Critical charge fluctuations and scaling at the Kondo break-down of Heavy Fermion systems
One of the challenges in strongly correlated electron systems, is to understand the anomalous electronic behavior that develops at an antiferromagnetic quantum critical point (QCP), a phenomenon that has been extensively studied in heavy fermion materials. Current theories have focused on the critical spin fluctuations and associated break-down of the Kondo effect. Here we argue that the abrupt change in Fermi surface volume that accompanies heavy fermion criticality leads to critical charge fluctuations. Using a model one dimensional Kondo lattice in which each moment is connected to a separate conduction bath, we show a Kondo breakdown transition develops between a heavy Fermi liquid and a gapped spin liquid via a QCP with scaling, which features a critical charge mode directly associated with the break-up of Kondo singlets. We discuss the possible implications of this emergent charge mode for experiment.
Introduction - The relation between valence fluctuations and the Kondo effect has long fascinated the physics community Schrieffer and Wolff (1966). A partially occupied atomic state, weakly hybridized with a conduction sea, forms a local moment Anderson (1978) and its virtual valence fluctuations give rise to low frequency spin-fluctuations, while leaving its charge essentially frozen. On the other hand, in heavy fermion systems, the Kondo-screening of the local moments gives rise to an enlargement of the Fermi surface, a phenomenon that is well established both theoretically Yamanaka et al. (1997); Oshikawa (2000) and through Hall coefficient Paschen et al. (2004), quantum oscillation Shishido et al. (2005), ARPES and STM measurements Chen et al. (2017); Aynajian et al. (2012). The large Fermi surface of a Kondo lattice is believed to partially collapse when Kondo screening is disrupted Coleman et al. (2001a); Si et al. (2001); Senthil et al. (2003, 2004); Pépin (2007); Coleman et al. (2005a); Nejati et al. (2017) at an antiferromagnetic (AFM) quantum critical point (QCP), a phenomenon known as “Kondo breakdown” (KBD).
Recently, a number of experiments have observed a coincidence of critical charge fluctuations at the magnetic quantum critical points in CeRhIn Ren et al. (2017) YbRhSi Prochaska et al. (2018) and -YbAlB Kobayashi et al. (2018).
Watanabe and Miyake have argued that the development of soft charge fluctuations near a heavy fermion QCP is likely a result of a quantum-critical end-point, in which a first-order valence changing transition line is suppressed to low temperatures Watanabe and Miyake (2013); Miyake and Watanabe (2014); Watanabe and Miyake (2014, 2015); Scheerer et al. (2018). Here we present an alternative view, arguing that the coincidence of soft charge fluctuations and Kondo breakdown is a natural consequence of the Fermi surface collapse.
In the eighties, Anderson introduced the concept of a nominal valence to distinguish the valence of a rare earth ion infered from the apparent delocalization of f-electrons Anderson (1983); Aeppli and Fisk (1992), from the core-level valence, infered from spectroscopy. From this perspective, a shift in nominal valence is associated with formation of a large Fermi surface, even in a strict Kondo lattice where the core-level valence is fixed. Interpreted literally, this implies a kind of many-body ionization in the Kondo lattice, in which a fractionalization of local moments into charged heavy electrons, leaves behind a compensating positive background of Kondo singlets Lebanon and Coleman (2008). Taken to its logical extreme, such an interpretation would then imply that at KBD quantum critical point, degenerate fluctuations in the nominal valence will give rise to an observable soft charge mode.
While Kondo Breakdown has been extensively modelled at an impurity-level Pixley et al. (2012); Chowdhury and Ingersent (2015) and simulated using dynamical mean-field theory Si et al. (2001); De Leo et al. (2008); Martin et al. (2010), a possible link with charge fluctuations has not sofar been explored in the lattice. To examine this idea, we introduce a simple field-theoretic framework for Kondo breakdown, emplying a Schwinger boson representation of spins that permits us to treat Kondo screening and antiferromagnetism Parcollet and Georges (1997); Arovas and Auerbach (1988). Early application of this method demonstrated its efficacy for describing a ferromagnetic quantum critical point Komijani and Coleman (2018) in a Kondo lattice. Here we consider a Kondo screened one dimensional (1D) AFM [Fig. 4a], examining the quantum phase transition transition between a spin-liquid and a Fermi liquid [Fig. 4(b)]. The conduction electron phase shift (related to the Fermi surface size) jumps at , indicating that QCP is a KBD transition. Additionally, we find that the KBD features a zero point entropy [Fig. 4(d)]. In our calculations we observe that the KBD is linked to the emergence of a gapless charge degree of freedom at the QCP which occurs in natural coincidence with a divergent charge and staggered spin susceptibility.
