Topological Quantum Phase Transition from Fermionic Integer Quantum Hall Phase to Bosonic Fractional Quantum Hall Phase through P-Wave Feshbach Resonance

TL;DR

This study investigates quantum phase transitions in a fermionic atom system under p-wave Feshbach resonance, revealing a transition from fermionic integer quantum Hall to bosonic fractional quantum Hall phases, with an intermediate coherent superposition phase.

Contribution

It demonstrates the existence of a new intermediate phase and characterizes the order of phase transitions in a fermionic system with p-wave pairing.

Findings

Second order transition from fermionic IQH to intermediate phase.

First order transition from intermediate phase to bosonic FQH.

Emergence of a coherent superposition phase at strong pairing.

Abstract

We use exact diagonalization to study the quantum phases and phase transitions when a single species of fermionic atoms at Landau level filling factor $\nu_f = 1$ in a rotating trap interact through a p-wave Feshbach resonance. We show that under weak pairing interaction, the system undergoes a second order quantum phase transition from $\nu_{f} = 1$ fermionic integer quantum Hall (FIQH) state at positive detuning, to $\nu_{b} = \frac{1}{4}$ bosonic fractional quantum Hall (BFQH) state at negative detuning. However, when the pairing interaction increases, a new phase between them emerges, corresponding to a fraction of fermionic atoms stay in a coherent superposition of bosonic molecule state and an unbound pair. The phase transition from FIQH phase to the new phase is of second order and that from the new phase to BFQH phase is of first order.

Equations

$$H$$

$$+\left(\sum _{m_1,m_2,m_3}{g}_{m_1,m_2,m_3}{b}_{m_1}^{†}{f}_{m_2}{f}_{m_3}+h.c.\right)$$

$$+\sum _{m_1,m_2,m_3,m_4}{v}_{m_1,m_2,m_3,m_4}^{(0)}{b}_{m_1}^{†}{b}_{m_2}^{†}{b}_{m_3}{b}_{m_4}$$

$$+\sum _{m_1,m_2,m_3,m_4}{v}_{m_1,m_2,m_3,m_4}^{(2)}{b}_{m_1}^{†}{b}_{m_2}^{†}{b}_{m_3}{b}_{m_4}$$

$$⟨\Delta m,M|m_1,m_2⟩\equiv$$

$$\frac{1}{\sqrt{(2\pi {)}^4\cdot 2^{\left(\Delta m+M+m_1+m_2\right)}\cdot \Delta m!\cdot M!\cdot m_1!\cdot m_2!\cdot l^8}}$$

$$\int d^2{z}_1\int d^2{z}_2{\left(\frac{z_1^{\ast }-z_2^{\ast }}{\sqrt{2}l}\right)}^{\Delta m}{\left(\frac{z_1^{\ast }+z_2^{\ast }}{\sqrt{2}l}\right)}^M{\left(\frac{z_1}{l}\right)}^{m_1}{\left(\frac{z_2}{l}\right)}^{m_2}$$

$$\times e^{-\frac{|z_1{|}^2+|z_2{|}^2}{2l^2}}$$

$$=\sqrt{\frac{\Delta m!M!}{{\left(2\pi \right)}^4\cdot 2^{\Delta m+M}\cdot m_1!\cdot m_2!}}$$

$$\sum _n(-1{)}^{\Delta m-n}{C}_n^{m_1}{C}_{\Delta m-n}^{m_2}{\delta }_{\Delta m+M,m_1+m_2}$$

$$N_{tot}=2N_b+N_f=\sum _m(2b_m^{†}{b}_m+f_m^{†}{f}_m).$$

$$M_{tot}=\sum _m\left[(m+1)b_m^{†}{b}_m+m{f}_m^{†}{f}_m\right].$$

$$M_{gs}=\frac{N_{tot}\left(N_{tot}-1\right)}{2}.$$

$${\psi }_{\nu =\frac{1}{4}}\left(z_1,z_2\cdots \right)=\left[\prod _{i<j}^{N_b}{\left(z_i-z_j\right)}^4\right]e^{-\sum _{k=1}^N\frac{|z_k{|}^2}{4l^2}}.$$

$$\psi$$

Full text

Introduction. Topological phases of matter are of tremendous current interest. Equally important, but perhaps less studied thus far (especially on the experimental side), are quantum phase transitions (QPTs) between different topological phases. This is because tuning the system through such QPTs requires great control of certain knobs, which are not always available in electronic condensed matter systems. Recently there has been intense activities in topological phases realized in trapped cold atom systemsLin and et al. (2009); Gemelke et al. (2010); Aidelsburger et al. (2011); Dalibard et al. (2011); Goldman et al. (2016). Advantages of these systems include availability of qualitatively different types of inter-particle interactions, and great abilities experimentalists have to control them. This allows for detailed studies of QPTs in such systems. Among such activities are attempts to realize quantum Hall (QH) states(Cooper, 2008; Roncaglia et al., 2011), which were the very first topological phases discovered.

