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) † † iσ0Ah (i) HJ ∝ e α b b e α bi+ˆασ0 bi−σ0 σ−1 = σ−1 + π2σ . (1) i−σ i+ˆασ h s iασσ0 A microscopic derivation of this composition rule will be and the hopping term into given later. X s † h † For the doping δ sufficiently small, the antiferromag- iAα(i) −iσAα(i) Ht ∝ e hi hi+ˆαe bi+ˆασbiσ + h.c., netic (AF) phase is reached where the spinons are de- iασ confined and form superfluids, i.e., σ → ∞. 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 ˆ =x, ˆ yˆ denotes the unit vector in Indeed, single holon cannot appear in the excitation spec- † the α-direction. The spinon (biσ, biσ, σ =↑, ↓) and holon trum. Rather, holons are excited in pair and are logarith- † 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 b† b − b† b mod 2π, is reached. Beyond this QCP holons become deconfined. x∈C α C x inside C ↑ ↑ ↓ ↓ (2) P h α P † For sufficiently large δ, a dual scenario applies. The x∈C AαJC ≡ π x inside C h h mod 2π. holons are deconfined and form superfluids, i.e., σ → ∞. h Here, J α(x) = +1 (−1) for the link x → x+α ˆ (x+α ˆ → 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 † † P P −iσAα(i) integer spin excitations.) According to Eq. (1) this phase simplified to H = − iα(Js σ e bi−σbi+ˆασ + s e † i(Aα(i)+Aα(i)) is superconducting (SC), i.e., σ → ∞. Upon decreasing the hi hi+ˆα) + 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 → ∞ 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 (→ ∞) 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 † D0↑ + λ −JsDα b↑ Ls = (b↑, b↓) h h s † , occurs. In particular, they are generally separated by −JsDα D0↓ + λ b↓ the BI phase, with the AF (SC)-BI transition as a holon † s h s
Lh = h D0 + λ − thDα h. (spinon) deconfinement QCP. We now turn to present some technical details. Here and below, the Einstein’s summation convention Lattice field theory.–Formally, we start from the Hamil- is implied and the summation over the indices µ, ν, λ tonian of the t-J model, which consists of the superex- includes both the imaginary time and spatial compo- change and hopping terms, describing the spin-flip and nents. The on-site repulsive potential V softens the 3

† † µ hard-core boson condition and depends on h h and bσbσ. πJC is absent), two phase vortices of the holon super- 0µν s We shall not further present its details to which the re- fluid carrying opposite vorticity ±2π dµNν are log- sults below are insensitive. µνλ is the totally antisym- arithmically confined. Therefore, no free phase vortices RR s µ metric tensor. Finally, the notation “ D” stands for exist with Nα set to zero. Then, πJC mimics an exter- the integral over the fields: b†, b, h†, h, As,h and the La- nal dipole which may be produced by a pair of excited s,h s,h grange multipliers λ , and the summation over Nµ , spinons with identical or opposite spin polarizations. In s e s e h s i(A +A ) −dα dα −i(A +A ) h iA −dα the latter case, a vortex with a vorticity of −2π is excited Dα = e α α e + e e α α , Dα = e α e + d −iAh s s h h from the background and bound to a spinon, forming a e α e α , D = d − iA , D = σ(d − iA ) with d 0 0 0 0σ 0 0 µ dipole. Taking these considerations into account, we find the lattice derivative. With these preparations, the pair of order parameters are defined as h ln Wδ [C] ∼ −πthδ T ln R, (6) ZZ s,h P s,h µ −1 − x(L −iAµ JC ) Wδ [C] ≡ Z e , (4) which shows that the spinons are logarithmically con- D s h fined. To calculate Wδ [C], we ignore aα. Integrating out where C is a spacetime rectangle with length T (R) in the matter fields, we find the holon deconfinement, the imaginary time (spatial) direction and T  R. Z P s sµ P s µ s s −thδ x AµA +i x AµJC Differing from the prototypical field theory [9], L ln Wδ [C] ∼ ln D(A ) e keeps firm track of the compact nature of As,h, which α 1 affects profoundly the ground state properties, as will be ∼ − (T + R). (7) shown below. In particular, the lattice mutual Chern- 4thδ Simons term, namely the last term in Eq. (3) (Such To further probe the SC long range order, we con- a term was found previously in a study of Josephson s,h sider the response to a static small magnetic field. For junction arrays [15].), is periodic under a shift: Aα → s s e s,h s,h s,h s,h s,h s,h this purpose we need to substitute Aα by Aα + Aα in Aα + 2πmα , Nα → Nα + 2πmα with mα ∈ Z. L . If Ae = 2mπ (m ∈ ), the partition function Z re- s,h 0µν h,s h α Z Moreover, summing up N enforces  dµA to be e 0 ν mains unchanged. For Aα = (2m+1)π, the additional π- mπ with m ∈ . s Z phase may be absorbed into Aα, leaving Lh (and thereby Superconducting phase.