) phase is reached where the spinons are de- iασ confined and form superfluids, i.e., σ ! 1. According s with the coefficients omitted and h:c: being the Hermi- to Eq. (1) this phase is insulating, giving a vanishing σ. tian conjugate. Here,α ^ =x; ^ y^ denotes the unit vector in Indeed, single holon cannot appear in the excitation spec- y the α-direction. The spinon (biσ; biσ; σ ="; #) and holon trum. Rather, holons are excited in pair and are logarith- y mically confined. These pairs are bound to the vortices of (hi ; hi) operators are both bosonic. The mutual statis- spinon superfluids and are thus immobile. Upon increas- tics obeyed by spinons and holons is accounted for the ing δ, the typical size of holon pairs becomes larger and following topological constraints satisfied by the gauge larger and, as a result, the confinement becomes weaker fields. For each loop C in the lattice plane, and weaker. Eventually, a quantum critical point (QCP) P As J α ≡ π P by b − by b mod 2π; is reached. Beyond this QCP holons become deconfined. x2C α C x inside C " " # # (2) P h α P y For sufficiently large δ, a dual scenario applies. The x2C AαJC ≡ π x inside C h h mod 2π: holons are deconfined and form superfluids, i.e., σ ! 1. h Here, J α(x) = +1 (−1) for the link x ! x+α ^ (x+α ^ ! x) In this phase, single spinon cannot be excited. Rather, C on C, and is zero otherwise. spinons are excited in pair, logarithmically confined and In the underdoped regime, it suffices to invoke the bound to the vortices of holon superfluids, i.e., σ = 0. s mean field approximation [7, 8], with the Hamiltonian (Consequently, the excitation spectrum is composed of h y y P P −iσAα(i) integer spin excitations.) According to Eq. (1) this phase simplified to H = − iα(Js σ e bi−σbi+^ασ + s e y i(Aα(i)+Aα(i)) is superconducting (SC), i.e., σ ! 1. Upon decreasing the hi hi+^α) + h:c: (Js; th > 0). The exter- e δ, the confinement becomes weaker and weaker. Even- nal gauge field Aα(i) generates an electromagnetic field tually, another QCP is reached: for smaller δ spinons and couples to the holon degree of freedom. The Hamil- are deconfined. Thus, an intermediate phase can appear tonian H and the constraints (2) constitute our exact where both spinon and holon vortices are condensed. In starting point. this phase, σs;h ! 1 implies σ = 0. This brings us to To proceed further, we consider the coherent state the term, Bose insulating (BI) phase. path integral representation of the partition function, The zero temperature phase diagram described above Z = Tr e−βH , where β (! 1) is the inverse tempera- is summarized in Fig. 1. All three phases are char- ture. The technical challenge arises mainly from taking acterized by confinement or deconfinement of spinons the topological constraints (2) into account. Such a task and holons, formally implemented by unconventional or- was fulfilled in Ref. [9] in the continuum limit. Here, we der parameters{a pair of Wilson loops W s;h[C] with C extend substantially the previous results to a more real- δ P RR − x L a spacetime rectangle. Specifically, they are defined as istic lattice model. We are able to show Z = D e i H As;hdxµ the expectation values of e C µ that probes the in- with s teraction between a pair of test holons (spinons). Wδ [C] h L = Ls + Lh + V (Wδ [C]) displays nonanalyticity at the holon (spinon) de- i µνλ s s h h confinement QCP. The existence of these nonlocal \or- + Aµ − 2πNµ dν Aλ − 2πNλ : (3) der parameters" suggests the non-LGW nature of the π s;h quantum phase transitions: in contrast to an LGW type Here, Nµ are integer fields and Ls;h is the spinon and scenario of order parameter competing, the AF and SC holon Lagrangian, respectively, phases are intrinsically incompatible such that they can- h s h not be turned into unless confinement or deconfinement y D0" + λ −JsDα b" Ls = (b"; b#) h h s y ; occurs.
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