The simplified 1D Kondo lattice is a chain of antiferromagnetically coupled spins each individually screened by a conduction electron bath:
Here is the spin at the -th site, coupled antiferromagnetically to its neigbor with strength . describes the conduction bath coupled to the -th moment in the chain, where is the momentum of the conduction electron. is the spin density at site , where creates an electron on the chain at site .
Global phase diagram - Numerical and experimental studies of heavy-fermion systems are often interpreted Si et al. (2001); Coleman et al. (2001b); Coleman and Nevidomskyy (2010) within a global phase diagram of the Kondo lattice, with two axes: a Doniach parameter Doniach (1977), where is the Kondo temperature, and a frustration parameter representing the magnitude of quantum fluctuations, controlled by geometrical or dimensional frustration. The 1D limit provides a way to explore the two extremes of : on the one hand, the uniform magnetization of a 1D FM commutes with the Hamiltonian and has no quantum fluctuations, corresponding to Komijani and Coleman (2018), whereas a 1D AFM never develops long range order, loosely corresponding to . When the magnetic coupling is Ising-like, both models can be mapped via bosonization to the dissipative transverse-field Ising model Lobos et al. (2012); Lobos and Cazalilla (2013), but a Heisenberg coupling changes the behavior significantly.
The method is a large- approach, obtained by enlarging the spin rotation group from SU(2) to SP(), representing the spin local moments using Schwinger bosons (“spinons”), according to Read and Sachdev (1991); Flint and Coleman (2009), where , and is the number of bosons per site. Each moment is coupled to a -channel conduction sea, with Hamiltonian
Here we have adopted a summation convention for the repeated greek spin and roman channel indices. The Lagrange multiplier imposes the constraint : we take for perfect screening, where is kept fixed.
We carry out the Hubbard-Stratonovich transformations:
where is a Grassmanian “holon” field that mediates the Kondo effect at site in channel , while describes the development of singlets between site and .
A mean-field RVB description of the 1D magnetism is obtained from a uniform mean-field theory where , and , giving rise to a bare spinon dispersion , with . Both and fields have non-trivial dynamics Parcollet and Georges (1997); Parcollet et al. (1998); Coleman et al. (2005b); Rech et al. (2006); Komijani and Coleman (2018), with self-energies
Here, and , and are the local propagators of the holons, spinons and conduction electrons, respectively. The conduction electron self-energy is of order and is neglected in the large- limit, so that is the bare local conduction electron propagator. The holon Green’s function , is purely local, whereas the spinons are delocalized by the RVB pairing with propagator . The self-energy is diagonal in Nambu space, while the momentum sum in can be done analytically SM ().
Stationarity of the free energy with respect to enforces the mean-field constraint , and with respect to determines the relation SM (). We solve these self-consistent equations numerically on the real-frequency axis using linear and logarithmic grids.
The two limits - In absence of Kondo screening (when is small) the constraint is satisfied with . This Schwinger boson model describes a bipartite spin chain, in which each sublattice is in the symmetric spin- representation of SP() Read and Sachdev (1991); Flint and Coleman (2009): Each spin can form singlets with its neighbors in an RVB state for any value . This, together with the Gutzwiller projection treated by a soft constraint leads to a U(1) gapped spin liquid Arovas and Auerbach (1988), closely analogous to the integer-spin Haldane chain Haldane (1983). A lattice with closed boundary condition has a unique ground state and corresponds to a symmetry protected topological phase Pollmann et al. (2010).
The large limit corresponds to a local Fermi liquid Rech et al. (2006); Komijani and Coleman (2018) at each site of the chain, in which the electrons and spinons form bound, localized singlets, protected by a spectral gap of the size ; the remaining electrons are scattered with a phase shift . Fig. (4b) summarizes the phase diagram as is varied between the above two limits, which we discuss in the following.