–Consider the case of dilute Z) unaffected. According to Eq. (2), we conclude that spin excitations where we may ignore the spinon field, a single spin, with two possible polarization directions, † h 2 i.e., b = b = 0. Then, λ |h| + V gives rise to the holon is locally excited and nucleated at the magnetic vortex superfluid. More precisely, factorizing the holon field as core [10]. As such, the external magnetic field mπ flux iθ(x) h(x) = |h|e and inserting it into Lh +V , we find that is fully screened–a profound result of the integer field the fluctuation of |h| is massive, whereas the Goldstone h N0 . If the flux value is not mπ, the magnetic field is mode θ(x) is massless. Therefore, we ignore the terms excluded by the superconductor. Indeed, in this case, Z associated with the spatial fluctuations of |h| and obtain is merely determined by Ae modulo π (denoted as A˜e), 2 s h 2 P s Lh = i|h| (d0θ − A − iλ ) − 2th|h| cos(dαθ − A ) 2t δβ P cos d θ−A˜e −t δβ P |A˜e (q)|2 0 α α Z ∼ R D(θ) e h iα ( α α) ∼ e h q ⊥ , in the absence of Ae , which is further simplified to α ˜e i|h|2(d θ − As − iλh) + t |h|2[(d θ − As )2 − 2]. (By the which justifies the Meissner effect. Here A⊥(q) is the 0 0 h α α e s transverse part of the Fourier transform of A˜ . definition of H, a 2π-shift in dαθ is absorbed into Aα.) To calculate W h[C], we separate Ah into the back- Antiferromagnetic phase.–We turn now to the case δ † ground value and the fluctuation. The former leads to where holons are dilute and, likewise, set h = h = 0. s 2 2 a uniform flux, πδ, at each plaquette and does not con- Then, λ (|b↑| + |b↓| ) + V gives rise to a√ two-component h spinon superfluid, with |b↑| ≈ |b↓| ≈ n implying a tribute to Wδ [C]. Then, we introduce the unitary gauge s magnetization in the transverse direction. Here n is so as to incorporate dµθ into A , and insert the simpli- µ the concentration of condensed spinons depending on fied expression of Lh into L . Integrating out the matter and As fields, we find δ. Similar to the discussions on the SC phase, we factorize the two-component spinon field, (b† (x), b (x)), Z ↑ ↓ X 1 P h h µν −iθ(x) h h − 4 x Fµν F as (|b |, |b |)e and insert it into L + V . Ignor- Wδ [C] ∼ D(a )e ↑ ↓ s {N s} ing the spatial fluctuations of |b|, we obtain Ls = √ P 2 h s P h P h µ µνλ s |bσ| [i(σd0θ − σA − iλ ) − 2Js cos(dαθ − A )]. iπ 2thδ a (J +2 dν N ) σ 0 α α ×e x µ C λ (5) Integrating out the matter and Ah fields gives h h h upon appropriate rescaling, where F = dµa − dν a is Z µν ν µ s X s − 1 P F s F s µν h W [C] ∼ D(A )e 4 x µν the Maxwell√ tensor with a the fluctuating component of δ Ah, and 2t δ is the bare “charge”. In the subsequent {N h} h √ h iπ 4nJ P As (J µ+2µνλd h) step, we integrate out a by using the Feynman gauge, ×e s x µ C ν Nλ , (8) which leads to important consequences. First of all, we s s s find that at the ground state (where the external source where Fµν = dµAν − dν Aµ. 4

Eq. (8) has far-reaching consequences. First of all, by is a consequence of i) that the two pieces involved are integrating out the As field, we find that at the ground vortices, and ii) that they obey mutual statistical inter- µ state (πJC = 0) the phase vortices of the spinon super- action. (This fact has been established in a completely 0µν h fluid carrying opposite vorticity ±2π dµNν are loga- different context [14].) For the BI phase, σs → ∞ be- µ rithmically confined. Most importantly, πJC mimics an cause of spinon condensation, implying σ = 0. external dipole produced by a pair of the holon and anti- Crossover from Mott’s law to activation law.