In electronic systems QPTs between different QH phases are usually driven by changing magnetic field, and disorder plays a dominant role at these transitions. In fact the field-driven transition can often be mapped to a transition at fixed field but driven by disorder strength instead. The presence of disorder, especially when combined with electron-electron interaction, makes such transitions extremely hard to study theoretically. Over the years theorists instead studied possible QH transitions driven by interactions without disorderWen and Wu (1993); Chen et al. (1993); Wen (2000); in most cases these require an additional periodic potential that is not available in electronic systems.

It was pointed out by Yang and co-workersYang and Zhai (2008); Barlas and Yang (2011), that in cold atom systems one can realize some of the closely related QH transitions with neither disorder nor periodic potential. Instead what drives the transitions are Feshbach resonances, across which two fermions bind into a bosonic molecule. The two phases involved are a fermionic integer QH phase on the atomic side (positive detuning), and bosonic fractional QH phase on the molecular side (negative detuning). One such phase transitionYang and Zhai (2008) driven by s-wave Feshbach resonance has been confirmed numbericallyCécile Repellin (2016). The other transition from $\nu_{f}=1$ fermionic integer quantum Hall (FIQH) phase to $\nu_{b}=\frac{1}{4}$ bosonic fractional quantum Hall (BFQH) was proposed by Barlas and YangBarlas and Yang (2011), where they studied this phase transition using quantum field theory methods, and argued that it is a second-order QPT in the (2+1)-dimensional Ising universality class.

In this paper we use exact diagonalization (ED) method to study the model of Barlas and YangBarlas and Yang (2011). The purpose of such numerical study is three-fold. First of all, while field theory approaches may determine the possible phases, it cannot determine the actual location of the phase boundary; the latter needs to be determined by calculations based on microscopic models. Secondly while field theory can tell if a second-order phase transition is possible, in general it does not guarantee it must happen in a specific microscopic model; in fact first order transitions are always possible. Lastly since field theory only keeps certain long-wavelength/low-energy degrees of freedom, it might miss certain phases. Indeed through our numerical work we (i) determine the phase diagram quantitatively; (ii) confirm that the direct transition from FIQH phase to BFQH phase is 2nd order; and (iii) perhaps most importantly, find an intermediate phase between FIQH and BFQH phases in certain region of the phase diagram.

To investigate this QPT, we consider a single species of fermions confined to two-dimensions and under either rotation or synthetic gauge field, whose effects mimic that of a strong magnetic field. We focus on the case where the Landau level filling factor $\nu_{f}=1$. When the system is tuned through p-wave FR, two fermions can pair up and form a p-wave bosonic molecule with twice the “charge”. We assume the Landau level spacing is large enough such that we can assume all particles are confined to their respective lowest Landau level. The system can thus be described by the following Hamiltonian on a disc (which is the geometry of our choice for finite-size numerical study, where total angular momentum is a good quantum number due to rotational symmetry):

[TABLE]

where $\left\{m_{i}\right\}$ are angular momentum quantum numbers of the lowest Landau level single particle orbitals, with $b_{m}$ ($b^{\dagger}_{m}$) and $f_{m}$ ($f^{\dagger}_{m}$) the corresponding annihilation (creation) operators for bosons and fermions. The first term above is the chemical potential term in which $\delta$ is the detuning corresponding to the energy difference between a pair of fermions and a boson; this term determines whether the atoms should pair up to form molecules or stay unbound energetically.

