The OP did start this thread asking about portable amps.
Though the portables will not extract the optimum sound quality from the 701s -- and the pair I have sounds delicious via XRay transport > Muse DAC > RKV-MkII w/OPA627 mods + Eddie Current EC/DC impedance matcher -- BUT I've had very happy listening times with my Portaphile PV2^2 Maxed (LT1210/1500BGx2), and also my RSA Hornet.
My Tomahawk doesn't really have enough juice for these phones (…gotta say, it's my most-carried portable due to its diminutive size, enormous battery life, good looks/fun handling, external gain switch, perfect volume knob, smooth/airy/refined sound for relaxed listening in hysterical NYC subways and in the late night quiet at home…). Remember, as has been said, the 701s aren't hard to drive loud, but they're very hard to drive well for the great sound quality they're capable of -- even moreso than the 501 (as Asr noted). BTW - the Portaphile's deep bass helps the 501s, but some prefer a warmer midrange where this amp is more "neutral/open" than warm. (More warmth would probably close up the amp's transparent window and inner detail.)
My Portaphile has, to my ears, great transparency in that it lets you hear deep into the music space, almost "around" the performers with great recordings and phones, and that transparency offers up much inner detail and subtleties that many other amps don't retrieve. Since some listeners have said they don't find it very transparent, perhaps language is a problem here. I wonder what "they" mean by transparent -- and what they're hearing!?
It has excellent extension down and up, deep and very well-defined/articulated bass, very rich/complex harmonic structure, rock solid placement of instruments in a palpable space, great dimension with the 701s, great inner detail (as I said), a more "lifelike" presentation than many portables (I haven't heard them all), and certainly very discrete separation of instruments. As one example of its timbral excellence, drums sound more like full/dimensional drums rather than the surface strikes that most small amps present.
The Portaphile does present the featured performers a bit further back in the space with the 701s, since it makes the most of the huge spatiality of these phones. The Hornet offers warmer mids (many will enjoy this with the 701s) and a more forward presentation in place of the Portaphile's more open window/great sense of space. If you prefer more warmth and forward punch, the Hornet would be recommended over the Portaphile -- but, oops … it's over 300.
Another caveat about the Portaphile -- and it's not an issue for me -- but the amp is a power hog. I often use it at work and I almost never use a wallwart … always rechargeable batteries. The best battery life it can get with a rechargeable is with the iPower 500mAh Lithium-Polymer 9V. Portably I keep Class-A on and Wide-Bandwidth off and get 8-8.5 hours of constant play (a day's work) and always carry a charged (lightweight) spare for long days. It takes me well under a minute to swap batteries. Some find this unacceptable but it never bothered me. At shows I use an UltraLife 1200mAh (disposable) 9V for 20-hr. life. If one insists on keeping Wide Bandwidth on portably, there's only 4-4.5hrs. per iPower 500mAh charge, but in portable venues the subtleties added by Wide Bandwidth are obliterated by even modest ambient noise so there's really no point in that.
At HeadFest I listened to Justin's AE-2 for awhile (loved it a lot), but didn't hear it with the 701s. It might also perform very well with the 701s, but I can't say yet, and it is over the 300 mark, as is the Hornet.
The Xin amps mentioned might be fine, but I still have not been able to audition one so I can't comment.
So Portaphile Maxed (only with the LT1210 and 2x1500BG caps) gets my very strong recommendation.
Of course, these are just my humble impressions -- and we all have different ear/brain systems.