–Finally, holon. The latter is a −π-fluxoid, formed out of a −2π we discuss a possible experimental observation of resis- phase vortex and a π-flux carried by the holon. Such a tivity in the AF phase at sufficiently low but nonvanish- holon-anti-holon pair is logarithmically confined, ing temperatures. In this regime, a holon-anti-holon pair s bound to the vortex of spinon superfluids displays two- ln Wδ [C] ∼ −2πJsn T ln R. (9) dimensional variable-range hopping conduction known as −1 −1/3 In other words, a holon pair nucleates in a phase vor- the Mott’s law, ln σh ∼ T [16]. On the other hand, tex of the spinon superfluid of vorticity −2π, forming spinon superfluid is actually the so-called strongly en- a “neutral” object. Furthermore, since the phase vor- tangled vortex phase [17]. Carrying the arguments of tex is static, the pair of holons is spatially localized and Ref. [17] to the present context, we find that the spin the AF phase is insulating (see below for further expla- transport, in response to Eh, displays an activation law, −1 nations). It should be noticed that without the integer i.e., ln σs ∼ T . According to Eq. (1), the resistivity dis- h field Nµ , such an insulating phase cannot be established. plays a crossover from Mott’s law to activation law upon Instead, the SC phase is pushed all the way to δ = 0 [8]. decreasing temperatures, which may serve as a probe of Repeating the derivation of Eq. (7), we find the spinon holon (spinon) confinement (deconfinement) in the AF deconfinement, phase. We thank J. Zaanen for discussions and M. Garst, h 1 ln Wδ [C] ∼ − (T + R). (10) V. Gurarie, and L.-H. Tang for conversations. Work 8Jsn supported by NSFC Nos. 10688401 and 10834003, by Bose insulating phase.–The analytic results, namely MOST National Program for Basic Research, by DFG Eqs. (6), (7), (9) and (10), allow us to make an important SFB/TR12, and by HK RGC No. HKUST3/CRF/09. observation. The asymptotic behavior, Eqs. (6) and (10), signal a critical concentration separating the spinon con- h finement and deconfinement phases, at which Wδ [C] is nonanalytic in δ. Indeed, in the SC phase, the spinon ex- citations become progressively important as δ decreases: [1] Z. Zou and P. W. Anderson, Phys. Rev. B 37, 627 (1988). they cause a renormalization of the bare “charge” in [2] For a review, see, P. A. Lee, N. Nagaosa, and X. G. Wen, Rev. Mod. Phys. 78, 17 (2006). Eq. (6) and eventually drive the system to the spinon [3] T. Senthil, et. al., Science 303, 1490 (2004). deconfinement QCP where the “charge” vanishes. For [4] T. Senthil and M. P. A. Fisher, Phys. Rev. B 62, 7850 smaller δ the SC long range order disappears. Likewise, (2000). Eqs. (7) and (9) signal another critical concentration, at [5] S. Sachdev, Rev. Mod. Phys. 75, 913 (2003). s [6] J. Zaanen and B. J. Overbosch, arXiv:0911.4070. which Wδ [C] is nonanalytic, separating the holon confine- ment and deconfinement phases. Renormalizing the bare [7] Z. Y. Weng, et. al., Phys. Rev. B 55, 3894 (1997). “charge” in Eq. (9) drives the system towards this QCP. [8] See, for a review, Z. Y. Weng, Intl. J. Mod. Phys. B 21, 773 (2007). For δ larger than this critical value, the AF long range or- [9] S. P. Kou, X. L. Qi, and Z. Y. Weng, Phys. Rev. B 71, der disappears. Because these two renormalization mech- 235102 (2005). anisms are independent, these two QCPs are generally [10] V. N. Muthukumar and Z. Y. Weng, Phys. Rev. B 65, not identical, giving an intermediate phase where both 174511 (2002). the AF and SC long range orders vanish and both the [11] J. W. Mei and Z. Y. Weng, Phys. Rev. B 81, 014507 holon and spinon are deconfined (condensed). (2010). This intermediate phase does not support dc electric [12] C. Xu, Phys. Rev. B 83, 024408 (2011). [13] The spacetime lattice constant, the Planck’s constant, transport. To prove this, we notice that the composition the speed of light in vacuum, and the electron charge are rule (1) is valid for the entire underdoped regime. Indeed, set to unity. δLs,h i αµν s,h s,h minimizing L gives h,s = −  dµ(Aν − 2πNν ), [14] V. M. Galitskii, et. al. Phys. Rev. Lett. 95, 077002 δAα π s,h 1 0αβ s,h s h (2005). i.e., jα = π  Eβ . By definition, jα ≡ σsEα and [15] M. C. Diamantini, et. al., Nucl. Phys. B 474, 641 (1996). h s jα ≡ σh(Eα + Eα). Noticing that the electric and holon [16] B. I. Shklovskii and A. L. Efros, Electronic properties of currents are identical, we obtain Eq. (1) from these three doped semiconductors (Springer, Berlin, 1984). relations. The striking structure of the composition rule [17] M. V. Feigelman, et. al. Phys. Rev. B 48, 16641 (1993).