The second term depicts the pairing interaction through p-wave FR. The matrix element $g_{m_{1},m_{2},m_{3}}=g\,\delta_{m_{1},M}\,\left\langle 1,M|m_{2},m_{3}\right\rangle$, where $g$ represents the strength of the pairing interaction in the p-wave channel (corresponding to width of the p-wave FR), $|m_{2},m_{3}\left.\right\rangle$ is a two-body state with the two particles having angular momentum $m_{2}$ and $m_{3}$ respectively, and $|1,M\left.\right\rangle$ is a two-body state with their relative angular momentum equal to 1 and center-of-mass angular momentum $M$. This matrix element indicates that only those pairs of fermions whose relative angular momentum $\Delta m=1$ can pair up, and the consequently composite boson has orbital angular momentum $m_{1}=M=m_{2}+m_{3}-1$. This term conserves angular momentum because the p-wave bosonic molecule also carries an internal momentum $+1$. The matrix element $\left\langle 1,M|m_{2},m_{3}\right\rangle$ is a special case of

[TABLE]

where $z_{a}\equiv x_{a}+iy_{a}$, is the complex coordinate of the $a$-th particle on a disc in the lowest Landau level, $d^{2}z_{a}=dx_{a}dy_{a}$, $l$ is the corresponding magnetic length, and $\Delta m$ is the relative angular momentim of a pair. The sum of $n$ is summing over all natural numbers bounded at max$\left(0,\Delta m-m_{2}\right)\leq n\leq$ min$\left(\Delta m,m_{1}\right)$. Note that the bosonic magnetic length square $l_{b}^{2}$ is half of the fermionic magnetic length square $l_{f}^{2}$ ($l_{b}^{2}=l_{f}^{2}/2$) in our model due to the doubled charge of bosons.

The last two terms correspond to repulsive interactions between bosons. In order to stabilize the BFQH state with $\nu_{b}=\frac{1}{4}$, we consider the zero-th order and the second order of Haldane pseudo-potentials Haldane (1983). The matrix element, $v_{m_{1},m_{2},m_{3},m_{4}}^{(\alpha)}=v^{(\alpha)}\sum\limits_{M}\left\langle m_{1},m_{2}|\alpha,M\right\rangle\left\langle\alpha,M|m_{3},m_{4}\right\rangle$, where $v^{(\alpha)}$ is the strength of the $\alpha$-th order Haldane pseudo-potential. In our calculation, we simply use $v^{(0)}=v^{(2)}=1$.

In this model, the total charge ($N_{tot}$) and the total angular momentum ($M_{tot}$) are good quantum numbers. The total charge is the sum of the number of fermions and twice the number of bosons:

[TABLE]

The prefactor 2 in the bosonic part comes from the fact that each bosonic molecule has twice the charge of the fermion. And the total angular momentum is the sum over all angular momentum of orbitals occupied by bosons and fermions plus the number of bosons since each boson has one internal angular momentum:

[TABLE]

In our numerical calculation, we use these two quantum numbers to label the sector in which we perform our calculations.

There are two limits of this Hamiltonian in Eq. (1). For $\delta>0$ and $|\delta|\gg g$, a boson costs more energy than an unbound pair of fermions, so the ground state will be dominated by fermions and the system forms a FIQH state at $\nu_{f}=1$, which does not require interactions between fermions to stabilize (and that is the reason we do not include fermion interaction in our model). On the other hand, when $\delta<0$ and $|\delta|\gg g$ it is energetically favorable for fermions to pair into bosonic molecules. The resultant boson filling factor is $\nu_{b}=1/4$, resulting in a Laughlin-type BFQH state with the boson-boson interaction we introduced. We can easily distinguish between these two phases by inspecting their low-energy spectra, as we now turn to.

In Fig. 1, we show the energy spectra for a system with $N_{tot}=10$ given 10 fermionic orbitals and 20 bosonic orbitals with $M_{tot}$ from 44 to 48. Under FIQH limit, the system will have the lowest energy state only when it forms a FIQH state in which there are no bosons and all fermionic orbitals are occupied, namely $M_{tot}=M_{gs}$ with

[TABLE]

$M_{gs}$ is 45 in this case for $N_{tot}=10$. Because we only give the least number of fermionic orbitals which is 10 here for FIQH state, no edge states should show up at $M_{tot}>M_{gs}$. In Fig. 1 (a) and (b) with $\delta=6$ (FIQH limit) at $g=0.6$ (weak pairing) and $g$ = 3 (strong pairing), the lowest energy states indeed appear at $M_{tot}=45$ and the expectation values of boson numbers in the lowest energy states, $\left\langle N_{b}\right\rangle$, are close to zero as well. Furthermore, the first excited state is a state with one boson. The energy difference between the ground state and the lowest energy excited states, mainly due to the chemical potential term, is about 3$\delta$ (losing two fermions and gaining one boson) as found in Fig. 1 (a) and (b).