Ward identity, entropy, Phase Shifts - At large , the many body equations can be derived from a Luttinger Ward functional, leading to an exact relation between the conduction electron and holon phase shifts and a closed form formula for the entropy Coleman et al. (2005b); Rech et al. (2006). Fig. 4(c) shows the conduction electron phase shift as a function of . In the Fermi liquid, is equal to , equivalent to a large Fermi surface, but it is zero in the spin liquid regime. Extrapolating the calculations to , the phase shift appears to jump at the QCP separating the spin-liquid (decoupled electrons) and the Fermi-liquid. From the perspective of conduction electrons, both SL and FL phases are Fermi liquids and the transition in is a measure of change in the Fermi surface, a manifestation of Kondo breakdown (KBD).
Entropy - Fig. 4(d) shows the colormap of the entropy across the phase diagram. The gray dashed line indicates a second order phase transition for the internal variable that separates a local Fermi liquid () from a de-localized regime (). The collapse of the energy scale from both sides are visible. Unlike the 1D ferromagnetic QCP Komijani and Coleman (2018), the antiferromagnetic QCP develops a residual entropy at a spin of per moment (inset of Fig. 4d),
Magnetic excitations - Fig. (5a) shows the spinon spectrum vs. at . Approaching the transition from the Fermi liquid side (right), the spinon spectrum shifts to positive frequencies and, maintaining overall gap size, brings the gap edge close to the chemical potential and only then, the hard gap closes at the QCP. Passing through the critical point, the gap re-opens due to development of short-range RVBs in the spin liquid regime. Fig. 5(b) shows the temperature dependence of the staggered spin susceptibility , which acquires a logarithmic temperature dependence at the QCP. The Fermi liquid (blue) exhibits a crossover from Curie law to a Pauli form , with a characteristic peak at . As is reduced the peak position is unchanged (unlike the 1D FM case Komijani and Coleman (2018)) whereas the low temperature susceptibility develops a logarithmic divergence. Similar divergence is observed in local spin susceptibility but the uniform susceptibility is only suppressed by magnetism SM ().
The holon spectrum - shows a striking behavior at the QCP (Fig. 6a). Most of the spectral weight is contained in a sharp holon mode which crosses the chemical potential as is tuned from Fermi liquid (right) to spin-liquid (left). In the critical regime at a finite temperature, the holon mode is pinned to the Fermi energy over a finite range of Doniach parameter, which shrinks to a point as , forming a strange metal regime at finite temperature with deconfined critical holon and spinon modes.
scaling - At the QCP, the holon mode lies at zero energy. Fig. 6(b) shows that the holon spectra at different temperatures collapse onto a single scaling curve . For we find consistent with a scaling analysis SM (). The universality class of the QCP appears to be that of an overscreened impurity modelParcollet and Georges (1997) , with an effective number of channels .
The holon modes have an emergent coupling to the electromagnetic field, mediated via the internal vertices of the lattice Kondo effect. In particular, the field theory implies an vertex correction that couples to the electric potential as shown in Fig. 6(c). At low energies the vertex can be approximated by . This quantity perfectly cancels the wavefunction renormalization of the holon propagator , where , so that the holon couples to the electric potential with a net charge . Fig. (6)(d) shows the charge susceptibility calculated using this vertex corrections. At the QCP, the temperature dependence of the holon charge susceptibilty acquires a Curie-like temperature dependence .
Discussion - We have studied a simplified Kondo lattice model in the large- limit, enabling us to extract, for the first time, the KBD physics directly on a lattice. One of the striking features of the KBD quantum critical point is the presence of an emergent, spinless critical charge mode with a Curie-like charge susceptibility. Our model calculations can be extended in various ways, by going to higher dimensions, by generalizing to the mixed valence regime, and with considerable increase in computation, to a model in which a single bath is shared between all moments. In the general Kondo lattice, the charge conservation Ward identity links the change in the volume of the conduction electron Fermi surface to the charge density of the Kondo singlets, described by the holon phase shift Coleman et al. (2005b)
Quite generally, the holon phase shift is zero or in the localized magnetic, or Fermi liquid phases respectively, but must pass between these two limits at the quantum critical point. This strongly suggests that the gapless holon excitions seen in our model calculation will persist at a more general Kondo breakdown quantum critical fixed point.