In BFQH limit, if the system which has $N_{b}$ bosons forms a BFQH state with $\nu=\frac{1}{4}$, it will have Laughlin wave function as

[TABLE]

The total angular momentum is composed of two parts: the angular momentum of the orbitals occupied by bosons and the internal angular momentum of bosons, $M_{tot}=4\times N_{b}\left(N_{b}-1\right)/2+N_{b}$. In this limit, $N_{b}=N_{tot}/2$, so $M_{tot}=M_{gs}$, the same as that in FIQH state. When $M_{tot}<M_{gs}$, there is no state with lower energy than the Laughlin state at $M_{gs}$. However, when $M_{tot}>M_{gs}$, edge states Wen (1992) degenerate with the Laughlin state exist. Therefore, by counting and comparing the numbers of the low-lying states with the numbers of edge states of Laughlin type states at various $M_{tot}$, we can demonstrate the system forms a BFQH state. In Fig. 1 (c) and (d) with $\delta=-6$ at $g=0.6$ and $g=3$ we demonstrate this is indeed the case. Notice that the low-lying states are no longer exactly degenerate because of the existence of the pairing interaction. Besides, the boson number in the low-lying state at $M_{tot}=45$ is very close to 5, the maximum number of bosons, as expected. In Fig. 1 (d), we see that $\left\langle N_{b}\right\rangle$ has bigger deviation from 5. This is due to the fluctuation induced by the strong pairing interaction namely fermions have bigger matrix element to go back and forth between paired and unpaired states. This is also the reason why the average boson number is bigger in the FIQH state with larger $g$, as shown in Fig.  1 (b). The other thing to notice is in the BFQH regime the low-energy excited state still have essentially all particles as bosons, as a result their excitation energies in Fig. 1 (c) and (d) are much smaller than $3|\delta|$ and determined by the boson-boson interaction instead.

In our model for a specific $N_{tot}$, we need at least $N_{tot}$ fermionic orbitals and $2N_{tot}-3$ bosonic orbitals respectively in order to access the appropriate ground states in the FIQH and BFQH limits. Furthermore, $M_{gs}$ is the same in these two limits. Therefore we will focus on the $M_{gs}$ sector in our calculations from now on using the minimum orbital numbers mentioned above, unless noted otherwise. To identify phase boundaries, we drive a system from FIQH phase to BFQH phase by changing $\delta$ at various $g$. Since quantum phase transitions between different gapped phases must involve gap closing, we plot the gap $\Delta$ of systems in Fig. 2 (a) and (b) as functions of $\delta$, where the gap is defined as the energy difference between the first excited state and the ground state in the $M_{gs}$ sector. Four system sizes with $N_{tot}=8,10,12,14$ are studied. There is one gap-closing point at $g$ = 0.6 (weak pairing regime, Fig. 2 (a)), indicating a single phase boundary separating the FIQH phase and the BFQH phase. Closer inspection (see inset of Fig. 2 (a)) the gap is not exactly zero in these finite size calculations, but instead approaches zero with increasing systems sizes, indicating the transition is due to level anti-crossing, consistent with a 2nd order QPT predicted by Barlas and Yang Barlas and Yang (2011). Unfortunately the limited system sizes and obvious fluctuations that are present do not allow us to perform finite-size scaling to extract critical exponents.

The situation is quite different at $g$ = 2.5 (strong pairing regime). In this case the gap closes twice, implying that the systems undergo two phase transitions in the strong pairing regime. We provide a phase diagram in Fig. 3, based on the locations of gap-closing points. As predicted by Barlas and Yang Barlas and Yang (2011), there is a direct transition from FIQH state to BIQH state in the weak pairing regime (small $g$). However, it is clear that an unexpected new phase shows up between FIQH phase and BFQH phase in the strong pairing regime (large $g$). We now study the properties of this new phase, and phase transitions involving it.

As discussed earlier, one can distinguish between FIQH and BFQH states by the average number of bosonic molecules in the ground state $\langle N_{b}\rangle$. To gain intuition into the new phase, we inspect $\langle N_{b}\rangle$ for both the ground and first excited states. Fig. 4 (a) shows that for $N_{tot}=14$ at $g$ = 0.6 (weak pairing regime), $\langle N_{b}\rangle$s in the ground state and the first excited state increase monotonically and smoothly from FIQH state toward BFQH state, consistent with a second order phase transition happening at the critical point, $\delta_{c}$. In the strong pairing regime in Fig. 4 (b), $\langle N_{b}\rangle$ still changes smoothly at $\delta_{c1}$. However, $\langle N_{b}\rangle$s in the ground state and the first excited state have jumps at $\delta_{c2}$ with opposite signs. This suggests a first-order phase transition triggered by crossing between the ground and first excited states at $\delta_{c2}$. Motivated by this we examine the energy spectrum for $N_{tot}=14$ case at $g$ = 2.5 shown in Fig. 5. It clearly shows that a high energy state comes down and eventually becomes the ground state after a sequence of level crossings with other lower energy states as $\delta$ varies from $\delta_{c1}$ toward $\delta_{c2}$. This suggests the phase transition occurring at $\delta_{c2}$ is a first order phase transition, triggered by crossing of the Laughlin-like state with the ground state of the new phase between the BFQH state and FIQH state.

Returning to the boson number expectation value, we expect the ground state in the intermediate phase contains a finite fraction of fermions that stay in a coherent superposition of bound molecular state and unbound scattering state, even when $\delta$ is quite negative. Such a superposition state gains energy from the pairing term, and when $g$ is much larger than $v^{(0)}$ and $v^{(2)}$, the pairing term dominates, and the energy gain from such coherence dominates the energy cost from putting fermions in the unpaired state. The physics of this intermediate phase is perhaps most clearly revealed by considering the limit $g\rightarrow\infty$. In this limit there is no way to tell whether a fermionic atom is in an (open channel, unbound) scattering state, or forms a bound (closed channel, molecular) state with another fermion, because the $g$-term in the Hamiltonian forces all p-wave fermion pairs to be in a coherent superposition of these two states. For this reason we call it coherent Bose-Fermi mixture phase. A natural variational wave function for the ground state with total angular momentum $M_{tot}=\sum\limits_{\left\{M\right\}}(M+1)$ is

[TABLE]

where $\left\{M\right\}$ is a set of boson angular momenta, with $u_{M}$ and $v_{M}$ variational parameters that describe the coherence between boson and fermion pair states. Since there exist many different configurations of $\left\{M\right\}$, the system is most likely compressible. Of course the actual ground state in the $M_{tot}$ sector can be written as linear superpositions of the state above with different configurations of $\left\{M\right\}$. Phy (2009)

To test the analysis above, we examine the energy spectrum for $N_{tot}=14$ of an extreme case with $g=1$ and all $\delta$ and $v^{(0)}$ and $v^{(2)}$ equal to 0 at various $M_{tot}$ in Fig. 6(a). We see the lowest energy state locates at $M_{gs}$ which is 91 in this case, just like the neighboring BFQH and FIQH phases. The fermion fraction of the total conserved charge is about 0.6 and independent of the system size, as shown in Fig. 6(b). This supports the argument that the $g$-term enforces the ground state to be a superposition of a scattering state and a bound state with $u_{M}\approx v_{M}$. On the other hand the gap $|\Delta E|$ decreases significantly as system size $N_{tot}$ increases; it plausibly extrapolates to zero upon approaching thermodynamic limit. This strongly suggests that the coherent Bose-Fermi mixture phase is compressible.

Conclusion. We performed a systematic numerical study of the topological phase transition from an integer quantum Hall (FIQH) state made of fermionic atoms, to a bosonic fractional quantum Hall (BFQH) state made of bosonic molecules, driven by a p-wave Feshbach resonance. The phase diagram can be separated into two regimes: weak pairing and strong pairing. In the weak pairing regime corresponding to narrow resonance, we demonstrate the existence of the second order quantum phase transition from FIQH phase to BFQH phase which is consistent to earlier theoretical work Barlas and Yang (2011). In the strong pairing regime corresponding to wide resonance, a new phase appears which contains a finite fraction of fermions that stay in a coherent super-position of bound molecular state and unbound scattering state. According to the energy spectra and the behavior of $\langle N_{b}\rangle$, we conclude that the phase transition from FIQH state to the new phase is of second order; from the new phase to BFQH phase it is a first order transition due to the energy level crossing behavior.

Acknowledgments. S.-F. Liou and K. Yang are supported by DOE grant No. de-sc0002140. Z-X. Hu is supported by NSFC under Project No. 11674041, 91630205 and FRF for the Central Universities No. CQDXWL-2014-Z006.