This raises the fascinating question how the predicted critical charge modes at KBD might be observed experimentally. One mode of observation, is via the coupling to nuclear Mossbauer lines Komijani and Coleman (2016). A recent observation of the splitting of the Mossbaüer line-shape Kobayashi et al. (2018), characteristic of slow valence fluctuations, may be a fingerprint of these slow charge fluctuation.
Another interesting question is whether the residual entropy of the QCP might survive beyond the independent bath approximation. A residual ground-state entropy is a signature of infinite-range entanglement, and has been seen in various quantum models, such as the two channel Kondo model Andrei and Destri (1984); Tsvelik and Wiegmann (1984); Ludwig and Affleck (1994) or the SYK model Sachdev and Ye (1992); Georges et al. (2001); Maldacena and Stanford (2016). In the single-channel Kondo problem, the Kondo screening length Affleck (2010) plays the role of an entanglement length-scale, beyond which the singlet ground-state is disentangled from the conduction sea. If the collapse of the Kondo temperature at the QCP of a Kondo lattice involves a divergence of the entanglement length , the corresponding quantum critical point would be expected to exhibit an extensive entanglement entropy. Such naked quantum criticality is likely to be censored by competing ordered phases that consume the entanglement entropy of the critical regime, concealing QCP beneath a dome of competing phase, such as superconductivity.
Lastly, a peculiar feature of strange metals in heavy-fermions is that the resistivity tends to be linear in over a wide range of tempreature. In quenched disordered models it is possible Parcollet and Georges (1999); Patel et al. (2018); Chowdhury et al. (2018) to derive such behavior from model calculations. One of the fascinating implications of a Curie-law charge susceptibility seen in our calculations, is that if combined with a temperature-independent diffusion of incoherent holon motion, it would give rise to a Curie conductivity (linear resistivity ) via the Einstein relation , where is the holon diffusion constant. This raises the interesting possibility that linear resistivities are driven by an emergent critical charge mode.
This work was supported by NSF grant DMR-1309929 (Piers Coleman) in its early stages, and by a Rutgers University Materials Theory postdoctoral fellowhsip (Yashar Komijani). We would like to thank S. Nakatsuji, H. Kobayashi, A. Vishwanath, J. Pixley, C. Chung and A. Georges for stimulating discussions and M. Oshikawa for helpful correspondence.
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Appendix A Supplementary material
This supplementary section contains additional details, proofs and calculations that are not included in the paper. For completeness, in section A, we include a bosonization mapping between the Ising limit of our model and the dissipative transverse field Ising model. In section B, we review the Arovas-Auerbach solution to the 1D antiferromagnet using the Schwinger boson representation of the spin. In C we provide the details of the dynamical large- equations, as well as formulas used to evaluate various thermodynamical properties. In section D we present a perturbative analysis of the large- equations showing that the phase shift jumps between and . In section E we establish that at the QCP the renormalized energies of spinon and holon go to zero, and show that the critical exponent we found numerically is consistent with the scaling limit of large- equations. In section F we provide additional data on the holon phase shift and charge susceptibility to support the statements of the paper.
a.1 A. Mapping to the dissipative Transverse-field Ising model
In this section we use the bosonization technique to show that in the limit of purely Ising coupling between the magnetic moments (both ferromagnetic and antiferromagentic), our model can be mapped to a dissipative transverse-field Ising model. See also Lobos et al. (2012); Lobos and Cazalilla (2013). This is a property of the independent screening approximation and does not depend on the dimensionality of the spin lattice. The Hamiltonian is
Let us write the Hamiltonian in the general anisotropic form
In the continuum limit, is
At we have the boundary condition . It is useful to unfold the semi-infinite wire into a single set of right-movers which interact (via a Kondo interaction) with the impurities at the origin. Next, we bosonize:
We can introduce charge and spin bosons and only spin-bosons appear in the Hamiltonian. The conduction-band spin is given by
The bosonized Hamiltonian is then
We also note that
Next we apply the Emery-Kievelson transformation to simplify the Kondo interaction
Using the identities
and choosing , the bosonic part of the transverse Kondo coupling is eliminated, and we then have
We have only explicitly written the Ising part of Ferromagnetic coupling and have moved the remaining terms into . If we choose , the Ising part of Kondo interaction then drops out and the transverse Kondo interaction appears as a Zeeman field acting on the “dressed spin”. This is called the “Decoupling point” and constitutes an additional solvable point of the single-impurity Kondo model, similar to the “Toulouse point”. The Kondo temperature of this highly anisotropic Kondo system is . Therefore, assuming , we have
which is the transverse-field Ising model. This model can be re-written using a Jordan-Wigner transformation in terms of two Majorana fermions with a gap that changes sign at the QCP separating the disordered and ordered phases. Moving away from the Decoupling point introduces an additional term into the Hamiltonian which implies that the spins are now dissipatively coupled to a gapless bosonic bath.
If , we have
Because of the additional bosonic factors these additional terms can only be treated perturbatively Lobos et al. (2012).
a.2 B. Review of Arovas-Auerbach
Here, we review Arovas-Auerbach Arovas and Auerbach (1988) approach to 1D antiferromagents. In the momentum space we have
Using a Bogoliubov-de Gennes (BdG) transformation, we have
We can diagonalize the Hamiltonian by tuning with energy , so that
Constraint - We can write
At zero temperature we have
In 1D we have . Since , and using we find
where is the complete elliptic integral of first kind. Parametrizing , near we have
so that the gap in the spectrum is proportional to
a.3 C. Details of large- compuations
The self-energies of spinons and bosons
in real-frequency become
where all the functions are retarded () in this expression and
The propagator for the holon is local:
By contrast, the p-wave pairing in the spinon Hamiltonian causes them to delocalize and develop anomalous pairing components in their Green’s function. The appearing in the self-energies is the local Green’s function in which the anomalous components cancel out due to momentum-summation,
To see this explicitly, the spinon Greens function is given by
This momentum-sum can be performed analytically which significantly accelerates the computation. First we write
where we have used the fact that the self-energy given in Eq. (28) is diagonal in the Nambu space, involving two components, and ,
and we have defined
Using this short-hand notation, we have
where we have defined
The momentum-sum gives
which is diagonal in Nambu space with the particle-particle element proportional to .
Saddle-point equations - Stationarity of the Free energy with respect to and then leads to the saddle-point equation
which determine and self-consistently. The momentum sum can be done analytically and we find
The calculations start at high temperature and the temperature is gradually reduced. Due to development of sharp features in the spectrum at low-temperatures and the importance of high-frequency components, more frequency points are required to achieve lower temperatures, a time/memory cost that eventually limits the lowest temperatures achieved.
Having computed the Green’s functions, we can use them to calculate various thermodynamical properties:
Susceptibility - The spin correlation function in SP() is
The static susceptibility can be obtained by doing the imaginary-time integral off this correlation function which can be written in the form
Again the momentum-sum can be performed analytically. For uniform and staggered susceptibility this gives
For the local susceptibility, we sum Eq. (47) over and find
Entropy - The derivation of the entropy has been discussed in Rech et al. (2006) and Komijani and Coleman (2018). The free energy worked out in previous section is a stationary functional with respect to , and . Keeping those constant, we take derivative w.r.t to obtain the entropy. The result is Rech et al. (2006)
Here, all the greens functions are retarded and is the bare Green’s function of conduction band (30). And where is the self-energy of the conduction electrons, which in the frequency domain is
The function is part of the spinon contribution to the entropy that requires a momentum-summation.
Again, this momentum sum can be done analytically to obtain
The specific heat is obtained from numerical differentiation of the . Fig. 5 shows the specific heat coefficient as a function of and . The collapse of energy scale from both sides are visible. The suppression of the at the QCP implies a zero-point entropy due to entropy balance.
a.4 D. The leading order solution
Although, we have numerically obtained the solution to the self-consistent large- equations, it is interesting to study the results of a single-iteration of these equations.
In a simple antiferromagnet, the particle-particle component of the spinon Green’s function is
where , , and . Substituting this into (29), we obtain
Using a flat density of states for , at we find
To obtain a rough estimate of this quantity, we can replace by the typical energy of a spinon, so that
where we have used , . The phase shift at zero frequency is given by
While this is a rough crude approximation, it captures the key feature that the holon field first develops a bound-state when becomes of order .
a.5 E. Scaling conditions
Fig. 6 shows the renormalized energies of spinons and holons . We find that and in the limit at the QCP, whereas in SL, or FL regimes, remains finite.
In the FL regime, becomes negative at a finite temperautre, indicating electron-spinon bound state formation. At the QCP, as meaning that , whereas in the SL regime, remains finite.